A method for drawing a fitting curve on the surface of a three-dimensional object

Abstract

The present invention provides a method for drawing a fitting curve on the surface of a three-dimensional object. On the basis of the existing projection method, the method identifies and processes points that may fall into a local minimum through a feature acceleration mechanism. When the center point that falls into the local minimum is detected, rays are uniformly emitted in all directions to find the direction vector with the minimum Euclidean distance. In order to accelerate the convergence speed of the algorithm, a momentum optimization mechanism is also introduced. A weight-based interpolation method is also designed to improve the smoothness of the curve and accelerate the algorithm's secondary search for surface points. Through this method, various features on complex surfaces can be processed more accurately, thereby achieving curves with higher fit on these surfaces. The present invention solves the problem that projection points are prone to fall into local minima on the surface, and improves the smoothness of the lines, effectively solving the challenges faced in drawing fitting curves on the surface of three-dimensional objects, and provides a new solution for the technical development in related fields.

Description

Technical Field

[0001] The invention relates to the field of three-dimensional object surface drawing, and in particular to a method for drawing a fitting curve on the surface of a three-dimensional object. Background Art

[0002] When drawing lines that fit the surface of a complex three-dimensional object, we often encounter some problems, such as the lines cannot fit the surface tightly, the lines penetrate the model or fall into the local minimum. These problems make it urgent to find a method that can calculate and obtain line segments that fit the surface when dealing with three-dimensional objects.

[0003] The curvature and change degree of the surface of complex three-dimensional objects are very high, which requires us to adjust the line trend in real time according to the surface change of the object when drawing lines. However, traditional line drawing methods are difficult to meet this requirement, so it is easy for lines to penetrate the model or fall into local minimum values.

[0004] Existing projection methods project a curve onto a surface. This is usually done by calculating the closest points from the curve points to the surface, and then moving the curve points to these closest points. Projection methods may cause unnatural distortions in the projection of the curve on the surface, especially in areas where the surface curvature changes greatly. In addition, the calculation of finding the closest point from the curve point to the surface can be mathematically complex and computationally intensive, especially for complex three-dimensional surfaces. It is also possible that the local minimum of the projected point on the surface may occur instead of the global optimal solution, resulting in an unsatisfactory curve fit. Summary of the invention

[0005] 1. Technical problems to be solved:

[0006] In existing methods for drawing fitting curves on the surface of three-dimensional objects, the projection method may cause unnatural distortions in the projection of the curve on the surface. The calculation of finding the closest point from the curve point to the surface may be mathematically complex and computationally intensive, especially for complex three-dimensional surfaces. It is also possible that the local minimum of the projected point on the surface may appear instead of the global optimal solution, resulting in unsatisfactory curve fitting.

[0007] 2. Technical solution:

[0008] In order to solve the above problems, the present invention provides a method for drawing a fitting curve on the surface of a three-dimensional object, comprising the following steps:

[0009] Step S01: interpolate between two points. The interpolation method is based on the number of interpolation points required between the two points. Each point is inserted in sequence according to the number of interpolation points.

[0010] Step S02: Calculate the points obtained in step S01 and the points on the surface.

[0011] Step S03: Solve the local minimum problem of the projection point on the surface, rather than the global optimal solution problem.

[0012] The specific steps are: Step S031: find the point trapped in the local minimum; Step S032: obtain all the point pairs trapped in the local minimum from Step S031, find the center points of these point pairs, emit rays from the center points, and then retain the rays that intersect with the three-dimensional surface; Step S033: intersect the ray that meets the conditions obtained in Step S032 with the surface of the three-dimensional object, calculate the length of the line segment from the starting point of the ray to the intersection, and select the ray with the smallest distance as the direction vector; Step S034: introduce the momentum function to intervene in the automatically calculated direction. The specific method is to introduce the manually set direction to calculate the intersection with the surface again. Two line segments will be obtained between the intersection and the point pair. Repeat Step S01, and interpolate in the middle of the line segment to find the nearest point on the surface; Step S035: repeat the above steps until the loop is exited.

[0013] Step S04: Design an interpolation method based on the distance. The larger the distance, the more points are interpolated, and vice versa. After the interpolation is completed, all the points are fitted once, and the fitting point is a smooth curve.

[0014] Step S05: Get the final result.

[0015] The specific method of step S01 is: calibrate two points on the surface and , connect the two points to form a line segment, perform linear distance interpolation between the two points, the number of interpolations is p, and define a parameter t such that , for each interpolation point between two points, its coordinates are expressed as:

[0016] ,

[0017] Among them, t is the interpolation parameter from point A to point B. In order to obtain p interpolation points, t is evenly distributed in the interval [0, 1], specifically:

[0018]

[0019] So each interpolation point The coordinates of are calculated as:

[0020] ,

[0021] We will get p linear interpolation points from A to B, including the starting point A and the end point B. hour, ,when hour, , all the points obtained by the above interpolation are .

[0022] The specific method of step S02 is:

[0023] By step S01 points to find the point on the surface triangle closest to the line segment, assuming that Define a surface S and find a point on the surface S. , so that Q is the shortest distance to point P, the distance function is expressed as:

[0024] ,

[0025] The minimum distance problem is described as , find the smallest surface point for each point and get the surface point , after removing duplicates, the set of points is , use this point as the base point for the next calculation.

[0026] The step S031 is specifically as follows:

[0027] Get all the points obtained, determine the number of times each key point appears, set the threshold T, and the nearest points of each two adjacent points , When the following conditions are met:

[0028] ,

[0029] in Represents Point When the number of occurrences is greater than the set threshold T, the local minimum alarm is triggered and the local minimum elimination calculation module is entered.

[0030] The step S032 is specifically as follows:

[0031] All the point pairs that fall into the local minimum are obtained from step S031, and the center point is found between every two pairs of points. and Point is a pair of points, then the center is:

[0032] ,

[0033] by is the starting point of the ray, the ray direction vector d, and the ray is emitted in all directions to obtain the ray

[0034] ,

[0035] Will Substitute into the surface equation of the object and solve the equation to find If there is a reasonable , a positive real number and satisfies the object equation, then the ray intersects the surface of the object, retain the ray, and the ray that satisfies the conditions is recorded as .

[0036] The step S033 is specifically as follows:

[0037] For the rays that meet the conditions, calculate the distance between them and the intersection point of the surface, which is recorded as , select the point with the shortest distance as the corresponding ray The final direction is expressed as follows:

[0038] ,

[0039] The corresponding rays is with The corresponding rays.

[0040] The step S034 is specifically as follows:

[0041] In order to avoid falling into the local minimum again, the parameter momentum is introduced and momentum is added as an influencing factor to the calculation of the ray direction. The corresponding ray direction , the momentum M is considered as a vector, and the adjusted ray direction It is expressed as a combination of the original direction and momentum, with the momentum vector added to the original direction:

[0042] ,

[0043] in is a scaling factor that controls how much momentum affects direction.

[0044] The new ray is formed by Centered, adjusted The intersection point H with the surface equation of the object is calculated again, and new line segments AH and HB are formed. Interpolate the line segments and find the nearest point on the surface and add them to Corresponding position.

[0045] The step S035 is specifically as follows:

[0046] Repeat the above steps again until there are no points that meet the local minimum. Let the final set of points be , whose length is the number of points ,in Indicates the number of points added to solve the problem of being stuck in a local minimum Indicates length.

[0047] The specific method of step S04 is: after obtaining a basic point of the surface, a weight-based interpolation method needs to be performed between every two points. The specific interpolation method is as follows:

[0048] In three-dimensional space, suppose two points and , the Euclidean distance D between them is calculated by the following formula:

[0049] .

[0050] Set a base distance , which is the distance used for reference. Corresponding to this benchmark distance, a benchmark interpolation point number is set For the actual distance D between any two points, the actual number of interpolation points is determined according to the ratio to the reference distance. , the formula is as follows:

[0051] ,

[0052] Indicates rounding up to ensure that there is at least one interpolation point. If D is greater than 0, finally insert q interpolation points between point A and point B, and then use the n points obtained by curve fitting.

[0053] The method of obtaining n points by curve fitting is:

[0054] set up and are the endpoints of the segments, then a cubic spline is drawn at each segment The above is expressed as:

[0055] ,

[0056] in: is the spline function on the ith segment,

[0057] are the coefficients of the spline function on this segment.

[0058] Each of the interpolation points Calculated by the following formula:

[0059] ,

[0060] Putting the above points together, we should get n points, and the calculation formula of n is as follows:

[0061] .

[0062] 3. Beneficial effects:

[0063] On the basis of the projection method, the present invention 1. proposes a distance linear interpolation method to address the problem that the projection method has fewer sampling points, which leads to jumps in the surface fitting lines. By setting the number of points, the nearest point on the surface can be accurately obtained; 2. In view of the problem that the projection method may have a local minimum value of the projection point on the surface, rather than a global optimal solution, which leads to an unsatisfactory curve fitting, an innovative solution is proposed. By searching for the position between every two points that may fall into the local minimum value, the center point is used to evenly emit rays to the surroundings to find the direction vector with the smallest Euclidean distance. On this basis, momentum is introduced to help the optimization algorithm converge to the optimal solution faster. At the same time, a weight-based interpolation method is designed to improve the smoothness of the curve and accelerate the secondary search of the algorithm surface points. Through the above steps, complex surfaces can be accurately processed for complex surface features to draw fitting curves on them. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 Schematic diagram of falling into a local minimum in an embodiment.

[0065] Figure 2 It is a schematic diagram of the final result in the embodiment.

[0066] Figure 3 It is the comparison of curve smoothness before and after curve fitting.

[0067] Figure 4 It is a comparison between the final result of the improved algorithm and the effect of the original algorithm. DETAILED DESCRIPTION

[0068] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0069] A method for drawing a fitting curve on the surface of a three-dimensional object, characterized in that it comprises the following steps:

[0070] Step S01: interpolate between two points. The interpolation method is based on the number of interpolation points required between the two points. Each point is inserted in sequence according to the number of interpolation points.

[0071] Since the projection method usually projects two points on a three-dimensional surface (that is, finding the nearest point on the surface), this will result in too few points, and on some surfaces, it will easily cause the line between the two points to penetrate the model. Therefore, our method is to interpolate between two points. The interpolation method is based on the number of interpolation points required between the two points. We insert each point in order according to the number of interpolation points, so that the above-mentioned penetration phenomenon will be reduced when projecting.

[0072] In one embodiment, the specific method of step S01 is: calibrate two points on the surface and , connect the two points to form a line segment, perform linear distance interpolation between the two points, the number of interpolations is p, and define a parameter t such that , for each interpolation point between two points, its coordinates are expressed as:

[0073] ,

[0074] Among them, t is the interpolation parameter from point A to point B. In order to obtain p interpolation points, t is evenly distributed in the interval [0, 1], specifically:

[0075] ,

[0076] Therefore, each interpolation point The coordinates of can be calculated as:

[0077] ,

[0078] You will get p linear interpolation points from A to B, including the starting point A and the end point B. Note that hour, ,when hour, .

[0079] Through the above interpolation, let all the points obtained by interpolation be .

[0080] Step S02: Calculate the points obtained in step S01 and the points on the surface.

[0081] The purpose of this step is to obtain the coordinate value of the point on the surface when the distance between the new marking point and the point on the surface is the smallest.

[0082] In one embodiment, the specific method of step S02 is:

[0083] By step S01 points to find the point on the surface triangle closest to the line segment, assuming for Define a surface S, then find a point on the surface S. , so that Q is the shortest distance to point P. The distance function can be expressed as

[0084] ,

[0085] The minimum distance problem is then described as , find the smallest surface point for each point and get the surface point , after removing duplicates, the set of points is , use this point as the base point for the next calculation.

[0086] Step S03: Solve the problem of finding the local minimum value of the projection point on the surface rather than the global optimal solution.

[0087] Usually, the curve falls into a local minimum because the span between two points is too large, such as Figure 1 As shown, the figure below is improved according to the present invention.

[0088] The purpose of this step is to solve the problem of poor curve fitting due to the local minimum of the projection point on the surface rather than the global optimal solution.

[0089] The specific steps are as follows: Step S031: finding which points fall into the local minimum;

[0090] This step is to find out which pairs of points fall into the local minimum. First, we need to find out which points fall into the local minimum, and then we can solve these pairs of points. In one embodiment, specifically:

[0091] Get all the points obtained, determine the number of times each key point appears, set the threshold T, and the nearest points of each two adjacent points , When the following conditions are met:

[0092] ,

[0093] in Represents Point When the number of occurrences is greater than the set threshold T, the local minimum alarm is triggered and the local minimum elimination calculation module is entered.

[0094] Step S032: Obtain all point pairs trapped in the local minimum from step S031, find the center points of these point pairs, emit rays from the center points, and then retain the rays that intersect with the three-dimensional surface; in one embodiment, specifically:

[0095] In the first stage, all the points that meet the conditions are found between each pair of points. and Point is a pair of points, then the center is:

[0096] .

[0097] by is the starting point of the ray, the ray direction vector d, and the ray is emitted in all directions to obtain the ray

[0098] ,

[0099] Will Substitute the surface equation into the object , solve the equation to find If there is a reasonable (positive real number and satisfies the object equation), then the ray intersects the surface of the object, and the ray is retained. The ray that satisfies the condition is recorded as .

[0100] Step S033: Obtain the intersection points of the rays satisfying the conditions in step S032 with the surface of the three-dimensional object, calculate the length of the line segment from the ray-obtaining point to the intersection point, and select the ray with the shortest distance as the direction vector;

[0101] In one embodiment, specifically:

[0102] For rays that meet the conditions , calculate the distance between it and the surface intersection point, recorded as . Select the point with the shortest distance as the corresponding ray The final direction is expressed as follows:

[0103] ,

[0104] The corresponding rays is with The corresponding rays.

[0105] In some cases, such as when the sharp corners are too jittery, the surface intersection point may fall into multiple searches for the point with the minimum value of the surface. Therefore, the momentum function is introduced to intervene in the automatically calculated direction. This parameter is set manually to affect the original direction vector to speed up the calculation. The specific method is to introduce the manually set direction to calculate the intersection point with the surface again. Two line segments will be obtained between the obtained intersection point and the point pair. Repeat the method in step S01 to interpolate between the line segments to find the nearest point on the surface.

[0106] Step S034: introducing a momentum function to intervene in the automatically calculated direction. The specific method is to introduce the manually set direction to recalculate the intersection with the surface. Two line segments will be obtained between the obtained intersection and the point pair. Repeat step S01 and interpolate between the line segments to find the nearest point on the surface.

[0107] In one embodiment, specifically: to avoid falling into the local minimum again, a parameter momentum (Momentum, M) is introduced, and momentum is added as an influencing factor to the calculation of the ray direction. The corresponding ray direction , the momentum M can be regarded as a vector, then the adjusted ray direction can be expressed as some combination of the original direction and momentum. A simple way to do this is to add the momentum vector to the original direction:

[0108] ,

[0109] in is a scaling factor that controls how much momentum affects direction.

[0110] The new ray is formed by Centered, adjusted is the ray direction, and the surface equation of the object is calculated again The intersection point H of the line segment AH and HB is formed. The line segment is interpolated and the nearest point on the surface is found and added to Corresponding position.

[0111] Assuming that after the calculations of step S01, step S02 and step S03, the system still falls into the local minimum, the above steps are repeated until the loop is exited.

[0112] Step S035: Repeat the above steps until the loop is exited.

[0113] In one embodiment, specifically: Repeat the above steps again until there is no point that meets the local minimum value, and set the final set of points to be , whose length is (number of points) ,in Represents the number of points added to solve the problem of falling into the local minimum. Represents the length. Avoid the effect of falling into local minima, such as Figure 1 shown.

[0114] Step S04: Design an interpolation method based on the distance. The larger the distance, the more points are interpolated, and vice versa. After the interpolation is completed, all the points are fitted once, and a smooth curve is formed at the fitting point.

[0115] Through the above steps S01 to S03, multiple data points have been obtained. However, the distance between two adjacent points is not constant. In order to achieve fair interpolation, an interpolation method based on distance size is designed. The core of this method is that the greater the distance between two points, the more data points are inserted; conversely, the smaller the distance, the fewer points are inserted. After the interpolation is completed, all points will be fitted to generate a smooth curve.

[0116] In one embodiment, on the acquired surface After the basic points are obtained, a weight-based interpolation method is required between every two points. The specific interpolation method is as follows.

[0117] In three-dimensional space, suppose two points and , the Euclidean distance D between them can be calculated by the following formula:

[0118] .

[0119] This step is to calculate the Euclidean distance between two points.

[0120] Set a base distance , which is the distance used for reference. Corresponding to this reference distance, set a reference interpolation point number For the actual distance D between any two points, the actual number of interpolation points is determined based on the ratio to the reference distance. The formula is as follows:

[0121] ,

[0122] here Indicates rounding up to ensure that there is at least one interpolation point (if D is greater than 0).

[0123] This step is to calculate the number of interpolations, which is determined according to the distance.

[0124] Finally, we need to insert q interpolation points between point A and point B. Each interpolation point It can be calculated by the following formula:

[0125] ,

[0126] Putting the above points together, we should get n points, and the calculation formula of n is as follows:

[0127] .

[0128] Where m is the newly added point to solve the problem of falling into the local minimum. The comparison effect before and after using the weighted interpolation method is as follows: Figure 2 As shown, the figure on the right is improved according to the present invention.

[0129] This step is to obtain the interpolation coordinate points.

[0130] Next, use the n points obtained by curve fitting.

[0131] This step uses the correlation method to fit the curve to all points in order to smooth the line.

[0132] In one embodiment, the fitting method includes spline function, least square method, etc. and are the endpoints of the segments, then a cubic spline is drawn at each segment It can be expressed as:

[0133] ,

[0134] in: is the spline function on the ith segment. are the coefficients of the spline function on this segment.

[0135] The key to cubic spline interpolation is to determine these coefficients This is usually done by solving a system of linear equations consisting of the above continuity conditions and endpoint conditions. In this way, cubic splines can generate a smooth curve that approximates the given data points. Figure 3 As shown, the figure on the right is improved according to the present invention.

[0136] Through the above method, the final result is obtained, such as Figure 4 As shown, the figure on the right is improved according to the present invention.

Claims

1. A method for drawing a fitting curve on the surface of a three-dimensional object, characterized in that: The following steps are involved: Step S01: interpolate between two points. The interpolation method is based on the number of interpolation points required between the two points. Each point is interpolated in sequence according to the number of interpolation points. Step S02: Calculate the points obtained in step S01 and the points on the surface; Step S03: solving the local minimum problem of the projection point on the surface, rather than the global optimal solution problem; The specific steps are as follows: Step S031: find the point trapped in the local minimum; Step S032: obtain all the point pairs trapped in the local minimum from Step S031, find the center points of these point pairs, emit rays from the center points, and then retain the rays that have intersections with the three-dimensional surface; Step S033: calculate the length of the line segment from the starting point of the ray to the intersection point of the ray obtained in Step S032 that meets the conditions and the surface of the three-dimensional object, and select the ray where the line segment with the smallest distance is located as the direction vector; Step S034: introduce the momentum function to intervene in the automatically calculated direction. The specific method is to introduce the manually set direction to calculate the intersection point with the surface again. Two line segments will be obtained between the obtained intersection point and the point pair, repeat Step S01, and interpolate between the line segments to find the nearest point on the surface; Step S035: repeat the above steps until the loop is exited; Step S04: design an interpolation method based on the distance. The larger the distance, the more points are interpolated, and vice versa. After the interpolation is completed, all the points are fitted once, and the fitting point is a smooth curve. Step S05: Get the final result.

2. The method for drawing a fitting curve on the surface of a three-dimensional object according to claim 1, characterized in that: The specific method of step S01 is: calibrate two points on the surface and , connect the two points to form a line segment, perform linear distance interpolation between the two points, the number of interpolations is p, and define a parameter t such that , for each interpolation point between two points, its coordinates are expressed as: , where t is the interpolation parameter from point A to point B. In order to obtain p interpolation points, t is evenly distributed in the interval [0, 1], specifically: ， So each interpolation point The coordinates of are calculated as: ， We will get p linear interpolation points from A to B, including the starting point A and the end point B. hour, ,when hour, , all the points obtained by the above interpolation are .

3. The method for drawing a fitting curve on the surface of a three-dimensional object as claimed in claim 2, characterized in that: The specific method of step S02 is: By step S01 points to find the point on the surface triangle closest to the line segment, assuming Define a surface S as the center point and find a point on the surface S , so that Q is the shortest distance to point P, the distance function is expressed as: ， The minimum distance problem is described as , find the smallest surface point for each point and get the surface point , after removing duplicates, the set of points is , use this point as the base point for the next calculation.

4. The method for drawing a fitting curve on the surface of a three-dimensional object as claimed in claim 3, characterized in that: The step S031 is specifically as follows: Get all the points obtained, determine the number of times each key point appears, set the threshold T, and the nearest points of each two adjacent points , When the following conditions are met: ， in Yes Point When the number of occurrences is greater than the set threshold T, the local minimum alarm is triggered and the local minimum elimination calculation module is entered; The step S032 is specifically as follows: All the point pairs that fall into the local minimum are obtained from step S031, and the center point is found between every two pairs of points. and Point is a pair of points, then the center is: ， by is the starting point of the ray, the ray direction vector d, and the ray is emitted in all directions to obtain the ray ， Will Substitute into the surface equation of the object and solve the equation to find If there is a reasonable , a positive real number and satisfies the object equation, then the ray intersects the surface of the object, and the ray is retained. The ray that satisfies the conditions is recorded as ; The step S033 is specifically as follows: For rays that meet the conditions , calculate the distance between it and the surface intersection point, recorded as , select the point with the shortest distance as the corresponding ray The final direction is expressed as follows: ， The corresponding rays is with The corresponding rays; The step S034 is specifically as follows: In order to avoid falling into the local minimum again, the parameter momentum is introduced and momentum is added as an influencing factor to the calculation of the ray direction. The corresponding ray direction , the momentum M is considered as a vector, and the adjusted ray direction It is expressed as a combination of the original direction and momentum, with the momentum vector added to the original direction: ， in is a scaling factor that controls how much momentum affects direction, The new ray is formed by Center, adjust The latter is the ray direction, and the surface equation of the object is calculated again The intersection point H of the line segment AH and HB is formed. The line segment is interpolated and the nearest point on the surface is found and added to corresponding position; The step S035 is specifically as follows: Repeat the above steps again until there is no point that meets the local minimum. Let the final set of points be , whose length is the number of points ,in Represents the number of points added to solve the problem of falling into the local minimum. Indicates length.

5. The method for drawing a fitting curve on the surface of a three-dimensional object as claimed in claim 4, characterized in that: The specific method of step S04 is: after obtaining a basic point of the surface, a weight-based interpolation method needs to be performed between every two points. The specific interpolation method is as follows: In three-dimensional space, suppose two points and , the Euclidean distance D between them is calculated by the following formula: ， Set a base distance , which is the distance used for reference. Corresponding to this benchmark distance, a benchmark interpolation point number is set For the actual distance D between any two points, the actual number of interpolation points is determined according to the ratio to the reference distance. , the formula is as follows: ， Indicates rounding up to ensure that there is at least one interpolation point if D is greater than 0; Finally, insert q interpolation points between point A and point B. Next, use the n points obtained by curve fitting.

6. The method for drawing a fitting curve on the surface of a three-dimensional object as claimed in claim 5, characterized in that: The method of obtaining n points by curve fitting is: set up and are the endpoints of the segments, then a cubic spline is drawn at each segment The above is expressed as: ， in: is the spline function on the ith segment, are the coefficients of the spline function on this segment.

7. The method for drawing a fitting curve on the surface of a three-dimensional object as claimed in claim 5, characterized in that: Each of the interpolation points Calculated by the following formula: ， Putting the above points together, we should get n points, and the calculation formula of n is as follows: 。

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