Positioning of objects in three-dimensional space

Abstract

A novel definition of the bearing of an arbitrarily shaped object in three-dimensional space, consisting of the coordinates of the position of the origin of the object and the two deflection angles of the first main axis of the object from the corresponding main axis of the space and of a further deflection angle of a second main axis of the object from the corresponding main axis of the space, makes it possible to directly and precisely transform the object from its own object coordinate system into the same bearing in the space coordinate system by means of a transformation matrix defined by the predetermined deflection angles, by applying the rotation angles computed according to the novel method on the basis of the predetermined deflection angles.

Description

[0001]Positioning of Objects in Three-Dimensional Space The invention relates to an optimized method for transforming three-dimensional objects into their own coordinate system and subsequently into an arbitrary position, consisting of position and orientation, in any (three-dimensional) space. To describe an arbitrarily shaped object with all known and identifiable reference points in any space, one must first know all the coordinates of these reference points in the object's own coordinate system and then transform these known coordinates of the object into the coordinates of the object in space using a transformation sequence. For this purpose, the coordinates of the object's origin point are required, as well as three angles to define the object's orientation in its position in space.All transformations of three-dimensional objects in space known from the literature have the disadvantage that the three rotation angles for the rotation sequence in the transformation to be performed cannot be directly derived from measurable angles of the object in space. Furthermore, each transformation has the additional disadvantage that different rotation angles result for the same descriptive position of the object in space for all possible transformation sequences. The present invention presents a new method by which, on the one hand, the angles of an object's position in space can be determined in a traceable manner, and on the other hand, the required rotation angles for the transformation sequence to be performed can subsequently be pre-calculated based on these angles, so that the object is precisely positioned in space after the transformation has been performed, based on these predetermined angles.To better understand the new method, some definitions are presented here first: The spatial definition of an object. An object that is at least partially known spatially has, by definition, its own origin point and, radiating from this, three orthogonal principal axes X, Y, and Z, which define the object's own coordinate system. The first principal axis direction of an object results from the line defined by two key reference points on the object's first body axis. The second principal axis direction is determined either by a third reference point outside this line and its perpendicular to it, or by the projection of a second body axis through two independent reference points onto the base plane, which is spanned perpendicular to the first principal axis direction and is therefore orthogonal to the first principal axis direction.The third principal axis direction is the vector product of the first two principal axis directions and is therefore orthogonal to them. The object's origin is defined, depending on its functionality, either at an existing reference point of the object or at its own center. The object's coordinate system is thus determined by this defined origin and the three principal axis directions, which define the three coordinate axes at this origin. Which of the three principal axes—X, Y, or Z—is defined as the first principal axis depends on the situation (object-related properties and functions, integration into a higher-level system, function within the higher-level system) and can, in principle, be redefined at any time.An explicit goal of the method presented here is to demonstrate a way for the precise coordinate transformation, independent of the individual transformation steps, for converting an object from spatial coordinates into its own coordinate system, which in every case delivers the same result in a reproducible manner, regardless of which of the three principal axes was defined as the first, the second and the third.Another explicit goal of the method presented here is to transform one or more arbitrary objects or groups of objects in space into an arbitrary position, consisting of position and orientation, based on their object-specific coordinates. This allows for the precise determination of their relative positions in space, the resulting distances between individual reference points of these objects or groups, and their relative direction vectors during given movements of these objects or groups. In the literature, the direction of motion for many moving objects (airplanes, cars, ships) is defined as the object's X-axis and first principal axis. However, in medicine, the Z-axis is considered the first principal axis for humans due to their upright position.In general, it can be stated that a single object usually consists of several sub-objects in a defined relationship, which we can define as an assembly of objects. Due to their position and function within the object, different principal axes must be defined as the first principal axis for each sub-object. To illustrate this, consider a car as the (superordinate) object in its own space, for which the X-axis is defined as the first principal axis. Then consider a single wheel of this car, for which its own axis of rotation functionally forms the first principal axis. Therefore, for the orientation of the single wheel within the car's space, it makes sense to define the Y-axis of the car as the first principal axis of this wheel.In general terms, it can first be stated that the first principal axis of an object results from its own primary function and / or its orientation within the larger space. Secondly, the direction of the first principal axis of the sub-objects that comprise this object does not necessarily coincide with the direction of the object's first principal axis. This means that it is not practical to define the same spatial principal axis as the first object-related principal axis for all (sub-)objects in the same larger space. Therefore, for each (sub-)object in a common (larger) space, its own first principal axis, determined by its function and / or its position in space, should be defined as the orientation axis and designated as the principal direction (X, Y, or Z) of the object that exhibits the smallest deviation from the corresponding principal direction of the space.To orient all individual (sub-)objects in space, a transformation defined by its own first principal axis must be performed for each of these (sub-)objects. For the sake of simplicity, the derivation of the method will subsequently use the z-axis of space as the first principal axis for orienting the object. However, all definitions and formulas for the other spatial axes as orientation axes are also explicitly described in detail. These can be used in the calculations with the same precision and always lead to the same computational result. Defining a space based on a known object: In order to describe multiple objects and / or groups of objects in a common space with the x, y, and z axes, a reference to this space must first be defined.For this purpose, an existing, known object is selected as a reference basis, and a decision is made as to which known reference points or landmarks of this object should be used to determine the origin and principal axes of the space. It follows implicitly from this that any sufficiently known object can be used at any time to define a new space as a reference coordinate system for all objects in that space, and furthermore, all known reference points or landmarks of this object can serve to determine the origin and axes of this reference coordinate system.This, in turn, implies that an object, initially only partially known, in a specific space, whose definition is further refined by calculating additional reference points and landmarks within that space, can later be used to define a new, referenced space based on these originally unknown reference points or landmarks. This allows for the determination of a new reference coordinate system for this object and all other objects and object groups in the same space. This interoperability between individual objects and the possible spaces based on them enables a more detailed determination of the spatial extent of only partially known objects based on other objects in the same space, thereby generating a more detailed overall virtual representation of all objects in that space.To ensure that multiple changes of the reference coordinate system for all objects in the same space can be performed as precisely as possible and without the accumulation of errors, systematic inaccuracies in the transformations between the coordinate systems must be avoided. If inaccuracies in the transformations were tolerated, a seemingly more detailed determination of an object would lead to imprecise calculations of the originally unknown reference points and landmarks of that object, which in turn would result in an inaccurate determination of a new reference coordinate system based on it. Therefore, when dealing with complex dependencies between objects in a shared space, it is crucial to determine each individual object with the best possible accuracy in its respective coordinate systems at all times and to systematically eliminate any potential inaccuracies.The Transformation of an Object into its Own Coordinate System In order for an object, partially known in a spatial coordinate system, to be referenced to itself and, if necessary, its shape to be described more precisely by further reference points or landmarks, it is first transformed into its own coordinate system in three steps: 1. The object is translated to its own origin point by translating all known reference points and landmarks. 2. The first principal axis of the object is aligned with the corresponding coordinate axis of space. This requires at least two rotations. a. In the Eulerian transformation (named after Leonhard Euler), the first rotation rotates the object around its first principal axis so that the line of intersection between the object's base plane and the base plane of space lies on the third principal axis of space.Then, in the second rotation, the object's first principal axis is rotated around the third principal axis of space until it aligns with the first principal axis. b. In the Cardano transformation (after Gerolamo Cardano), the object is first rotated around the third principal axis of space until its first principal axis lies in the plane formed by the first and third principal axes. Then, the object is rotated around the second principal axis until it aligns with the first principal axis. 3. Finally, the object is rotated around the now-common first principal axis until its second principal axis aligns with the second principal axis, and thus its third principal axis also aligns with the third principal axis. This rotation can also be described as the object's (natural) intrinsic rotation.This sequence of transformation steps 1 to 3 – in its two listed variants a and b – is the only one that prevents the parameters underlying each transformation step from constantly changing during the sequence. Nevertheless, even with this transformation sequence, it must be noted that the order of the two rotations for axis alignment affects the required rotation angles. However, the direction vector of the object's first principal axis is used to calculate the required rotation angles in both transformation types.The difficulty arising from each transformation lies in calculating the second and third rotation angles required to align the object with the first and second principal axes in space. This is because these angles depend on the type of transformation and the choice of the first and second rotation axes for axis alignment. The third rotation angle, in particular, is affected by each preceding transformation to such an extent that it is difficult to calculate directly if the direction of the object's second principal axis is not precisely known but can only be determined via one or two additional reference points, at least one of which must not itself lie on the object's second principal axis, as will be described in detail later.In Euler's transformation, the first rotation angle for orienting the object by rotating it about the first principal axis of space—in the following description, about the z-axis—is determined by the direction of the deflection of the object's z-axis (^^^^, ^^^^, ^^^^). The second rotation angle in this transformation is then the angle of this deflection of the object itself. Thus, the tangent of the first rotation angle is ^^^^^^ ^^^^ =^^. ^^ and the tangent of the second rotation angle The third intrinsic rotation angle ^^ ′′ For the Euler transformation for the last rotation of the object around the z-axis of space, both the measurable extrinsic rotation angle and, indirectly, its tangent are taken into account. ^^ = ^^ ^^ ^^ as well as from the first two rotations ^^ ^^ and ^^ ^^dependent. Its calculation is omitted here because this transformation is no longer relevant in the further considerations. In the gimbal transformation, the rotation angles are calculated for the orientation of the first principal axis, in the method presented here the Z-axis ^^(^^^^, ^^^^, ^^^^) of the object. ^^ ^^ not through its tangent ^^^^^^^^ ^^ = ^^ ^^ , for rotation around the x-axis of space using the associated rotation matrix ^^ and ^^ ^^ ^^ , calculated by its tangent ^^^^^^^^ ^^ = ^^ ^^ , used for rotation around the y-axis of space using the associated rotation matrix. Assuming that the second principal axis should be the X-axis ^^(^^^^ , ^^^^ , ^^^^) of the object, because rotation angles in the base plane are generally measured on it, it is necessary to define the first rotation about the y-axis of space with the rotation angle ^^ ^^ to perform. This first rotation also changes the length ratios in the new orientation vector ^^′(^^). ′ ^^ , ^^ ′ ^^ = ^^^^, ^^^ ′ ^ ) changes and the second rotation angle ^^ ^^ , or rather its tangent must be adjusted to ^^^^^^^^ ′ ^^ = ^^^^^^^^^^^^^^^^^^^^^^. The second rotation angle ^^^ ′^ can therefore be calculated directly from the rotation sequence and the two deflection angles of the original direction vector of the object's first principal axis. This results in the counterclockwise rotation matrix for the orientation of the object's z-axis to the z-axis of space with the first rotation about the y-axis of space. In the proposed rotation sequence for aligning the object's Z-axis, the first rotation around the y-axis of space rotates the object's rotationally neutral axis, which we call the O-axis ^^(^^^^ , ^^^^ , ^^^^) in the xz-plane of space onto the x-axis of space. The direction of this rotationally neutral O-axis of the object is ^^(^^^^ = ^^^^^^^^^^, ^^^^ = 0, ^^^^ = −^^^^^^^^^^). Due to the first two rotations for aligning the object's Z-axis with the z-axis of space, the orientation of the object's X-axis inevitably changes from the extrinsic rotation angle ^^ when rotated around the z-axis of space. ^^ with the ^^ tangent ^^^^^^^^ ^^ ^^ = ^^ ^^ for rotation around the z-axis of space using the ^^^^^^^^ ^^ −^^^^^^^^ ^^ 0 associated rotation matrix ^^^^(^^^^) = (^^^^^^^^ ^^ ^^^^^^^^ ^^ 0) to ultimately 0 0 1 intrinsic rotation angle ^^^ ′ ^ ′ with the tangent ^^^^^^^^ ^ ′ ^ ′ = ^^ ′′ ^^ . From the first two rotations, measured with respect to the rotationally neutral O-axis of the object and the x-axis of space, the following equations result for the X-axis of the object: the intrinsic rotation angle ^^ ^ ′ ^ ′ for the last rotation from the extrinsic rotation angle ^^ ^^ The X-axis and the rotations of the object's orientation along its Z-axis can be directly calculated (and vice versa). Therefore, the complete transformation of the object into its own coordinate system after translation to its origin can be performed using the left-rotating transformation matrix. ^^ ( ^^ ^^ ) ^^ ^^ ( ^^ ^ ′ ^ ) ^^^^ ( ^^ ′′ ^^) =^^^^^^^^ ^^ ^^^^^^^^ ′′ ^^ − ^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ −^^^^^^^^ ′′ ^^ ^^^^^^^^^^ − ^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^^ ( ^^^^^^^^ ′ ^^^^^^^^ ′′ ^^^^^^^^ ′ ^^^^^^^^ ′′ ^^^^^^^ ′ ^^ ^^ ^^ ^^^ ^^ ). −^^^^^^^^ ^^ ^^^^^^^^ ′′ ^^ − ^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ′′ ^^ ^^^^^^^^^^ − ^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^^It is explicitly pointed out again here that these derived equations are only valid for the direction of the X-axis of the object depending on the presented orientation transformation of the Z-axis, and that different values ​​for the extrinsic rotation angles apply to all reference points, landmarks and direction vectors of the object. ^^ from the transformation-dependent intrinsic rotation angle ^^ ^ ′ ^ ′This results in a paradoxical outcome, since the continuity of the rigid body must apply to the entire object. However, the displacement of the object away from the z-axis of space results in an apparent distortion of the object's axes in space. This special property of the object's coordinate system becomes apparent from the fact that the vector product of the first principal axis, here the Z-axis with the direction ^^(^^^^ = ^^^^^^^^^^^^^^^^^^^^, ^^^^ = ^^^^^^^^^^^^^^^^^^^^, ^^^^ = ^^^^^^^^^^) and the rotationally neutral O-axis with the direction ^^(^^^^ = ^^^^^^^^^^, ^^^^ = 0, ^^^^ = −^^^^^^^^^^) of the object, and thus the third axis of the seemingly rotationally neutral coordinate system of the object, has the direction ^^(^^^^ = ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^, ^^^^ = ^^^^^^^^^^^^^^^^^^^^^^^^^^ + ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^)It follows that, as a rule, neither the Z-axis nor its Y-axis of the object lies in a principal plane of space if the X-axis of the object lies in the xz-plane of space. The apparent distortion of the principal axis directions of the object's coordinate system in space illustrates that aligning the object with the first principal axis through the rotation sequence of the transformation results in a rotation of at least the third principal axis of the object around the first principal axis of space, even if the second principal axis of the object is not rotated around the first principal axis of space. It follows that every transformation to align the object with the first principal axis of space also entails a rotation of the object around this principal axis, which necessitates a correction of the object's orientation to enable precise alignment of the object in space.The perspective image as a snapshot of an object in space: A perspective image of an object in space is created by projecting individual points of the object onto a recording plate. This is achieved either by shining light generated by a specific light source through a lens with a defined focal point, passing through the object and onto the recording plate behind it, as in an X-ray machine, or by focusing indeterminate light reflected from the object onto the recording plate through a lens (or an objective consisting of several lenses) with a specific focal point, as in a standard camera. In both cases, the light on the recording plate creates image points of the object, which represent the corresponding object points as projections through the common focal point in the lens.The image produced on the recording plate can be enlarged or reduced as desired without changing the relative positions of the individual image points. According to the intercept theorem, the part of the object depicted in the photograph that is closer to the focal point of the lens during the exposure appears to be magnified in the image compared to the part of the object that is farther away. This fact can be used to unambiguously determine the position of the object in the coordinate system of the recording device, or rather, its image, as will be explained later in this description.Distance Measurement in Perspective Images: The distance between two points in space can be calculated directly in orthogonal images, such as plan drawings, from the differences in the respective coordinates of the two points in the coordinate system created by the two orthogonal images. Therefore, the question arises as to how the distances between the projections of the same points in two orthogonal perspective images can be used to calculate the distance in space. In order to calculate the actual distance between two points from the distance between their projections in a perspective image, the two points must lie in a plane parallel to the image plane, and the distance of this plane to the image plane must be known in relation to the distance of the focal point of the perspective image's lens.Since this condition cannot be met for two orthogonal perspective images, calculating the distance between any two points in space using the projections of these points into two orthogonal images is not possible and can only be approximated. To determine the distance between two points in space using two perspective images, the two coordinate systems of these perspective images must first be transformed into a reference coordinate system. Then, the coordinates of the two points can be determined by finding the intersection of their respective projection lines in the reference coordinate system, and the distance between these two points can be calculated. For this to work, the two perspective images do not need to be orthogonal to each other.It should be noted, however, that at small angles (less than 30°) between the perspective images, the depth of field influences the expected accuracy when virtualizing the calculated coordinates in the image direction. It follows that without near-perfect virtualization of the two coordinate systems of the perspective images in a reference space, measuring the distance between any two points in this reference space is impossible. Angle measurement in the perspective image: An angle between two intersecting lines in space can be calculated directly in orthogonal images from the two angles between the projections of these lines in two orthogonal perspective images. Therefore, the question arises as to how the angles between the projections of the same lines in two orthogonal perspective images can be used to calculate the angle in space.For the angle between the projections of two lines in the perspective image to coincide with the angle between the projections of the same lines in the corresponding orthogonal image in the same image plane, one of the following three conditions must be met: a. Both lines are parallel to the (common) image plane. b. One of the two lines is parallel to the image plane, and the projection of the lens's focal point in the perspective image lies on the other line. c. The projection of the lens's focal point coincides with the intersection point of the projections of the two lines in the perspective image. For two orthogonal perspective images, only the third condition can be fulfilled for both images, although in practice this is sometimes difficult and can only be achieved for a single angle.This raises the question of whether the deviation of the angle of the projections of any two lines in space in the perspective image from the angle of the projections of the same lines in the corresponding orthogonal image can be derived from the generation data of the perspective image. The answer is no if the position of the two lines in the coordinate system of the perspective image is not already sufficiently known. It inevitably follows that every angle measurement in the perspective image deviates from the angle measurement in the corresponding orthogonal image in an unknown way and is therefore unsuitable for the precise determination of real angles in three-dimensional space. This conclusion calls into question all common angle measurement methods in perspective images per se, and any accuracy claim for a common angle measurement method in images is unverifiable without near-perfect virtualization.To determine the angle between the direction vectors of two lines in space using two perspective images, the positions of the two lines in the coordinate systems of the two perspective images must be determined by finding the intersection of their respective projection planes. By transforming the coordinate systems of the two perspective images into a reference coordinate system, it is then possible to uniquely determine the positions of the two lines in the reference space and thus calculate the angle between their direction vectors. These two perspective images do not need to be orthogonal to each other.It should be noted, however, that at small angles (less than 30°) between the perspective images, the depth of field influences the expected accuracy in the virtualization of the lines in the image capture direction, since the projection planes of the two perspective images then intersect at a shallow angle. The object of the present invention is to fully describe the definitions and the optimizations derived therefrom for the transformation of an object in space into its own coordinate system for all six possible gimbal transformations, and also to derive from this the transformation of the object into an arbitrary position, consisting of position and orientation, in space, in order to be able to use this for all possible movement patterns of different objects in space.In a further step, the optimized transformations will be used to uniquely determine arbitrary objects in the coordinate system of a perspective image as a snapshot of these objects and their relative positions. This will enable the near-perfect virtualization of these objects and their relative positions in any space, based on at least two perspective images taken from different perspectives relative to the objects. The virtualization will completely eliminate the disadvantages of perspective images regarding the accuracy of distance and angle measurements, allowing the objects and their relative positions to be determined with the highest possible accuracy. Figures / Drawings: The optimized method for transforming three-dimensional objects is explained using an example shown in Figure 1.Figure 1 shows an object with its own coordinate system with axes X with direction vector ^^(^^^^ , ^^^^ , ^^^^), Y with direction vector ^^(^^^^, ^^^^, ^^^^) and Z with direction vector ^^(^^^^, ^^^^, ^^^^), which is located in a space with axes x, y, and z, where the origin of the object and the origin of the space are identical. In this example, the actual deviation of the Z-axis from the z-axis is 16°, which means that for the angle ^^^^, the value ^^^^ = 16° applies, and this deviation is measured in the direction of 120° to the x-axis, which in turn means that for the angle ^^^^, the value ^^^^ = 120° applies. Furthermore, the intrinsic rotation of the object about its Z-axis in this example is 20°, which in turn means that for the intrinsic angle ^^. ^ ′ ^ ′ The information is valid ^^ ′′ ^^ = 20°.Using the example in Fig. 1, the following section explains in detail how the various object axes are positioned in space and how the corresponding rotation angles for the six different rotation matrices must be calculated to obtain exactly identical results for all six possible transformation sequences when transforming the object into its own coordinate system. In principle, for each principal axis of the spatial coordinate system, two rotation angles around that principal axis can be determined, which typically differ in both sign and absolute value. For rotation around the x-axis, these are the rotation angles ^^ ^^ relative to the ^^ Y-axis of the object and calculated using its tangent ^^^^^^^^ ^^ ^^ = ^^ ^^ with the resulting associated rotation matrix ^^^^(^^^^) =and the Previously presented rotation angles ^^ ^^relative to the Z-axis of the object and calculated using its tangent ^^^^^^^^ ^^ = ^^ ^^ ^^ with the resulting associated rotation matrix For rotation around the y-axis, this is the rotation angle ^^ ^^ relative to the object's x-axis and calculated using its tangent ^^^^^^^^ ^^ ^^ = ^^ ^^ with the resulting and the Previously presented rotation angles ^^ ^^ relative to the Z-axis of the object and calculated using its tangent ^^^^^^^^ ^^ = ^^ ^^ ^^ with the resulting associated rotation matrix. For rotation around the z-axis, this is the rotation angle already presented ^^ ^^ relative to the object's x-axis and calculated using its tangent ^^ ^^^^^^^^ ^^ = ^^ ^^ ^^ with the resulting associated rotation matrix ^^^^^^^^ ^^ −^^^^^^^^^^ 0 = ( ^^^^^^^^ ^^ ^^^^^^^^ ^^ ) and the rotation angle ^^ ^^ relative to the Y-axis 0 0 1 ^^ of the object and calculated by its tangent ^^^^^^^^ ^^ = ^^ ^^ ^^ with the resulting For all possible (extrinsic) rotation angles, both the calculation of the rotation angle and the resulting rotation matrix establish a unique reference to a second principal axis, as this second principal axis is used for both calculating the rotation angle and determining the direction of rotation. It should be noted again that the two extrinsic rotation angles associated with a principal axis are generally not identical and do not share a common intrinsic rotation angle.Depending on how an object is oriented in space, and consequently which principal axis of the object represents its first principal axis (defined by its function and position), a clearly preferable rotation sequence for orientation in space results for each object. This sequence yields three preferred rotation angles, measured about the first and second principal axes of the object. The first two rotation angles denote the orientation angles about the first principal axis and have their corresponding subnotations, while the third rotation angle is the rotation about the first principal axis that has the subnotation of the second principal axis. To explain the rotation sequences in detail, all meaningful possibilities for transforming the object into its corresponding spatial coordinates are presented individually.It should be mentioned in advance that a viable approach exists only if each of the three rotations is suitable for transferring the object in the correct direction and into the next rotation, or rather, to axis overlap. Therefore, the goal of the first rotation is to transfer the object's first principal axis into the base plane of the second rotation axis, so that the subsequent rotation around the second rotation axis can align the object's first principal axis with the corresponding principal axis of space.By defining the first two rotations as alignment rotations of the first principal axis in space, it is logical and correct to use the deflection angles of the first principal axis relative to the corresponding spatial principal axis as the base values ​​for the first two rotations, and not the two other deflection angles with the same axis of rotation, which would not normally lead to an alignment of the first principal axis. Therefore, meaningful transformations are always performed using the deflection angles of the object's first principal axis relative to the corresponding spatial principal axis, whereby the second rotation angle must be corrected as described above to enable precise alignment with the first principal axis. This leads to the following alignment transformations for the respective first object principal axes. For the object's principal axis X as the first principal axis and thus the alignment axis, the resulting rotation sequence is clockwise. For the object's principal axis Y as the first principal axis and thus the alignment axis, the following clockwise rotation sequence results. For the object's principal axis Z as the first principal axis and thus the alignment axis, the following clockwise rotation sequence results. −^^^^^^^^ ^^ −^^^^^^^^ ^^ ^^^^^^^^ ^^ ^^^^^^^^ ^^ ^^^^^^^^ ^^As already mentioned, the rotation sequence of the orientation transformation is preferably determined depending on the position of the second principal axis, since this avoids rotating this second principal axis during the orientation transformation. This results in the six preferred transformation sequences depending on the choice of the first and second principal axes of the object, which are shown individually below. For the object's principal axis X as the first principal axis and the object's principal axis Y as the second principal axis, the preferred left-rotating transformation matrix is ​​therefore ^^. = For the object's principal axis X as the first principal axis and the object's principal axis Z as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^ ^^ ( ^^ ^^ ) ^^ ^^ ( ^^ ^ ′ ^ ) ^^^^ ( ^^ ′′ ^^) = ( For the object's principal axis Y as the first principal axis and the object's principal axis Z as the second principal axis, the preferred left-handed transformation matrix is ​​therefore obtained ^^ ^^ ( ^^ ^^ ) ^^ ^^ ( ^^ ^ ′ ^ ) ^^^^ ( ^^ ′′ ^^) = For the object's principal axis Y as the first principal axis and the object's principal axis X as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^ ^^ (^^ ^^ )^^ ^^ = For the object's principal axis Z as the first principal axis and the object's principal axis X as the second principal axis, the preferred left-handed transformation matrix results, as already explained in detail. ^^ (^^ ^^ )^^ ^^ (^^ ^ ′ ^ )^^ ^^ (^^ ′′ ^^) = ^^^^^^^^ ^^ ^^^^^^^^ ′′ ^^ − ^^^^^^^^^′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ −^^^^^^^^ ^^ ^^^^^^^^ ′′ ^^ − ^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^^ ( ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ′ ^ ^ ). −^^^^^^^^ ′′ ^^ ^^^^^^^^^^^^ − ^^^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ′′ ^^ ^^^^^^^^^^^^ − ^^^^^^^^^^^ ′ ^ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ^ ′ ^^^^^^^^ ^ ′ ^ ^^^^^^^^ ^^ For the object's principal axis Z as the first principal axis and the object's principal axis Y as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^ ^^( ^^ ^^ ) ^^ ^^ ( ^^ ^ ′ ^ ) ^^^^ ( ^^ ′′ ^^) = For the transformations explicitly presented here, a rotational symmetry is evident for both the left-handed and right-handed transformation matrices. The rows containing the analogous formulas shift downwards, and simultaneously, the corresponding columns shift to the right. Additionally, for each right-handed transformation matrix, there exists a symmetrical left-handed one with the complementary rotation angles. From this, it can be deduced that similar formulas result for calculating the respective rotation angle corrections for the corresponding transformation matrices, whereby the second and third rotation angles must always be corrected using the corresponding analogous formulas. The derivations for the second rotation angles are analogous to the formula already presented. ′ ^^ = ^^^^^^−1(^^^^^^^^^^^^^^^^^^^^) accordingly, and thus the formulas for ^^ ′ −1 are obtained. ^^ = ^^^^^^ (^^^^^^^^^^^^^^^^^^^^^^), for , für ^^ ′ ^^ = ^^^^^^−1(^^^^^^^^ ′ −1^^^^^^^^^^^^) and for ^^^^ = ^^^^^^ (^^^^^^^^^^^^^^^^^^^^). The derivations for the third (intrinsic) rotation angles are analogous to the formula already derived ^^. ′′ ^^ = meaningfully and therefore The formulas for ^^ are derived from this. ′′ ^^ = Objects with two non-orthogonal reference axes. For the sake of simplicity, the preferred rotation sequence with the Z-axis as the first principal axis and the X-axis as the second principal axis of the object will be used again in the following explanations. As already mentioned, the method discussed here favors the gimbal transformation with the first rotation of the object around the y-axis and the second rotation around the x-axis of space to align the Z-axis of the object with the z-axis of space, because this rotates the rotationally neutral O-axis of the object solely in the xz-plane onto the x-axis of space, thus determining the third rotation angle. ^ ′ ^ ′ to align the X-axis of the object with the X-axis of space by rotating it around the Z-axis of space in a unique relationship to the two deflection angles measurable in space ^^ ^^ and ^^ ^^the Z-axis and the orientation angle ^^ ^^The x-axis of the object can be calculated. This simplifies the determination of the necessary correction for the precise orientation of the object on the x-axis of space after its alignment on the z-axis. Furthermore, it generally reduces the error in the orientation angle relative to the x-axis induced by the orientation transformation. As described earlier, at least four reference points are sufficient for the spatial definition of an object: two for determining the first principal axis direction and two further reference points for determining a second principal axis direction orthogonal to the first. The third principal axis direction is calculated from the vector product of the first and second, according to the definition. However, this does not implicitly mean that the direction of the second reference line is the direction of the object's second principal axis.It is quite possible that one can determine the deflection angles relatively easily ^^. ^^ and ^^ ^^ The direction of the object's Z-axis in space can be uniquely determined by the direction from the first to the second reference point, but not the direction of the X-axis itself, only the direction of another non-orthogonal axis of the object with the direction vector ^^(^^^^, ^^^^ , ^^^^) in the object coordinate system, given by its direction vector ^^(^^^^ , ^^^^, ^^^^) in the spatial coordinate system at a given (intrinsic) angle ^^ = ^^^^^^ −1 The direction from the third to the fourth reference point can be determined relative to the object's XZ plane. In this typical case, the formula shown above for the direct conversion of the extrinsic rotation angle ^^ = −1 ′′ −1 ^^^^ ^^^^^^ to the intrinsic rotation angle ^^^^^^ = ^^^^^^ Not valid, nor is their inverse. However, what remains binding in any case is the fact that, after the two orientation rotations for the orientation of the Z-axis in space, the object rotates in space by the intrinsic rotation angle ^^′′ = ^^^^^^−1 of the object around the z-axis of space. From the two fundamental properties described for any transformation of an object from known spatial coordinates into its own coordinate system—namely, the unavoidable rotation of the object around the orientation axis due to the orientation transformation and the resulting intrinsic rotation around the orientation axis, dependent on the orientation transformation, for the final orientation of the object in space—it follows that the necessary correction of the rotation angle for the final orientation must always result directly from the orientation transformation. This applies to any known axis of the object that is not parallel to the first principal axis of the object, since the first principal axis itself becomes the rotation axis of the final orientation due to the orientation transformation.Furthermore, for any axis of the object, given by its direction vector ^^(^^^^, ^^^^, ^^^^) at the origin of the object, its intrinsic rotational deviation ^^. ^^^^ known from the XZ plane of the object, this intrinsic rotational deviation ^^ ^^^^ during the transformation of the object into its own coordinate system, it must be preserved, and therefore the following also applies: This also implies that this arbitrary The axis is ultimately adjusted by the sum of this intrinsic rotational deviation ^^ ^ ′ ^ ′ ^ ^ and the rotation angle resulting from the orientation transformation of the object on the z-axis of space ^^ ^ ′ ^ ′ ^ ^ The object must be rotated around the z-axis of space to precisely align it with the x-axis of space and thus transform it into its own coordinate system. This results in the intrinsic rotation angle.^ ′ ^ ′ of the direction vector ^^′′(^^ ′′ ^^, ^^ ′′ ^^, ^^^ ′ ^ ′ ) by the orientation transformation of the direction vector ^^(^^^^ , ^^^^, ^^^^), which depends on both the type of orientation transformation and the order of the transformation steps. Thus, for the direction vector ^^(^^^^, ^^^^ , ^^^^) of the object, which is not parallel to the Z-axis of the object, its intrinsic rotation angle ^^ ^ ′ ^ ′ , ^ ′′ or its tangent ^^^^^^^^ ^ ′ ^ ′ ^ b = ^^ ^^ ′′ ^^ , in the base plane of the z-axis of space by the orientation transformation on this z-axis of space depending on the orientation transformation with first orientation rotation around the y-axis, which is the already known orientation matrix This can be calculated. This results in the following for the last rotation angle ^^ ′′ ^^ = ^^ ′′^^^^ + ^^^ ′ ^ ′ the complete equation ^^ ′′ ^^ = ^^^ ′ ^ ′ ^ ^ + Analogous formulas can be derived for all other transformations based on the existing geometric symmetry, and For any known or measurable intrinsic angle of a vector ^^(^^^^, ^^^^ , ^^^^) in the object, the same property holds: after the object's orientation transformation, this angle must coincide with the intrinsic rotation angle in space. It follows that the following equation also holds true. Analogous formulas can also be derived for all other transformations based on the existing geometric symmetry, and thus the following apply to In order to perform a precise and complete transformation of an object into its own coordinate system, one needs the known origin of the object in order to transform it in a first step by translation into the origin of the object's coordinate system, a known first principal axis of the object in order to align the object with the corresponding principal axis of space by means of an orientation transformation using the two deflection angles of this principal axis of the object, and a further axis of the object whose direction vector in space and whose intrinsic rotational deviation in the object's coordinate system are known.in order to finally align the object in the base plane of the first principal axis of space by means of the sum of the intrinsic rotational deviation of the second axis of the object and the rotation angle resulting from the orientation transformation of the direction vector of this second axis of the object on the corresponding second principal axis of space. Since the intrinsic rotation angle ^^, ^ ′ ^ ′The intrinsic rotational deviation of the second principal axis, resulting from the alignment of the object's first principal axis with the first principal axis of space, is relatively small when the first principal axis of the object deviates from the first principal axis of space. Therefore, it is often neglected in calculations, and only the intrinsic rotational deviation of the second principal axis in the base plane of the first principal axis is considered for aligning the object with the second principal axis of space. While this may not be significant when calculating a single transformation of a single object into its own coordinate system, neglecting the intrinsic rotations caused by these transformations leads to an unknown accumulation of errors without a predictable trend.Provided that the transformation of an object or a previously defined space into its own coordinate system has been precisely performed, other objects and groups of objects in the shared space can also be transformed into the coordinate system of that object or the new shared space using the same transformation. This allows their relative position and orientation to the new base object or reference space to be precisely calculated. Thus, entire groups of objects can also be transformed, depending on a single object from the group, into the coordinate system of another object within the same group or an object from a different group. The transformation of the (known) object into an arbitrary position in space: To transform an object known in its own coordinate system into an arbitrary position in space, defined by its position and orientation, an inverse transformation is performed.In this process, all steps involved in transforming the object from spatial coordinates to its own coordinate system are performed in reverse order and with reversed signs for the rotation angles, or the inverse transformation is directly carried out using the transposed transformation matrix. The following explanations will initially focus only on the transformation sequence in the Cardan coordinate transformation with alignment to the z-axis and orientation to the x-axis, which includes the following steps: 1. The object is rotated in the base plane of the first principal axis of space by the angle ^^. ^ ′ ^ ′ 1. The object is deflected (use a negative angle if the rotation matrix is ​​the same, or the same angle if the rotation matrix is ​​transposed). 2. The object is deflected by two successive rotations about the second and third principal axes of space by the deflection angles ^^^ ′ ^ and ^^ ^^ deflected from the first principal axis of space. 3. The object is finally moved into the specified position by a translation. Or: 1. The object is directly moved using the transposed transformation matrix. 1. The object is deflected into its position at the origin of space. 2. Finally, the object is moved into the predetermined position by a translation. To ensure completeness, the transposed transformation matrices resulting from the six preferred transformation sequences, depending on the choice of the first and second principal axes of the object, are presented individually below. For the object's principal axis X as the first principal axis and the object's principal axis Y as the second principal axis, the preferred left-handed transformation is therefore obtained. For the object's principal axis X as the first principal axis and the object's principal axis Z as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^^ ^ ′′ ^ ^ (^^ ^^ )^^ ^ ^ ^ ^ (^^ ^ ′ ^ )^^ ^ ^ ^ ^ (^^ ^^) = ^^^^^^^^ ^^^^^^^^ ′ ^^^^^^^^ ′ ′ ^ ^ ^^ ^^ ^^^^^^^^ ^^ ^^^^^^^^ ^^ ( ^^^^^^^^ ′′ ^^^^^^^^^^^^ − ^^^^^^^^^ ′ ^ ′ ^^^^^^^^ ^^ ^^^^^^^^ ^ ′ ′′ ^ ^^^^^^^^^^^^^^^^^^^^′ −^^^^^^^^ ′′^^^^^^^^^^^^ − ^^^^^^^^^ ′ ^ ′ ^^^^^^^^ ′ ^ ^ ^^^^^^^^ ^^ ). −^^^^^^^^′′^^^^^^^^ − ^^^^^^^^ ′′ ^^^^^^^^ ^^^^^^^^ ′ ′′^^^^^^^^ ′ ^^^^^^^^ ′′^^^^^^^^ − ^^^^^^^^ ′′ ′ ^^ ^^ ^^ ^^ ^^^^^^^^^^^^ ^^ ^^ ^^ ^^^^^^^^^^ ^^ ^^^^^^^^ ^^For the object's principal axis Y as the first principal axis and the object's principal axis Z as the second principal axis, the preferred left-handed transformation matrix is ​​therefore obtained ^^ ^ ^ ^ ^( ^^^^ ) ^^^ ^ ^ ^( ^^^ ′ ^ ) ^^^ ^ ^ ^( ^^ ′′ ^^) = ( For the object's principal axis Y as the first principal axis and the object's principal axis X as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^ ^ ^ ^ ^ (^^ ^^ )^^ ^ ^ ^ ^ (^^ ^ ′ ^ )^^ ^ ^ ^ ^ (^^ ′′ ^^) = ( For the object's principal axis Z as the first principal axis and the object's principal axis X as the second principal axis, the preferred left-handed transformation matrix results, as already explained in detail. ^ ^ ^ ^( ^^^^ ) ^^^ ^ ^ ^( ^^^ ′ ^ ) ^^^ ^ ^ ^( ^^ ′′ ^^) = ( For the object's principal axis Z as the first principal axis and the object's principal axis Y as the second principal axis, the preferred right-handed transformation matrix is ​​therefore obtained ^^ ^ ^ ^ ^( ^^^^ ) ^^^ ^ ^ ^( ^^^ ′ ^ ) ^^^ ^ ^ ^( ^^ ′′ ^^) = Since the preceding explanations lead to the conclusion that, as a rule, no transformation for orienting an object in space can occur without also involving a rotation of the object in the base plane, this rotation of the object in the base plane, resulting from the subsequent orientation of the object along the first principal axis of space, must be subtracted from the object's initial rotation in the base plane. It has already been shown that intrinsic changes in the rotation angle have no effect on the subsequent orientation of the object along the principal axis if they occur before this orientation. Therefore, the object must already be rotated to the intrinsic rotation angle specified by the subsequent transformations in the first step of the transformation for orientation in the base plane. ^ ′ ^ ′can be correctly converted. If one takes the direction vector ^^(^^^^, ^^^^ , ^^^^) of an arbitrary axis of the object in its own coordinate system and the desired extrinsic rotation angle ^^^^^^ = ^^^^^^ −1 ^^ ( ^^ ^^^^) this axis ^^(^^^^ , ^^^^, ^^^^) around the first principal axis of space in the desired position and orientation of the object in space is known, the rotation angle for the first rotation must be ^^ ^ ′ ^ ′ to calculate the difference ∆^^ ^^^^ between the extrinsic rotation angle ^^ ^ ′ ^ ′ ^ ′ ^ of the direction vector ^^′′′(^^ ′′′ ^^ , ^^ ′′′ ^^ , ^^^ ′ ^ ′′ ) after the orientation transformation and the intrinsic rotation angle ^^ ^ ′ ^ ^^ of the direction vector ^^′(^^ ′ ^^ , ^^ ′ ^^ , ^^^ ′^) are corrected before the final alignment transformation. Thus, the corrected first rotation angle is given by the equation... Since the transposed orientation transformation is related to the Since the principal axis of space must ultimately be identical for any previously performed rotation around this first principal axis, it also holds that ∆^^ ^^ can depend solely on this transposed orientation transformation and therefore for ∆^^ ^^ also the equation must. This results in the following for Thus, the corrected rotation angle can be ^^ ^ ′ ^ ′The transposed transformation matrix can be uniquely determined depending on the known direction vector (^^^^, ^^^^, ^^^^), the desired extrinsic rotation angle of the vector (^^^^, ^^^^, ^^^^) in space, and the transposed orientation transformation to be performed for the desired position of the object in space. Analogous simplified formulas can also be derived for all other transposed transformations based on the existing geometric symmetry, and thus the following apply: As already explained several times and previously derived in detail, the calculation of the precise orientation angle in the transposed transformation matrix must be derived in direct dependence on both the type of orientation transformation and its sequence. This dependence also results in the previously demonstrated incompatibility of the different object transformations and their sequences, because different angles for the individual rotations to be performed in the sequence result depending on the chosen object transformation.However, the derivation presented demonstrates that by appropriately selecting the transformation sequences and calculating the individual rotations as a function of the chosen transformation sequence, the object's position after the rotational transformations can be precisely predicted, and the resulting object coordinates in space can thus accurately reflect its actual position. According to this new method, not only individual objects but also entire groups of objects can be transformed into any desired position in space, depending on a single object and ultimately on two known direction vectors and a single known extrinsic rotation angle.With the new method, it is not only possible to precisely transform an object or group of objects back into its previous position in space, but also to precisely transform this object or group of objects into any desired position in space. The universal coordinate system of an object in three-dimensional space: Through the analysis of all previously derived formulas, it becomes clear that three-dimensional space, in its original definition according to Descartes, contains conventional provisions that cannot do justice to the overarching symmetry of the orthogonal axes. One such provision is the definition of the three axes X, Y, and Z in a clockwise order.While this convention is useful in providing a framework for a shared understanding of space and the perception of real objects within a universally comprehensible system, it also limits the experience of space by precluding the possibility of universality. The convention restricts the possibilities of different "perspectives." By establishing, more generally and universally, that every object, by virtue of its form or function, possesses a uniquely definable primary axis and a secondary primary axis in a different direction, both determined by the object's inherent form and / or function, the view of the object can be decoupled from the direction of view from one of Descartes' principal axes of three-dimensional space.In this sense, Descartes' arbitrary definition of the three axes suddenly becomes an obstacle, because it restricts the perspective to a subset of the entire space. In the new, universal view of the object, the choice of axis designation becomes a matter of form and can be determined by the observer according to their own definition or a nomenclature common in their profession. On the other hand, according to Descartes, the measured angles in a principal plane of the coordinate system are predetermined in a specific direction: positive angles are always measured counterclockwise from one principal axis to the other. This convention is problematic because it is inconsistent with the symmetrical properties of three-dimensional space and does not follow a readily explainable analytical function.However, determining a measurable angle as a function of the first principal axis and the second principal axis, or as a function of the orthogonal coordinate system of an object resulting from these two principal axes, makes this measurement object-specific and thus direction-oriented, rather than convention-based. In the present definition, all angles are always determined as the tangent of a ratio in a direction vector and are therefore universally measured the same in all cases. The universal coordinate system of an object in object-related space is to be redefined by the three orthogonal principal axes P, S, and T, where P corresponds to the first object-specific principal axis (primary axis) and has the direction vector (^^^^, ^^^^, ^^^^).The second principal axis S (secondary axis) with direction vector ^^(^^^^, ^^^^, ^^^^) lies in the plane spanned by the first and second principal axes of the object and is orthogonal to the primary principal axis. The third principal axis T (tertiary axis) with direction vector ^^(^^^^ , ^^^^ , ^^^^) is orthogonal to the plane spanned by the first and second principal axes of the object. The orientation angles ^^ are used to orient the object in its own space with the axes p, s, and t. ^^ , calculated by its tangent ^^^^^^^^ ^^ = for the rotation about the tertiary t-axis of space using the associated rotation matrix ^^^^^^^^ ^^ −^^^^^^^^ ^^ 0 = ( ^^^^^^^^ ^^ ^^^^^^^^ ^^ 0), ^^ ^^ , calculated by its tangent ^^^^^^^^ ^^ for the 0 0 1 Rotation about the secondary s-axis of space using the associated calculated by Tangent ^^^^^^^^ = for the rotation about the primary p-axis of space using the associated rotation matrix ^^, from which with the corresponding required rotation angle corrections and ψ ′′ = direct alignment Transformation matrix ^^ (^^ ^^ ) = This results in the following: For every direction vector ^^(^^^^, ^^^^, ^^^^) of the object that is not parallel to the P-axis of the object, ψ also holds. ′′ = ψ ′′ ^^^^ + also For the transformation of the (known) object into an arbitrary position in space, the transposed transformation can be used, as already derived. = can be used and it also applies here that the corrected rotation angle ^^ ^ ′ ^ ′ with for these Transformation matrix can be determined depending on the known direction vector ^^(^^^^ , ^^^^, ^^^^) and the desired extrinsic rotation angle ^^^^^^ of the vector ^^(^^^^, ^^^^, ^^^^) in space. The universal coordinate system of the object in three-dimensional space can thus be understood as a summary of all previously derived formulas for the three-dimensional coordinate system according to Descartes, because by replacing the coordinates ^^(^^^^ , ^^^^, ^^^^) and ^^(^^^^, ^^^^, ^^^^) and the rotation angles ^^^^, ^^^^ and ^^^^ with the coordinates ^^(^^^^, ^^^^, ^^^^) and ^^(^^^^, ^^^^ , ^^^^) and the rotation angles ^^^^, ^^^^ and ^^^^ or with the coordinates ^^(^^^^, ^^^^ , ^^^^) ^^^^^^ ^^(^^^^, ^^^^, ^^^^) and the rotation angles ^^ ^^ , ^^ ^^ and ^^ ^^or one of the four other variants of the order of X, Y, and Z with the corresponding rotation angles, all of the previously derived formulas for any arbitrary vector in the Cartesian coordinate system can be represented. The advantages of the universal coordinate system over the Cartesian coordinate system become particularly apparent in the programming of algorithms for transforming objects in three-dimensional space, because a single transformation matrix in the universal coordinate system can replace all six possible transformations in the Cartesian coordinate system.Converting the Cartesian coordinates of an entire sequence of specific points, landmarks, and vectors into the universal coordinate system requires only a reversal of the order of these coordinates and the corresponding rotation angles according to the primary, secondary, and tertiary axes of the object. Even the sign for each of these three axes can be changed individually or in any combination, thus allowing both the classical right-handed Cartesian coordinate system in every possible variant and its left-handed mirror image to be equally convertible into the universal coordinate system. This results in the following general use of the derived formulas and transformations in the universal coordinate system for the coordinates of any object in three-dimensional space according to Descartes: 1.1. Conversion of the original Cartesian coordinates into the order of the object's primary, secondary, and tertiary axes, taking into account advantageous sign changes. 2. Determination of the resulting rotation angles. ^^ , ^^ ^^ and ^^ ^^ from their corresponding tangent and calculating their corrections ^^ ^ ′ ^ and ^For use in direct transformation for aligning the object in its own space or for transforming the (known) object into any position in space. 3. Reverse conversion of the universal coordinates after transformation into Cartesian coordinates by changing the order of the object's primary, secondary, and tertiary axes to the original axes according to Descartes, taking into account the sign changes. Determining an object in the coordinate system in a perspective image: The coordinate system of the recording device, or rather its image, is uniquely determined by the position of the recording plate and the focal point of the lens, or objective, in space.Since the recording plate is normally rectangular, it is natural to determine the direction of the first and second principal axes of the recording device's coordinate system using the image axes and to define the origin at a corner of the image. The direction of the third principal axis is then defined, by definition, by the direction vector of the image plane. Thus, the recording plate, or rather the image plane, and the focal point of the recording device's lens form a known object group consisting of two independent objects with a determinable relative position, because the coordinates of the lens's focal point ^^(^^^^, ^^^^, ^^^^) are uniquely defined in the space of the image plane, where ^^. ^^This results from the shortest possible distance between the focal point of the lens and the recording plate, which corresponds to the distance between the focal point and the image plate in the direction of the vector of the image plane. For any arbitrary point ^^(^^^^, ^^^^, ^^^^) of the object in the coordinate system of the image of the recording device with the origin in the image plane, the following applies to its projection ^^′ ( ^^ ′ ^^ , ^^ ′ ^^ , ^^^ ′ ′ ^ ) The image, shown in its original size, shows that ^^^^ = 0 if the image plane is defined as the base plane of the recording device's coordinate system. Furthermore, according to the intercept theorem, the coordinates of the object point and ^^ are given. ′ − ^^ ′ whose projection the equations ^^ ^^ ^^ ^^ ^ ′ ^ − ^^ ^^ = ^^ ^^ ^^ ^^ and ^^ − ^^ ^^ ^^ ^ ′ ^ − ^^ ^^ = ^^ ^^ or in another notation ^^^^ = ^^^ ′ ^^ ^ − ^^ ^^ ′ ^^ ) ^^ ^^ (^^^^ − ^^^^) and oder ^^ = ^^ ′ (1 −^^ ^^) + ^^^^ ^^ and also ^^ = ^^ ′ (1 −^^ ^^) + ^^ ^^ ^ ^ ^ ^^ ^^ ^^ ^^^^ ^^ ^^ ^^^^ ^^ ^^ ^^ ^ ^^ With ^^ ^^ = ^^ ^^ Thus, in simplified terms, ^^ ′^^ = ^^^^ (1 − ^^^^) + ^^ ′^^^^^^ and also ^^^^ = ^^^^ (1 − ^^^^) + ^^^^^^^^. This shows that every point of an object has its own determinable magnification factor ^^^^ = 1 − ^^^^ in the image projection at original size. If the image is not at original size, the image coordinates are determined using the image scale factor ^^ ^^ 1 (scale of the image is ^^^^) to be occupied and therefore ^^ ′^^ = ^^^^^^^^^^^^ + ^^^^^^^^ and ^^ ′^^ = ^^^^^^^^^^^^ + ^^^^^^^^. For any two points ^^(^^^^, ^^^^, ^^^^) and ^^(^^^^ , ^^^^ , ^^^^) of an object, whose distance is known, then ^^2 = ^^^^2 = With all known distances between the known points of the object with known projections in the image, the object's position and orientation, and thus its location in the image's coordinate system, can be determined iteratively. Another way to determine the object's position in the image's coordinate system is to iteratively manipulate its position and orientation within the image's coordinate system, shifting and rotating it until the rays from the lens's focal point through the object points converge as precisely as possible on the corresponding projection points in the image. For this, the formulas derived above for the projection points of the image are used. ^^ −^^ ^^ ^^ ^^ ^^ ^^ −^^ ^^^ object rewritten to ^^ ^ ′ ^ = ^^ ^^ ^^ ^^ , ^^ ^ ′ ^ = ^ ^^ ^^ ^^ ^^ ^^ and ^^ ′ ^^ = 0, where it is still true that ^^ ^^ ^^= ^^^^ and ^^^^ = 1 − ^^^^, and thus these two variables can be directly determined for each object point, provided the object's coordinates in the image's coordinate system have been determined with sufficient precision through iteration. This implies, metaphorically, that at least three points of an object in its own coordinate system must be known, and their projections in the generated image must be unambiguously identifiable, in order to fully determine the object's position and orientation in the coordinate system of the recording device or the image, because three different equations can be derived from the distances between these three points in different combinations, and three further equations can be derived from the plane formed by the three points.Using these six equations, based on the coordinates of the lens's focal point and the projections of the known object points onto the image plane, the three different magnification factors for the object points, and thus the actual coordinate values ​​of the individual object points, can be uniquely determined mathematically. However, if only an image is available, and neither the exact construction of the recording device nor the actual magnification factor of the image is known, at least four points of an object with their projection points in the image are required to derive enough equations to determine all unknown magnification factors and the position of the lens's focal point.If several (partially) known objects in the same image are known according to the defined specifications, all of these objects can be used to calculate the position of the lens's focal point. This then reduces the number of points required for a single object back to three, provided the relative position between the individual objects is at least approximately known. The second method of determining the object in the coordinate system of the perspective image, namely the iterative manipulation of the object's position in the perspective image's coordinate system, has the advantage that this iteration simultaneously averages out the inaccuracies in the only imperfectly known coordinates of the object in terms of their influence on the accuracy of the iteration.Specifically, this means that with this method for determining the object in the coordinate system of the perspective image, the theoretical values ​​of the coordinates of an object or group of objects are sufficient for iteration, even if it must be assumed that these coordinates are only approximately known and therefore differ from the virtualized object in the real object. While the accuracy of determining the object or group of objects in the coordinate system of the perspective image is affected in this case, the expected error is simultaneously averaged out by the iteration and can even be determined in terms of its standard deviation. If several objects are suitable for determination in the coordinate system of the same perspective image, the parameters of the perspective image can be averaged from all iterations of the different objects in order to determine them as precisely as possible.The object's position in the image's universal coordinate system: For the iterative manipulation of the object's position to determine its location in the image space, it is advantageous to use the universal coordinate system and define the orthogonal viewing direction on the image as the primary axis of the image coordinate system. This yields the coordinates ^^(^^^^, ^^^^, ^^^^) for the lens's focal point and the image coordinates of the corresponding projection point ^^′(^^) for any point ^^(^^^^ , ^^^^, ^^^^) of an object. ′ ^^ , ^^ ′ ^^ , ^^^ ′ ^ ) with ^^ −^^ ^^ ^^ −^^ the values ​​^^ ′ ^^ = 0, ^^^ ′ ^ = ^^ ^^ ^^ and ^^ ^ ′ ^ = ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ , whereby it still holds true that for every point of the object ^^ ^ ^= ^^^^ and ^^^^ = 1 − ^^^^ are. The iterative manipulation aims to reduce the average deviation of the projection points calculated from the object's position in the image. ^ ′ ^ ( ^^ ′ ^^ , ^^ ′ ^^ , ^^^ ′ ^ ) from the projection points identified in the image ^^ ^ ′ ^ ( ^^ ′ ^^ , ^^ ′ ^^ , ^^^ ′ ^ ) to minimize and at the same time keep the standard deviation of these deviations as small as possible in order to avoid individual outliers. The corresponding formula for this deviation is therefore also somewhat √ ′ 2 ′ 2 (^^ −^^ ^^ −^^ ^^ ^^ ) +(^^ −^^ ^^ −^^ ^^ ^^ ) ∆^^′ = ^^ ′ ′ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^ = ^^ ^^ ^^ ^^The calculation is performed for all projection points simultaneously for each individual change of one of the parameters for the object's position, in order to then calculate both the resulting average deviation and the standard deviation of the deviations, in order to determine the lowest values ​​through iteration. While, during iteration involving a translation of the object parallel to the image surface, the distances between the current projection points and the determined image points depend directly on the magnitude of this translation, rotations of the object around one of the three image axes always result in a translation of the projection points in different directions and of varying magnitudes, which essentially depend on the projected distance of the object point from the object's origin in the plane of rotation.Rotations therefore tend to affect the iteration abruptly and should always be performed inversely to the greatest projected distance of the object points from the object's origin in the plane of the intended rotation. This ensures that the resulting translations of the projection points are of the same order of magnitude as a translation parallel to the image plane. Since the projection points may be quite far from the corresponding image points at the beginning of the iteration, no changes should be made to the initially approximate position of the lens's focal point, which can be considered the origin of all rays, as these changes have a significant impact on the progress of the iteration. It is advisable to begin changing the lens's focal point position relatively late in the iteration to increase the precision of the achievable approximation to reality.Changing the distance of the lens's focal point also tends to result in abrupt changes in the projections if the object's distance to the image plane is not changed proportionally. Therefore, any change in the lens's focal point's distance from the image plane should always be accompanied by a proportional change in the object's distance from the image plane. Shifting the lens's focal point parallel to the image plane affects both the translation of the projection points in a similar proportion to translating the object parallel to the image plane, and the corresponding rotation of the object relative to the change in the beam angle.Therefore, a shift of the lens's focal point parallel to the image plane must always be compensated for by a corresponding translation of the object parallel to the image plane in the same proportion and by a rotation adapted to the change in the beam angle. The unambiguous determination of unknown objects in space: As previously described, an object or group of objects can be uniquely determined in space, using a single image, for all its known object points and landmarks, with respect to the space spanned by the image. This also applies to several independent objects and groups of objects in the same image. If an unknown object is present in a known image, and individual landmarks on the image can be uniquely referenced, the corresponding projection ray paths can be calculated as lines in space.If a second image, taken at a sufficiently different angle from the first, is analyzed using the described method, and the known objects in the space of this second image are determined analogously to those in the first image, the initially unknown object can be uniquely located in space by intersecting the projection ray paths of both images using its landmarks. Its spatial extent can then be measured based on the corresponding and unambiguously definable projections of the landmarks, or represented in a 3D model. For image analysis, it is therefore possible to unambiguously determine the position of all objects appearing in the images within (shared) space, even if only some of these objects are effectively known, at least partially, in terms of their spatial extent.It logically follows that, for determining the space defined by the image, the objects used should ideally be those whose dimensions are known as precisely as possible and which have sufficiently unambiguously identifiable projections in the respective image. Rigid bodies or groups of objects with rigid connections are particularly suitable for this purpose. Groups of objects in which the individual objects can move relative to one another are less suitable. Nevertheless, it is also possible to use such internally movable groups of objects for spatial analysis. However, it must be noted that a separate, independent spatial analysis must be performed for each known object, as they can be located in different relative positions and orientations in space within any given image.This implies that for all unknown objects, a relationship must be established between the two images and the known object that is as firmly connected as possible, or at least the known object closest to the unknown object, so that they can be fully measured within the respective space of the two images. In principle, this means that separate corresponding spaces must be created for all known objects in the images, within which the (corresponding) unknown objects are measured by determining the landmarks in space as precisely as possible. The relative position of individual objects to each other at different times: The new method for transforming the position of objects in space presented here opens up new possibilities for calculating complex temporal dependencies between different objects in space.As an example, let us explicitly mention an object group combination based on a hexapod construct, also known as a Stewart platform or, in medical applications, as a Taylor frame or MAXframe. In its basic form, a hexapod construct suitable for medical applications consists of two rings, possibly of unequal sizes, and six extendable supports of varying lengths fixed to the two rings between predetermined mutual support points. These supports are, in turn, rigidly attached to two different structures, which can be considered individual object groups.To calculate the relative position of two structures or object groups at a specific point in time, the second object group must be transformed into the coordinate system of the first. This can be achieved by transforming all known coordinates of both object groups into the coordinate system of the first. By subsequently calculating the position of the second object group within the coordinate system of the first, the relative position between these two structures at a known point in time can be determined. As already described, however, an object (or object group) must be known in its own coordinate system in order to transform it into any position in space.Therefore, it is not sufficient to determine the position of the second object group only within the coordinate system of the first object group, because it is also necessary to transform at least all known coordinates of the second object group into its own coordinate system in order to also know the object-specific coordinates of this structure. Assuming that at a specific point in time the two object groups have a determinable position in space, then the object-specific coordinates of both object groups, as well as their relative positions to each other, can be uniquely determined from the spatial coordinates of both object groups.If a further relative position between these two groups of objects is defined as a target, the new method allows for the calculation of a precise movement sequence with an exact timeline, and thus a plan for the temporal specification of the length changes of the six supports between the two hexapod rings. This is because the relative position of the two hexapod rings, and therefore the relative position of the mutual support points of the six supports between these hexapod rings, can be precisely calculated at any point in time between the initial situation and the target. Calculating the relative position of two hexapod rings using the known lengths of the six supports and the dimensions of their support points is not technically new and is already widely used in various hexapod applications.However, depending on the use of the support points, which are not ideally suited to the structure, it is not always easy to calculate this relative position directly. Here, too, the perfect relative position of the rings to each other, corresponding to the measured support lengths, can be determined quite easily and quickly by iteratively manipulating the position of one hexapod ring in the universal coordinate system of the other hexapod ring and approximating the respective resulting support lengths to the measured support lengths. For calculating a support length, it is defined that it corresponds to the distance ^^^^ between the support point ^^(^^^^, ^^^^, ^^^^) on the primary hexapod ring and the support point ^^(^^^^, ^^^^, ^^^^) on the secondary hexapod ring, which means that the calculation of the support length is ^^^^ = √(^^^^ − ^^^^)2 + (^^^^ − ^^^^)2 + (^^^^ − ^^^^)2. If the measured support length is given with the value ^^, then the deviation of the support length ∆^^ = ^^^^ − ^^ =. The iteration for approximating the calculated support length to the measured (or desired) support length can thus be performed by manipulating the position of the secondary hexapod ring in the reference space of the primary hexapod ring. While, when iterating by translating the secondary hexapod ring in any direction, the calculated support lengths can be directly derived from the magnitude of this displacement, rotations of the secondary hexapod ring around one of the three reference space axes always result in translations for the bearing points of the secondary hexapod ring in different directions and of different magnitudes, which essentially depend on the distances of the bearing points from the origin of the secondary hexapod ring in the plane of rotation.Rotations in this iteration also tend to cause abrupt changes and should always be performed inversely to the largest projected distance of the bearing points from the origin of the secondary hexapod ring in the plane of the intended rotation. When iterating the position of the secondary hexapod ring, it is logical to define the z-axis as the primary axis for conversion to the universal coordinate system if the primary hexapod ring lies in the xy-plane of the reference space and its origin is the origin of the reference space. In this case, the secondary hexapod ring should be assumed to be parallel to the primary hexapod ring at the start position of the iteration, at an expected distance with its origin on the z-axis of the reference space.The new method presented here does not require precise knowledge of the support lengths to determine the coordinates of the hexapod rings in a universal image coordinate system. Instead, the theoretical lengths of the supports between the two hexapod rings are determined by directly calculating the relative positions of the support points on the two hexapod rings. Thus, only a single image from a snapshot of a few measurable reference points in any given space is needed to calculate all the necessary coordinates of both object groups in their own coordinate systems and their relative positions.If not all objects in the construct described above are clearly identifiable, a second snapshot taken in space from a sufficiently different angle can be used to precisely determine and measure the unknown objects in relation to at least one known object in the two snapshots. From this, a new method can be derived by which any construct, and in particular a hexapod construct, can be completely and unambiguously determined and measured spatially, provided that at least two images taken from sufficiently different angles and possibly at different times are available: The two rings of the hexapod construct are determined with respect to their mass and the existing attachment positions of the six extendable supports to these rings.From the known dimensions of the rings and the available measurements, the corresponding theoretical attachment points of the supports to the rings are calculated. Using two images of the hexapod structure taken from different angles and possibly at different times, the positions of the two rings are determined relative to the two images by identifying the projections of at least three attachment points each. The two images are transformed into the respective coordinate systems of the two rings, and their positions within these systems are calculated. Additionally, landmarks of other (unknown) objects referenced in the images are used to calculate the spatial position and orientation of these objects relative to at least one of the two rings in each image.The position and orientation of one of the now known objects are used with respect to the first ring to determine the reference coordinate system. All known objects and object groups with respect to the first ring are transformed into the reference coordinate system. The position and orientation of another known object are used with respect to the second ring to determine the movable coordinate system. All known objects and object groups with respect to the second ring are transformed into the movable coordinate system. The position and orientation of the movable coordinate system and all objects related to it are determined in the reference coordinate system. The movable coordinate system and all objects related to it are aligned in the reference coordinate system by transformation. 12.The position and orientation of the movable coordinate system and all objects related to it, aligned in the reference coordinate system, is used as the starting point to transform the movable coordinate system and the objects related to it into the initial orientation and position in the reference coordinate system. 13. The position and orientation of the movable coordinate system and all objects related to it, aligned in the reference coordinate system, is used as the starting point to transform the movable coordinate system and the objects related to it into any position between the initial position and the aligned position in the reference coordinate system. 14. The position of the movable coordinate system in the reference coordinate system is used to calculate the length of the supports that connect the corresponding attachment points of the two rings. 15.Steps 13 and 14 are repeated as often as necessary to determine a specific predefined movement process, in order to simulate each individual position of the moving coordinate system within the reference coordinate system and to calculate the corresponding support lengths. The relative position of multiple constructs in relation to each other over time: In certain medical applications, two hexapod constructs are connected in such a way that they either share the central ring or one ring of each hexapod construct is attached to the same bone segment. In this configuration, it is possible to apply the new method to both hexapod constructs separately, as is already done with existing systems.This has the disadvantage, however, that the second hexapod construct constantly shifts depending on the movements of the first hexapod construct, and consequently, the reference coordinate system of the second hexapod construct also moves in space. This dependence of the second reference system on the first can also be used by considering the entire second hexapod construct as an object group of the first moving coordinate system at a specific point in time and transforming it as a whole, using the known coordinates of that same point in time in the second reference coordinate system, into the first reference coordinate system. To do this, however, the common object group to which both hexapod constructs are attached must first be transformed into a unified coordinate system.It should be noted that the movable coordinate system of the first hexapod construct does not necessarily have to coincide with the reference coordinate system of the second hexapod construct, even if both coordinate systems were defined in relation to the same group of objects. If different landmarks were used to determine the orientation of this group of objects, this will inevitably lead to different orientation angles and different origins for the two coordinate systems. If both hexapod constructs use a common ring, the already known parameters of the respective transformations from the ring coordinate system to the object coordinate system can be used to transform both object coordinate systems into the common ring coordinate system via inverse transformations.If different rings are used to attach the two hexapod constructs to the same group of objects, both rings must be defined in space, at least in one image, by their projection lines, and then their position and orientation relative to each other must be uniquely calculated. In this case, a further transformation of the entire second hexapod construct into the coordinate system of the second ring of the first hexapod construct is necessary so that both hexapod constructs are known in a common coordinate system based on the same common object. This now common coordinate system, based on the common ring attached to the same group of objects, can then be transformed into any new definition of the orientation of the common object.For example, the connection between the two origin landmarks can now be used to determine the new first principal axis of the object group, while the second principal axis is still determined by the same axis used to define the movable coordinate system of the first hexapod construct. The origin of the new object group's coordinate system can also be defined by either the origin landmark of the first or the second original coordinate system of the respective hexapod constructs. By transforming the new common object coordinate system and the entire second hexapod construct within it into the reference coordinate system at different times, the movable coordinate system of the second hexapod construct is automatically determined within the higher-level reference coordinate system of the first hexapod construct and thus within the higher-level reference space.By performing several such transformations at different times of the entire second hexapod construct into the higher-level reference coordinate system of the first hexapod construct, the movements of the movable coordinate system of the second hexapod construct within the higher-level reference space can now be directly traced. If the situation in which both movable coordinate systems are aligned within their respective reference coordinate systems is used as the starting point to transform the movable coordinate system of the second hexapod construct and the objects related to it into an arbitrary position, consisting of orientation and position, within the higher-level reference coordinate system, then this higher-level system can also be considered the reference coordinate system of the second hexapod construct, thus redefining the common object of both hexapod constructs as the movable coordinate system of both hexapod constructs.This changes the interdependence of the object groups. This new interdependence can now be used to transform the common object of both hexapod constructs into an arbitrary position in the higher-level space, depending on the relative positions of the two non-common object groups. This allows the lengths of all supports of both hexapod constructs to be calculated under the new constraints. By directly manipulating the two movable coordinate systems in their positions within the common higher-level space, new dependent movement patterns can be defined for these two movable coordinate systems, which are only indirectly dependent on each other, and calculated sequentially. From this, a new method for controlling a combination of two or more hexapod constructs can be directly derived: 1.1. For all combined hexapod constructs, all objects in the respective reference coordinate system are determined individually according to the predefined method for a single hexapod construct at the time of complete alignment. 2. For all combined hexapod constructs, all objects in the respective reference coordinate system are determined individually according to the predefined method for a single hexapod construct at a different predefined time common to all hexapod constructs. 3. For all hexapod constructs that share a common object group, the common object coordinate system is defined for this common object group. 4.For all hexapod constructs, the object coordinate systems common to other hexapod constructs are defined as only indirectly dependent movable coordinate systems in the common higher-level space and are jointly determined at the time of complete alignment, depending on their orientation and position relative to each other. 5. For all hexapod constructs, the object coordinate systems common to other hexapod constructs in the common higher-level space are determined at the first common time, depending on their orientation and position relative to each other. 6. For the movement of all independent movable object coordinate systems, a common movement pattern is defined based on the time of complete alignment of all hexapod systems, taking into account the position, consisting of orientation and position, of these object coordinate systems at the first common time. 7.For all points in time of the shared movement pattern, the relative positions of the individual, interdependent object coordinate systems are determined, and the respective lengths of the supports are calculated based on the distances between the corresponding attachment points. This new method thus allows even very complex combinations of multiple hexapod systems to be precisely coordinated and controlled.

Claims

AMENDED CLAIMS received by the International Bureau on 20 September 2025 (20.09.2025) 1. Method for calculating the three rotation angles of an object-related transformation matrix for converting all known object-related coordinates of an object of arbitrary shape with known origin, known first principal axis direction and differing second axis direction into an arbitrarily predefined position in a predefined three-dimensional space with known position of the object's own origin and the two known deflection angles of the first principal axis of the object from the corresponding first principal axis of space as well as the known deflection angle of the second axis of the object from a second principal axis of space in the base plane of the first principal axis of space, characterized in that this object-related transformation matrix is ​​selected depending on the given deflection angles of the object.The first rotation angle corresponds to one of the object-related deflection angles, and the calculation of the second rotation angle is performed as a function of the first rotation angle, and the calculation of the third rotation angle is performed as a function of the first and second rotation angles based on the given object-related deflection angles.

2. Method according to claim 1 characterized in that the object-related deflection angles of the axes of the object are determined based on the direction values ​​of the vectors of the object-related axis directions as the arctangent of the ratios of the two basis values ​​of these vectors measured in the basis plane of the principal axis of space corresponding to these vectors to the primary axis value of the same vectors measured in the direction of the same principal axis of space.

3. Method according to claim 2 characterized in that the first rotation angle corresponds to the first object-related deflection angle of the first principal axis of the object, which is determined in the plane of the first two principal axes of space.

4. Method according to claim 3, characterized in that the second rotation angle corresponds to the arctangent of the multiplication by the tangent of the second 49 AMENDED SHEET (ARTICLE 19) The object-related deflection angle of the first principal axis of the object corresponds to the cosine of the first rotation angle.

5. Method according to claim 4 characterized in that the second axis of the object is one of the two other principal axes of the object and the known deflection angle of this second principal axis of the object to the corresponding principal axis of space in the base plane of the first principal axis is determined.

6. Method according to claim 5 characterized in that the third rotation angle corresponds to the arctangent of the ratio between the multiplication of the tangent of the object-related deflection angle of the second principal axis of the object in the base plane of the first principal axis of space with the cosine of the first rotation angle and the sum of the cosine of the second rotation angle and the multiplication again of the tangent of the object-related deflection angle of the second principal axis of the object in the base plane of the first principal axis of space with the sine of the first rotation angle and with the sine of the second rotation angle.

7. Method according to claim 4 characterized in that the third rotation angle corresponds to the difference between the object-related deflection angle of the second axis of the object in the base plane of the first principal axis of space and the difference between the projection angle of the second axis of the object in the base plane of the first principal axis of space after the deflection transformation of the first principal axis of the object to the corresponding first principal axis of space and the projection angle of the second axis of the object in the base plane of the first principal axis of the object.

8. Method according to one of claims 6 and 7 characterized in that the transformation matrix selected depending on the given deflection angles of the object corresponds to the multiplication of the deflection rotation matrix of the first rotation angle with the deflection rotation matrix of the second rotation angle and with the deflection rotation matrix of the third rotation angle and with the object-related translation matrix. 50 AMENDED SHEET (ARTICLE 19) 9. Method according to claim 8 characterized in that the deflection rotation matrices of the rotation angles correspond to the transposed alignment rotation matrices of the rotation angles.

10. Method according to claim 9 characterized in that the alignment rotation matrices of the rotation angles each correspond to a rotation of the object about the principal axis of space corresponding to the respective rotation angle from the principal axis of the object in the direction of the corresponding principal axis of space.

11. Method according to one of claims 1, 2 and 10 characterized in that a plurality of objects are independently transformed into a mutually dependent position by object-related transformations in a common three-dimensional space.

12. Method according to claim 11 characterized in that the coordinate system of the common three-dimensional space corresponds to the coordinate system of a selected object from the plurality of objects.

13. Method according to one of claims 11 and 12 characterized in that the plurality of objects includes at least two images which are used to determine the coordinates of these reference points in common space from projections of reference points recognizable on the images.

14. Method according to claim 13 characterized in that the calculated coordinates of reference points of an object are used to define the coordinate system of this object.

15. Method according to one of claims 11 and 12 characterized in that the plurality of objects are combined as an object group to form a new superior object.

16. Method according to claim 15 characterized in that the superior object corresponds to a snapshot of the state of a group of related objects at a specified time. 51 AMENDED SHEET (ARTICLE 19) 17. Method according to claim 16 characterized in that a plurality of snapshots of the same group of related objects correspond to the different states of this group of related objects in a temporally defined sequence.

18. Method according to claim 17 characterized in that the multitude of snapshots of the same group of related objects is used to control individual modifiable objects from the group of objects in such a way that a predetermined state of the group of related objects is generated.

19. Method according to claim 18 characterized in that a plurality of predetermined states of the group of related objects are used to control the state of the group of objects according to a plan in temporal sequence.

20. Method according to one of claims 15 to 19 characterized in that the group of connected objects includes one or more hexapods, each consisting of two rings and 6 extendable supports, each of which is firmly connected to the two rings at predetermined attachment points.

21. Method according to claim 20 characterized in that individual or all adjacent hexapods have a common ring.

22. Method according to one of claims 20 and 21 characterized in that the changes in the lengths of the supports of one or more hexapods are used to control the state of the superior object.

23. Method according to claim 22 characterized in that the control of the state of the superior object by means of changes in the lengths of the supports of one or more hexapods is used to transform the state of the superior object from an undesirable initial state to a desired improved state in a planned temporal sequence.

24. Method according to patent claim 23 characterized in that the superior object is a body part of a living being and the unwanted 52 AMENDED SHEET (ARTICLE 19) The initial state corresponds to a deformity, which is transformed into a state approaching normality by the planned temporal sequence of the control of the extendable supports of the hexapods. 53 AMENDED SHEET (ARTICLE 19)

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