Apparatus and method for designing polynomial curve

Abstract

The present invention relates to a polynomial curve design technology and, more specifically, to an apparatus and a method for designing a polynomial curve by using a hodograph control polygon. According to an embodiment of the present invention, even when the order of the polynomial curve increases, the shape of the curve can be easily predicted. In addition, the shape of the curve can be adjusted by moving the position of a control point, and simultaneously, a tangent vector at a specific point on the curve can be controlled through each side of the hodograph control polygon, whereby the shape of the polynomial curve can be easily adjusted. Since the sum of the lengths of respective sides of the hodograph control polygon corresponds to a numerical integral of a speed function of the polynomial curve, the hodograph control polygon and the curve have similar lengths, and thus the shape and length can be simultaneously controlled when the polynomial curve is designed.

Description

Polynomial curve design device and method

[0001] The present invention relates to a polynomial curve design technology, and more particularly, to a device and method for designing a polynomial curve using a hodograph control polygon.

[0002] In areas such as computer graphics, font design, and computer-aided design / computer-aided manufacturing (CAD / CAM), various types of curves are used to express geometric shapes or the movement trajectories of objects. The most basic of these curves are polynomial curves, and the most frequently used polynomial curve expression in the design process is the Bezier curve. curve). The Bezier curve is a simple mathematical form, but it is a Bezier control polygon line ( Bézier curves have the advantage of enabling intuitive design using control polygons (CSPs). However, as the shape of the desired curve becomes more complex, more control points are required, which increases the degree of the curve. However, the CSPs for higher-order polynomial curves become too large compared to the curve itself, making curve design increasingly difficult. A representative method proposed to improve the problems of Bézier curves is the spline curve. A spline curve is a method of expressing a complex curve by connecting multiple pieces of low-order polynomial curves. It requires calculations that reflect various constraints to ensure the desired regularity at the connecting points. Another method proposed to improve the problems of Bézier curves is the rational curve. While rational curves can take on a wider variety of shapes than Bézier curves of the same degree, as the desired curve shape becomes more complex, the required CSPs also increase, making control difficult. Therefore, a device and method are required to overcome the limitations of Bezier control polygons and improve the usability of polynomial curves by using a new concept of control polygons.

[0003] The background technology of the present invention is described in U.S. Patent No. 5,726,896.

[0004] The present invention provides a polynomial curve design device and method.

[0005] The technical problems to be solved by the present invention are not limited to the technical problems mentioned above, and other technical problems not mentioned can be clearly understood by a person having ordinary skill in the technical field to which the present invention belongs from the description below.

[0006] According to one aspect of the present invention, a polynomial curve design device is provided.

[0007] A polynomial curve design device according to one embodiment of the present invention may include a node determination unit for determining nodes, a weight determination unit for determining weights, a polynomial calculation unit for calculating a hodograph control polynomial using the determined nodes and weights, a control polygon generation unit for generating a control polygon using input points, and a polynomial curve generation unit for generating a polynomial curve using each control point of the generated control polygon and the hodograph control polynomial.

[0008]

[0009] According to another aspect of the present invention, a polynomial curve design method and a computer program for executing the same are provided.

[0010] A polynomial curve design method according to one embodiment of the present invention may include a step of determining nodes, a step of determining weights, a step of calculating a hodograph control polynomial using the determined nodes and weights, a step of generating a control polygon using input points, and a step of generating a polynomial curve using each control point of the generated control polygon and the hodograph control polynomial.

[0011] According to an embodiment of the present invention, even when the degree of a polynomial curve increases, the shape of the curve can be easily predicted, and the shape of the curve can be adjusted by moving the position of the control point, and at the same time, each side of the hodograph control polygon can control the tangent vector at a specific point on the curve, so the shape of the polynomial curve can be easily adjusted. In addition, since the sum of the lengths of each side of the hodograph control polygon corresponds to the numerical integral of the speed function of the polynomial curve, the hodograph control polygon and the curve have similar lengths, so when designing a polynomial curve, it has the advantage of being able to control the shape and length at the same time. In addition, since it is composed of a single polynomial curve rather than a form in which small curve pieces are connected, even though it is a curve with a complex shape, it has the advantage of being efficient in the calculation process.

[0012] The effects of the present invention are not limited to the above-described effects, and should be understood to include all effects that can be inferred from the composition of the invention described in the description or claims of the present invention.

[0013] FIG. 1 is a drawing for explaining a polynomial curve design device according to one embodiment of the present invention.

[0014] FIG. 2 is an exemplary drawing for explaining a polynomial curve corresponding to a conventional Bezier control polygon represented by a dotted line in comparison with a polynomial curve design device according to one embodiment of the present invention.

[0015] FIG. 3 is an exemplary drawing for explaining the deformation of a corresponding polynomial curve according to the deformation of a Bezier control polygon compared to a polynomial curve design device according to one embodiment of the present invention.

[0016] FIG. 4 is an exemplary drawing for explaining the form of a 9th order Bernstein polynomial compared to a polynomial curve design device according to one embodiment of the present invention.

[0017] FIG. 5 is a diagram expressing a velocity vector according to one embodiment of the present invention and an exemplary diagram for explaining a generated hodograph curve.

[0018] FIG. 6 is an exemplary drawing for explaining the form of a 9th-order Gauss-Legendre polynomial using a polynomial curve design device according to one embodiment of the present invention.

[0019] FIG. 7 is an exemplary drawing for explaining a Gauss-Legendre control polygon (a) using a polynomial curve design device according to one embodiment of the present invention and a 16th-order polynomial curve (b) designed with the control polygon.

[0020] FIG. 8 is an exemplary drawing for explaining a Bezier control polygon and an enlarged curve observed when the polynomial curve of FIG. 7(b) is expressed as a Bezier curve using a polynomial curve design device according to an embodiment of the present invention.

[0021] FIG. 9 is an exemplary drawing for explaining the form of a 9th-order Gauss-Lobarto polynomial using a polynomial curve design device according to one embodiment of the present invention.

[0022] FIG. 10 is an exemplary drawing for explaining a Gauss-Lobarto control polygon (a) for the curve of FIG. 7(b) using a polynomial curve design device according to an embodiment of the present invention and a polynomial curve (b) designed with the control polygon.

[0023] FIG. 11 is an exemplary drawing for explaining a space curve designed using a Gauss-Legendre control polygon using a polynomial curve design device according to one embodiment of the present invention, a Gauss-Lobarto control polygon, and a Bezier control polygon.

[0024] FIG. 12 is a drawing for explaining a polynomial curve design method according to one embodiment of the present invention.

[0025] The present invention is susceptible to various modifications and embodiments. Specific embodiments are illustrated in the drawings and described in detail. However, this is not intended to limit the present invention to specific embodiments, but rather to encompass all modifications, equivalents, and alternatives falling within the spirit and technical scope of the present invention. In describing the present invention, detailed descriptions of related known technologies will be omitted if they are determined to unnecessarily obscure the gist of the present invention. Furthermore, the use of the singular forms "a," "an," and "the" in this specification and claims should generally be construed to mean "one or more," unless otherwise specified.

[0026] Throughout the specification, when a part is said to be "connected (connected, contacted, or coupled)" to another part, this includes not only cases where it is "directly connected," but also cases where it is "indirectly connected" with another part in between. Furthermore, when a part is said to "include" a component, this does not exclude other components, but rather implies that it may include other components, unless otherwise specifically stated.

[0027] The terminology used herein is merely used to describe specific embodiments and is not intended to limit the present invention. The singular expression includes the plural expression unless the context clearly indicates otherwise. In this specification, it should be understood that the terms "comprises" or "has" indicate the presence of a feature, number, step, operation, component, part, or combination thereof described in the specification, but do not exclude in advance the possibility of the presence or addition of one or more other features, numbers, steps, operations, components, parts, or combinations thereof.

[0028] Hereinafter, the present invention will be described with reference to the attached drawings. However, the present invention can be implemented in various different forms and is therefore not limited to the embodiments described herein. In the drawings, irrelevant parts have been omitted for clarity of description, and similar parts have been designated with similar reference numerals throughout the specification.

[0029]

[0030] FIG. 1 is a drawing for explaining a polynomial curve design device according to one embodiment of the present invention.

[0031] Referring to FIG. 1, the polynomial curve design device includes a node determination unit (110), a weight determination unit (130), a polynomial calculation unit (150), a control polygon generation unit (170), and a polynomial curve generation unit (190).

[0032] The node determination unit (110) determines the node. At this time, the node determination unit (110) can determine the node according to the selected numerical quadrature. The node determination unit (110) can use n different nodes (n is a natural number) given within the definition domain of the curve. For example, a polynomial curve design device can be defined as a function defined in the interval [a, b]. Definite integral of n nodes used to obtain ,..., can be selected. In addition, the node decision unit (110) can use a Gauss-Legendre node according to the selected numerical quadrature, or a Gauss-Robat node that expresses the velocity vector at the curve endpoint as the end edges of the hodograph control polygon.

[0033] The weight determination unit (130) determines the weight. At this time, the weight determination unit (130) can determine the weight according to the selected numerical quadrature. The weight determination unit (130) can determine n (n is a natural number) weights corresponding to n different nodes (n is a natural number) given within the definition domain of the curve. The nodes determined in this way ,..., and weight ,..., Using the interval Functions defined in Definite integral of When calculating, the numerical quadrature method of Equation 1 below can be used.

[0034]

[0035] At this time, depending on the method of setting the location and weight of the node, various quadrature methods such as trapezoidal quadrature, Simpson quadrature, Gauss-Legendre quadrature, Gauss-Lobatto quadrature, and Newton-Cotes quadrature can be utilized. The polynomial curve design device according to an embodiment of the present invention can change the integration interval through appropriate substitution using a linear function, for example, and thus the integration interval can be changed. It can be assumed that. In addition, the weight determination unit (130) can determine the weight in various ways depending on the selected quadrature method, but in particular, when a quadrature method belonging to the category of Gaussian quadrature is selected, the weight is defined as a value obtained by integrating the Lagrange interpolating polynomial for n different nodes determined by the node determination unit (110).

[0036] For example, a polynomial curve design device uses weights calculated by integrating Lagrange polynomials to generate hodograph control polynomials. When calculating , given node ,..., Lagrange polynomials for ,..., can be defined by the following mathematical formula 2.

[0037]

[0038] At this time, the defined Lagrange polynomial is an n-1 degree polynomial, and the ith polynomial among them is a node has a value of 1 only, In other nodes except , the value is 0, which means can be expressed as a weight in Gaussian quadrature. ,..., can be calculated using the Lagrange polynomial as shown in the following mathematical formula 3.

[0039]

[0040] Therefore, in Gaussian quadrature, the node ,..., Once the location of the field is determined, the corresponding weight is automatically determined by mathematical formula 3.

[0041]

[0042] The polynomial calculation unit (150) calculates a hodograph control polynomial using the nodes determined in the node determination unit (110) and the weights determined in the weight determination unit (130).

[0043] Hodograph controlled polynomial of degree n can be defined through the following process. First, each Lagrange polynomial The corresponding weight Divided by the normalized Lagrange polynomial Let this be the cumulative Lagrange polynomial as an integral of this polynomial. second , can be defined as, Is Among the negative ones When It has the value of . Also, Since the integral over the interval [-1,1] is 1, It has the properties of . Additionally The cumulative Lagrange polynomial for the cases 0 and n+1 is After defining the hodograph control polynomial Is It can be defined as follows. If the hodograph control polynomial defined by the above process is expressed as a Lagrange polynomial, it can be defined as the following mathematical expression 4.

[0044]

[0045] At this time, since all Lagrange polynomials are n-1 degree polynomials, the hodograph control polynomials defined by integrating them are n-th degree polynomials. Therefore, n+1 hodograph control polynomials is a basis for a vector space composed of polynomials of degree n or less. Also, the hodograph control polynomial They are partition of unity satisfies. The unity property plays a very important role in the process of generating a polynomial curve from a control polygon. If the polynomial bases do not have the unity property, the shape of the polynomial curve generated using them will not be maintained but will be deformed according to the parallel translation of the control polygon.

[0046] The most important property of the hodograph control polynomial of the polynomial curve design device according to one embodiment of the present invention is the form of its derivative. The derivative of the hodograph control polynomial can be calculated using the following mathematical equation 5.

[0047]

[0048]

[0049] Here, the derivatives of Equation 5 are the nodes used when generating the hodograph control polynomial. has a special value in . For example, node If we calculate the derivative in

[0050] can be calculated as

[0051] The control polygon generation unit (170) generates a control polygon using points input by the user. For example, a hodograph control polygon is a control polygon of n+1 control points input by the user on a plane or space. By sequentially connecting them, a polygon with n sides can be created. By utilizing the polynomial curve design device, the user can adjust the shape of the polygon by changing the position of each control point of the hodograph control polygon.

[0052] The polynomial curve generation unit (190) generates a polynomial curve using each control point of the generated control polygon and the hodograph control polynomial. The polynomial curve generation unit (190) can generate a polynomial curve by combining n+1 control points with their centers of gravity using the hodograph control polynomial calculated by the polynomial calculation unit (150). For example, the polynomial curve generation unit (190) generates a hodograph control polygon having n sides. Polynomial curve corresponding to nth order hodograph control polynomial Control point in the field It can be defined as the barycentric combination of the fields as shown in the following mathematical formula 6.

[0053]

[0054] At this time, the reason why mathematical expression 6 becomes a center-of-gravity combination is because of the polynomial They are This is because it has the unit division property.

[0055] A polynomial curve design device according to one embodiment of the present invention is a control polygon When designing a polynomial curve of Equation 6 using , the endpoint interpolation property can be included, just like a Bezier curve. This means that the starting and ending points of the curve coincide with the two endpoints of the hodograph control polygon, respectively. That is, It can be derived as a hodograph control polynomial, not a Bernstein polynomial, in the polynomial curve generation unit (190). The polynomial curve defined using Equation 6 has several advantages over the Bezier curve. For example, it can include velocity vector interpolation properties, which means that the parameter t is a node This means that the velocity vector of the curve is determined by the corresponding side of the control polygon. This can be expressed in a specific formula as follows.

[0056]

[0057] Here, class refers to the control points of the control polygon, means the velocity vector of the curve. In this way, since the polynomial curve design device determines the tangent direction of the curve at the node point corresponding to each side of the control polygon, it can easily design the shape of the curve using the control polygon. In addition, the polynomial curve generation unit (190) of the polynomial curve design device according to an embodiment of the present invention has the advantage that the length of the polynomial curve designed as the hodograph control polygon is similar to the length of the control polygon. For example, the polynomial curve defined in the section [-1,1] The length of , and the length of the hodograph control polygon is . This is the integral representing the length of the curve at the node and weight Since it corresponds to a numerical quadrature method calculated using , it has a value similar to the length of the curve. In particular, for the Pythagorean hodograph curve whose speed function of the curve is a polynomial function, the length of the control polygon exactly matches the length of the curve.

[0058] Once the nodes and weights are determined through the node determination unit (110) and the weight determination unit (130), the process of calculating the basis of the hodograph control polynomial through the polynomial calculation unit (150) and combining it with the input control polygon through the control polygon generation unit (170) to generate a curve in the polynomial curve generation unit (190) is performed similarly. In particular, if the weight determination unit (130) creates a Lagrange interpolation polynomial using [Mathematical Expression 2] from the nodes selected by the node determination unit (110) in order to make the hodograph control polygon have an endpoint interpolation property, and determines the weights through [Mathematical Expression 3], the form of the basis polynomial is determined by how the nodes are selected in the node determination unit (110). Important techniques for determining the arrangement of nodes include Gauss-Legendre quadrature, Gauss-Lobarto quadrature, and Newton-Coates quadrature.

[0059] For example, the Gauss-Legendre quadrature is defined through the following process. First, the nth-order Legendre polynomial is defined by the following mathematical formula (8).

[0060]

[0061] Nodes of Gauss-Legendre quadrature with n nodes are selected as different real roots of the nth-order Legendre polynomial in Equation 8. In addition, the weights corresponding to each node can be determined using mathematical formula 3. The nth-order Gauss-Legendre quadrature defined in this way can accurately calculate the integral of all polynomials with degrees 2n-1 or less. The Gauss-Legendre quadrature is a quadrature method developed to accurately calculate the integral of the highest-degree polynomial when the number of nodes is fixed by utilizing the orthogonality of Legendre polynomials. Using the Gauss-Legendre quadrature, the Lagrange polynomial of mathematical formula 2 is created and the weights are determined using mathematical formula 3. After calculating the hodograph control polynomial using mathematical formula 4, the calculated polynomial is called the Gauss-Legendre polynomial.

[0062] Another important quadrature used to generate hodograph control polynomials is the Gauss-Lobatto quadrature. In the Gauss-Lobatto quadrature with n nodes, the first and last nodes are each and fixed to the internal node Each of them is an n-1th degree Legendre polynomial Derivative of are selected as different real roots. Using the selected nodes, the Lagrange polynomial of Equation 2 is obtained, and the weights are calculated using Equation 3. The Gauss-Lobatto polynomial can be defined as the polynomial of mathematical formula 4 calculated using the nodes and weights configured while calculating. The hodograph control polygon using the Gauss-Lobatto polynomial can be selected to improve the part where it is difficult to control the velocity vector at the end point of the curve, which is a shortcoming of the Gauss-Legendre control polygon. When using the polynomial curve design device according to an embodiment of the present invention, each side of the hodograph control polygon determines the velocity vector at the corresponding point on the curve. In the case of the Gauss-Legendre quadrature, the node All of them are internal points of the interval [-1,1], and the end values ​​-1 and 1 are not nodal points. Therefore, the first edge of the Gauss-Legendre control polygon is a node that is not the starting point of the curve Controls the velocity vector in the same way as the last variable Nodes, not endpoints of the curve Determine the velocity vector in . Therefore, in order for the hodograph control polygon to control the velocity vector at both endpoints of the curve, the Gauss-Lobarto quadrature method can be selected, which has -1 and 1, which are the values ​​at both endpoints of the parameter interval, as node points.

[0063] In addition, the polynomial curve design device according to one embodiment of the present invention can use nodes and weights of various known quadrature methods, for example, the trapezoidal quadrature method that calculates the integral by approximating the equally spaced sections with straight lines, the Simpson quadrature method that calculates the integral by approximating the quadratic curve by combining two adjacent sections, and the Newton-Cotes quadrature method that divides the entire parameter section into equal parts to determine nodes and calculates the corresponding weights using mathematical expression 3, etc., can be selected to generate a polynomial curve.

[0064]

[0065] FIG. 2 is an exemplary drawing for explaining a polynomial curve corresponding to a conventional Bezier control polygon represented by a dotted line in comparison with a polynomial curve design device according to one embodiment of the present invention.

[0066] Fig. 2 is an example of a Bezier curve, in which points represent Bezier control points, and a polygon expressed as a dotted line connecting these points is a Bezier control polygon. As illustrated in Fig. 2, the solid line is a Bezier curve corresponding to the Bezier control polygon expressed as a dotted line. Using the Bezier control polygon, the shape of the curve can be adjusted by moving the position of the control point, thereby designing a desired curve. In the case of an n-th Bezier curve, the velocity vector at both end points is determined by the line segments at both ends of the corresponding Bezier control polygon, and in this case, the n-th Bezier curve The velocity vectors at each endpoint are is decided by

[0067]

[0068] FIG. 3 is an exemplary drawing for explaining the deformation of a corresponding polynomial curve according to the deformation of a Bezier control polygon compared to a polynomial curve design device according to an embodiment of the present invention.

[0069] Referring to Figure 3, Figure 3 is a diagram illustrating the design process of a Bezier curve, and it can be seen how the shape of the Bezier curve changes according to the movement of the position of one control point.

[0070]

[0071] FIG. 4 is an exemplary drawing for explaining the form of a 9th-order Bernstein polynomial compared to a polynomial curve design device according to one embodiment of the present invention.

[0072] Referring to Figure 4, the Bernstein polynomials used to define the conventional Bezier curve can include several useful properties, for example, the set of nth degree Bernstein polynomials is the basis of the polynomial vector space of degree n or less. At this time, the value of each polynomial is always positive, and if you add up all the n-th degree Bernstein polynomials, It has the partition of unity property. Through Figure 4, we can confirm the form for the set of 9th-order Bernstein polynomials.

[0073]

[0074] FIG. 5 is an exemplary drawing for explaining a velocity vector and a generated hodograph curve according to an embodiment of the present invention.

[0075] Referring to Fig. 5, (a) of Fig. 5 is a drawing for representing the velocity vector measured at the node point on the curve. (b) of Fig. 5 is an example drawing for representing the hodograph curve generated by connecting the velocity vectors measured at the node point on the curve. For example, a general curve including a polynomial curve When given, The hodograph of the velocity vector corresponding to each point It refers to a curve created by connecting the end points of vectors after moving the point of action of the fields to the origin.

[0076]

[0077] FIG. 6 is an exemplary drawing for explaining the form of a 9th-order Gauss-Legendre polynomial using a polynomial curve design device according to one embodiment of the present invention.

[0078] Referring to Figure 6, for example, among the 10 9th order Gauss-Legendre polynomials, the first and last Except for It can be confirmed that the positions move while all of them have similar maximum values ​​of about 0.87. This shows a similar phenomenon in orders other than the 9th order of Fig. 6. It can be confirmed that the Gauss-Legendre polynomial of Fig. 6 has a much larger maximum value than the 9th order Bernstein polynomial, which has a maximum value of about 0.3, and most of the values ​​are concentrated in a narrow region. This difference from the Bernstein polynomial allows the hodograph control polygon using the Gauss-Legendre polynomial to have improved operability than the Bézier curve using the Bernstein polynomial because each control point strongly influences the narrow region of the corresponding polynomial curve.

[0079] FIG. 7 is an exemplary drawing for explaining a Gauss-Legendre control polygon (a) and a 16th-order polynomial curve (b) designed using a polynomial curve design device according to one embodiment of the present invention.

[0080] Referring to Fig. 7, as an example of polynomial curve design using Gauss-Legendre polynomials, the shape of the English word “love” written in cursive is expressed as a single polynomial curve. Fig. 7 (a) is a hodograph control polygon used to design the curve, which consists of 17 control points and 16 line segments connecting them. Therefore, the generated polynomial curve becomes a 16th-degree polynomial curve and can be expressed as a shape as in Fig. 7 (b). At this time, it can be confirmed that the polynomial curve in Fig. 7 (b) is a complex curve, but is composed of a single polynomial curve and not a spline connecting small curve pieces. Theoretically, all polynomial curves can be expressed as Bézier curves, but it is very inefficient to design complex shapes as high-order Bézier curves. This is because, as the degree of the curve increases, the maximum value of the corresponding Bernstein polynomials due to the unitary partition property tends to decrease, so each Bezier control point has little influence on the shape of the curve.

[0081] FIG. 8 is an exemplary drawing for explaining a Bezier control polygon and an enlarged view of the curve observed when the polynomial curve of FIG. 7(b) is expressed as a Bezier curve using a polynomial curve design device according to an embodiment of the present invention.

[0082] Referring to Fig. 8, Fig. 8 shows (a) a Bezier control polygon when the polynomial curve of 7(b) is expressed as a Bezier curve, (b) a drawing that is magnified 20 times the control polygon of (a), and (c) a curve observed by magnifying (b) again 20 times. Fig. 8 (a) shows the Bezier control polygon for the “love” curve, and you can see that the actual curve is located small in the center of the figure. Fig. 8 (b) is magnified 20 times, and Fig. 8 (c) is magnified 20 times again. Therefore, you can see that it is practically almost impossible to control the subtle shape of the curve by adjusting a polygon that is about 400 times larger. On the other hand, when using a Gauss-Legendre control polygon, the influence of each control point on the shape of the curve is much stronger in a significantly smaller area, so you can design a polynomial curve much more easily than when using a Bezier curve.

[0083]

[0084] FIG. 9 is an exemplary drawing for explaining the form of a 9th-order Gauss-Lobarto polynomial using a polynomial curve design device according to one embodiment of the present invention.

[0085] Referring to Figure 9, a curve can be defined from a control polygon using Gauss-Lobatto polynomials. The velocity vectors at the two endpoints of the Gauss-Lobatto curve are determined by the edges of the two endpoints of the control polygon. This property allows for the design of more complex curves by connecting multiple Gauss-Lobatto curves, by matching the velocity vectors of the two curves at the connection points. It can be used conveniently when creating curves.

[0086] FIG. 10 is an exemplary drawing for explaining a Gauss-Lobarto control polygon (a) for the curve of FIG. 7(b) using a polynomial curve design device according to an embodiment of the present invention and a polynomial curve (b) designed with the control polygon.

[0087] Referring to Fig. 10, Fig. 10(a) shows a Gauss-Lobato control polygon for the “love” curve, and Fig. 10(b) is a diagram showing the Gauss-Lobato control polygon and the curve. As illustrated in Fig. 10, it can be confirmed that the curves are tangent at the two end points of the curves of the Gauss-Lobato control polygon.

[0088]

[0089] FIG. 11 is an exemplary drawing for explaining a space curve designed using a Gauss-Legendre control polygon using a polynomial curve design device according to an embodiment of the present invention, a Gauss-Lobarto control polygon, and a Bezier control polygon.

[0090] Referring to Fig. 11, Fig. 11(a) is a drawing showing a space curve designed using a Gauss-Legendre control polygon for a three-dimensional space curve, Fig. 11(b) is a drawing showing a Gauss-Lobatto control polygon for the same curve, and Fig. 11(c) is an exemplary drawing showing a Bezier control polygon for the same curve. As in the case of a plane curve, it can be confirmed that designing the shape of the curve is inefficient because the size of the Bezier polygon is very large compared to the Gauss-Legendre and Gauss-Lobatto curves.

[0091] FIG. 12 is a drawing for explaining a polynomial curve design method according to one embodiment of the present invention.

[0092] Referring to FIG. 12, in step S1210, the polynomial curve design device determines nodes and weights. The polynomial curve design device can utilize n different nodes (n is a natural number) given within the definition domain of the curve, and determine weights corresponding to each node. In addition, the polynomial curve design device can determine nodes and weights based on a selected numerical quadrature. The polynomial curve design device can calculate weights corresponding to the values ​​obtained by integrating Lagrange polynomials for the determined n different nodes (n is a natural number), so that the hodograph control polygon has endpoint interpolation properties. In addition, the polynomial curve design device can use Gauss-Legendre nodes, Gauss-Lobato nodes, and Newton-Coates nodes, etc., when selecting nodes.

[0093] In step S1230, the polynomial curve design device can produce a hodograph control polynomial using the nodes and weights determined in step S1210.

[0094] In step S1250, the polynomial curve design device generates a control polygon using points input by the user.

[0095] In step S1270, the polynomial curve design device generates a polynomial curve using each control point of the generated control polygon and a hodograph control polynomial. The polynomial curve design device can generate a polynomial curve by combining n+1 control points at their centers of gravity using the hodograph control polynomial calculated from the step of calculating the hodograph control polynomial using the nodes and weights determined in step S1230.

[0096]

[0097] Although all components constituting the embodiments of the present invention have been described as being combined or operated in combination as one, the present invention is not necessarily limited to such embodiments. That is, within the scope of the present invention, all of the components may be selectively combined and operated one or more times.

[0098] Although operations are depicted in the drawings in a particular order, it should not be understood that the operations must be performed in the particular order depicted, or in any sequential order, or that all depicted operations must be performed to achieve the desired results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of the various components in the embodiments described above should not be understood as requiring such separation, and it should be understood that the described program components and systems can generally be integrated together into a single software product or packaged into multiple software products.

[0099] The present invention has been described above, focusing on specific embodiments thereof. Those skilled in the art will appreciate that the present invention can be implemented in modified forms without departing from its essential characteristics. Therefore, the disclosed embodiments should be considered illustrative rather than limiting. The scope of the present invention is set forth in the claims, not the foregoing description, and all differences within the scope equivalent thereto should be construed as being encompassed by the present invention.

[0100] The mode for carrying out the invention is described together with the best mode for carrying out the invention.

[0101] The present invention relates to a method for designing a polynomial curve using a hodograph control polygon. The method generates a hodograph control polynomial using nodes and weights of a numerical quadrature method that approximately calculates the integral of a function defined in a parameter section of a curve, and combines control points with the polynomial to design a polynomial curve by centering the control points, and thus has industrial applicability because it can be used in various ways.

Claims

1. In the polynomial curve design device, Node decision unit that determines the node; Weight determination unit that determines the weight; A polynomial calculation unit that calculates a hodograph control polynomial using the above-determined nodes and weights; A control polygon generation unit that generates a control polygon using the input points; and A polynomial curve generation unit that generates a polynomial curve using each control point of the control polygon generated above and a hodograph control polynomial. A polynomial curve design device including:

2. In paragraph 1, The above node decision part Using n different nodes (n is a natural number) given within the domain of the curve. Polynomial curve design device.

3. In paragraph 1, The above weight determination unit Determine n (n is a natural number) weights corresponding to n different nodes (n is a natural number) within the domain of the above curve. Polynomial curve design device.

4. In paragraph 1, The above weight determination unit The mathematical expression is the value obtained by integrating the Lagrange polynomial at n different nodes determined in the above node determination section. Determine the weights from and set the end points of the hodograph control polygon to coincide with the end points of the curve. Here is the weight, represents the Lagrange polynomial Polynomial curve design device.

5. In paragraph 1, The above node determination unit and the above weight determination unit Using the nodes and weights of the Gauss-Legendre quadrature method Polynomial curve design device.

6. In paragraph 1, The above node determination unit and the above weight determination unit Using the nodes and weights of the Gauss-Lobarto quadrature method Polynomial curve design device.

7. In paragraph 1, The above polynomial curve generation section Using the hodograph control polynomial calculated in the above polynomial calculation section, a polynomial curve is generated by combining n+1 (n is a natural number) control points with their centers of gravity. Polynomial curve design device.

8. In the method of designing a polynomial curve by a polynomial curve design device, Step of determining the node; Step of determining weights; A step of calculating a hodograph control polynomial using the above-determined nodes and weights; A step of creating a control polygon using the input points; and A step of generating a polynomial curve using each control point of the control polygon generated above and a hodograph control polynomial. A polynomial curve design method including .

9. In paragraph 8, The step of determining the above node is Using n different nodes (n is a natural number) given within the domain of the curve. Polynomial curve design method.

10. In paragraph 8, The step of determining the above weights is Determine n (n is a natural number) weights corresponding to n different nodes (n is a natural number) within the domain of the above curve. Polynomial curve design method.

11. In paragraph 8, The step of determining the above weights is The mathematical expression is the value obtained by integrating the Lagrange polynomial at the n different nodes determined in the step of determining the above nodes. Determine the weights from and set the end points of the hodograph control polygon to coincide with the end points of the curve. Here is the weight, represents the Lagrange polynomial. Polynomial curve design method.

12. In paragraph 8, The step of determining the above node and the step of determining the weight are Using the nodes and weights of the Gauss-Legendre quadrature method Polynomial curve design method.

13. In paragraph 8, The step of determining the above node and the step of determining the weight are Using the nodes and weights of the Gauss-Lobarto quadrature method Polynomial curve design method.

14. In paragraph 8, The step of generating a polynomial curve using each control point of the control polygon generated above and the hodograph control polynomial is In the step of calculating the hodograph control polynomial using the nodes and weights determined above, a polynomial curve is generated by combining n+1 (n is a natural number) control points with the center of gravity using the calculated hodograph control polynomial. Polynomial curve design method.

15. A computer program for executing any one of the polynomial curve design methods of Articles 8 to 14 and recorded on a computer-readable recording medium.

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