Horse belly edge contour calculation method based on Bezier curve

Abstract

The invention relates to the technical field of stone cutting, in particular to a Bezier curve-based horse belly edge contour calculation method, which comprises the following steps of: (1) presetting a coordinate system, and enabling a constructed graph to take the origin of the coordinate system as a center, the chords of upper and lower arcs to be vertical to a y axis, and the chords of left and right arcs to be vertical to an x axis; (2) acquiring length, width, bow height and transition area range parameters input by a user; (3) calculating end point coordinates of long and wide side arcs according to the input parameters, constructing a circle through three points to obtain a circle center, and mirroring to obtain four large arcs; (4) calculating Bezier curves of four transition sections, and determining control points of the Bezier curves under the condition that the tangential directions of two end points are consistent with the tangential directions of adjacent arcs; and (5) converting the Bezier curve into arcs by adopting a biarc fitting method, obtaining all transition section tracks after mirroring, and finally outputting 12 sections of arcs. The method solves the problems of insufficient feasible solutions and low smoothness in the prior art.

Description

Technical Field

[0001] The present invention relates to the technical field of stone cutting, and in particular to a method for calculating a horse belly edge contour based on a Bezier curve. Background Art

[0002] In stone cutting, arc fitting technology has become crucial for tabletop contour design. Leveraging cutting-edge stone cutting machinery and this arc fitting strategy, we can achieve precise cutting of the horse-belly-shaped tabletop contour, ensuring smooth and precise cuts on every facet. Users simply input basic dimensional parameters to quickly complete the initial design process. Visual verification then instantly verifies the fit between the target shape and the design. From there, further fine-tuning of parameters allows for precise control of the detailed geometry. The use of arc fitting technology in this process not only ensures a fast design process but also ensures precise adjustments to every detail.

[0003] The currently widely used eight-segment arc fitting algorithm has very obvious flaws. It is obvious that when solving the problem of smoothly connecting any two arcs using a third arc, it is extremely common to fail to find the target arc. In practical applications, in order to find a relatively feasible target arc, the algorithm has to face a dilemma: either modify the input internally or reduce the curvature limit of the transition between the two arcs. The first option will cause the user to notice a significant gap between the input and output, and the user will not be able to obtain the desired size specifications; the second option will result in a less smooth final product. The market usually adopts a combination of the two methods to achieve a balance, supplemented by user interaction prompts, but there are still deficiencies in accuracy and user experience.

[0004] Therefore, a new technical solution is urgently needed to solve the above technical problems. Summary of the Invention

[0005] The purpose of the present invention is to overcome the problems of the above-mentioned prior art and provide a method for calculating the horse belly edge contour based on Bezier curves. By improving the transition section fitting method, the frequency of the non-existence of feasible solutions is reduced, the length and width arc size requirements are guaranteed, and a smoother transition and more convenient parameter control are achieved.

[0006] The above objectives are achieved through the following technical solutions: An animation simulation method for sampling different motion properties of an object, comprising: Step (1) Preset the coordinate system so that the constructed figure is centered on the origin of the coordinate system, the chords of the upper and lower arcs are perpendicular to the y-axis, and the chords of the left and right arcs are perpendicular to the x-axis, ensuring that the starting point and end point of the arc are symmetrical about the origin; Step (2) obtaining the length, width, bow height and transition zone range parameters input by the user; Step (3) Calculate the endpoint coordinates of the length and width arcs according to the input parameters, construct a circle by three points to get the center, and mirror to obtain four large arcs; Step (4) Calculate the Bezier curves of the four transition segments, and determine the control points of the Bezier curves under the condition that the tangent directions at the two end points are consistent with the tangent directions of the adjacent arcs; Step (5) uses the double arc fitting method to convert the Bezier curve into an arc, and then obtains all transition segment trajectories after mirroring, and finally outputs 12 arc segments.

[0007] As an optimization of this method, the endpoint coordinates of the length and width arcs are calculated according to the input parameters in step (3), specifically: Set the length and width entered by the user in step (2) as w and h, the upper and lower parts of the bow height as aw, the left and right parts of the bow height as ah, the transition zone range parameter as c, and set the left long side transition endpoint as B and the upper left transition starting point as F. The coordinates of B are as follows: Bx=-(w / 2-ah) By=h / 2-aw-c The coordinates of F are as follows: Fx=-(w / 2-ah-c) Fy=h / 2-aw.

[0008] And through the symmetry of B and F, we can obtain the transition endpoint C of the long side on the right and the starting point E of the upper right transition section.

[0009] As an optimization of this method, a circle is constructed using three points in B, C, F, and E. Using the theorem that a line perpendicular to a point on a circle must pass through the center of the circle, the intersection of the two perpendicular lines is found as the center of the arc.

[0010] As an optimization of this method, step (4) is specifically as follows: taking B and F as endpoints, calculating the slope of the tangent of the arc at the endpoints, obtaining the tangent expression through the point-slope equation, and taking the intersection point M of the two tangents as the control point of the Bezier curve.

[0011] As an optimization of this method, by setting the parameters rw and rh (value range 0 to 1), the percentage of control points B' and F' on the line segments BM and FM can be controlled to achieve fine adjustment of the smoothness of the transition segment.

[0012] As an optimization of this method, when fitting the double arc to the Bezier curve in step (5), the center of the inscribed circle of the triangle is selected as the middle point, and the center of the arc is determined by finding the intersection of the perpendicular line of the chord midpoint and the perpendicular line of the endpoint tangent.

[0013] As an optimization of this method, step (5) is specifically as follows: for a Bezier curve with endpoints A and D and control points B and C, find the midpoint E of the chord AD, calculate the slope of the perpendicular line r based on the slope of the chord, and solve the point-slope equation of the diameter at the endpoint A to obtain the center G.

[0014] As an optimization of this method, the Bezier curves of the four transition segments in step (4) and the double arc fitting trajectory in step (5) are mirrored with the x and y axes as the symmetry axes and obtained from the calculation results of a single transition segment.

[0015] The present invention provides a method for calculating the horse belly edge contour based on a Bezier curve. This method uses Bezier curves to fit transition segments, reducing the frequency of non-existent feasible solutions and ensuring design feasibility. This ensures that feasible solutions fully meet the required length and width arc dimensions, improving the precision of the finished product. This method is simple to implement. By improving the transition segment fitting method, it reduces the frequency of non-existent feasible solutions and ensures the required length and width arc dimensions, achieving a smoother transition and more convenient parameter control. It does not involve complex length and width arc parameter adjustments, making it easy for users to understand and use. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 This is a flow chart of an animation simulation method for sampling different motion attributes of an object according to the present invention; Figure 2 A schematic diagram of the coordinate system and parameter input and length and width arc calculation in an animation simulation method for sampling different motion properties of an object according to the present invention; Figure 3 A schematic diagram of constructing a Bezier curve transition segment in an animation simulation method for sampling different motion properties of an object according to the present invention; Figure 4 This is a schematic diagram of double arc fitting in an animation simulation method for sampling different motion attributes of an object according to the present invention; Figure 5 This is a sample diagram of the animation simulation method for sampling different motion properties of an object described in the present invention. DETAILED DESCRIPTION

[0017] The present invention will be further described in detail below with reference to the accompanying drawings and examples. The described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort shall fall within the scope of protection of the present invention.

[0018] like Figure 1 As shown, this solution provides an animation simulation method for sampling different motion properties of an object, including: Step (1) Preset the coordinate system so that the constructed figure is centered on the origin of the coordinate system, the chords of the upper and lower arcs are perpendicular to the y-axis, and the chords of the left and right arcs are perpendicular to the x-axis, ensuring that the starting point and end point of the arc are symmetrical about the origin; Step (2) obtaining the length, width, bow height and transition zone range parameters input by the user; Step (3) Calculate the endpoint coordinates of the length and width arcs according to the input parameters, construct a circle by three points to get the center, and mirror to obtain four large arcs; Step (4) Calculate the Bezier curves of the four transition segments, and determine the control points of the Bezier curves under the condition that the tangent directions at the two end points are consistent with the tangent directions of the adjacent arcs; Step (5) uses the double arc fitting method to convert the Bezier curve into an arc, and then obtains all transition segment trajectories after mirroring, and finally outputs 12 arc segments.

[0019] This method improves the conventional 8-segment arc fitting into 4-segment arcs as the length and width, supplemented by Bezier curves as transition segments, and finally converts the Bezier curves into 12-segment arcs as the algorithm output through double arc fitting.

[0020] Similar to the conventional solution, this solution also uses length, width, and bow height to determine the arc shape of the length and width sides. After determining the length and width sides, the conventional solution requires drawing straight lines perpendicular to the arcs at the endpoints of the two adjacent length and width arcs. According to the theorem that any straight line perpendicular to a point on a circle must pass through the center of the circle, these two straight lines must pass through the center of the transition arc. Therefore, the intersection of the two straight lines should be the center of the transition arc. However, conventional solutions often encounter the difficulties mentioned above. After determining the center of the circle, it is found that the two radius segments are not the same length, making it impossible to calculate the transition arc.

[0021] In contrast, this solution uses Bezier curves as transition segments. Since commonly used Bezier curves have four control points—two endpoints that control the position of the curve's ends and two intermediate points that control its direction—Bezier curves make it easier to calculate the actual shape of the transition segment after determining its start and end positions and direction, thus avoiding situations where a feasible solution does not exist. After this step, the algorithm uses a simple double-arc fitting Bezier method to convert the four Bezier curve segments into eight circular arcs with minimal error, forming the final output.

[0022] Specifically, in the algorithm, we need to presuppose a coordinate system. We assume that the constructed shape will be centered at the origin of the coordinate system, with the chords of the upper and lower arcs perpendicular to the y-axis, and the chords of the left and right arcs perpendicular to the x-axis. This assumption ensures that the starting and ending points of the arcs are symmetrical about the origin of the coordinate system.

[0023] The algorithm first obtains the length and width (denoted as w and h), bow height (denoted as aw for up and down, ah for left and right), and transition zone range parameter (denoted as c) input by the user.

[0024] like Figure 2 As shown in the figure, the length and width of the rectangle formed by the two sets of tangents are the input w and h. On this basis, the coordinates of the two points BF in the figure can be calculated according to the input transition section interval length and arch height: Bx=-(w / 2-ah) By=h / 2-aw-c Fx=-(w / 2-ah-c) Fy=h / 2-aw By symmetrically drawing points B and F, we can obtain the other endpoints of the arc, which are points C and E in the figure. After constructing a circle from these three points, we can obtain the centers of the two arcs respectively. By mirroring, we can obtain the four large arcs of the entire horse belly figure.

[0025] Next, we'll calculate the Bezier curves for the four transition segments, which is the core difference between this solution and the conventional approach. Taking points B and F as an example, we'll construct a Bezier curve with its endpoints at B and F, and the tangents at the endpoints aligning with the tangents of the two arcs at those points. To do this, we'll calculate two additional points, B' and F', which serve as the control points for the cubic Bezier curve.

[0026] The method is: like Figure 3 As shown, the slope of the tangent at point B is first calculated using the arc parameters. The tangent expression f is then calculated using the point-slope equation. Since ensuring that the tangent at point B is consistent with the arc at point B only requires that point B' satisfy the tangent expression f, the choice of B' is somewhat flexible. In this solution, the tangents at two points are calculated and deintersected, with the intersection point M serving as the location of the two control points (control points can be repetitive for a Bezier curve).

[0027] like Figure 4 As shown, next, we only need to mirror with the x and y axes as the axes of symmetry to obtain the Bezier curves of the four transition zones. It should be noted that since the selection of control points has a certain degree of freedom, in the advanced version of this solution we can also provide more refined parameter settings to finely control the smoothness of the transition segment. Set the parameters rw and rh, with a value range of 0 to 1, to control the percentage of the length of the control points BB' and FF' on the line segments BM and FM respectively. For example, if rw is set to 0.5, point B' is selected at the midpoint of BM. In this way, customers can control the transition segment more finely.

[0028] After the Bezier curve of the transition section is calculated, due to the limitations of the processing technology, it is necessary to use arc fitting of the Bezier curve as the actual processing trajectory. Also due to the nature of the mirror image, only one transition section needs to be calculated, and then the trajectory of all transition sections can be obtained by mirroring. The method used here is called double arc fitting of the Bezier curve. Figure 4 The Bezier curve (black) with endpoints A and D and control points B and C, according to research, we can choose point F as the middle point and generate arcs at both ends of AF and FD to replace the original Bezier curve. The geometric meaning of point F is the center of the inscribed circle of triangle AED. Figure 2 In the example, it is actually the intersection point E of the two tangent lines. Back to Figure 3 Next, we need to find the two centers, G and H. The methods for both are the same, so we'll use G as an example. First, we can find the midpoint E of the arc's chord. Then, based on the slope of the chord and the perpendicular relationship, we can find the slope of line r, and thus the point-slope equation. Similarly, we can find the perpendicular relationship between the endpoint tangent and the diameter to find the point-slope equation for the diameter at point A. These combined solutions provide the center G. This completes the calculation process for all 12 arcs using the Madubian algorithm.

[0029] As an embodiment of this solution, the method includes: 1. Coordinate system and parameter input 1. After starting the algorithm, first establish a plane rectangular coordinate system, set the figure to the origin (0,0) as the center, the chords of the upper and lower arcs are parallel to the x-axis and perpendicular to the y-axis, and the chords of the left and right arcs are parallel to the y-axis and perpendicular to the x-axis, ensuring that the arcs are symmetrical about the origin.

[0030] 2. Receive core user-entered parameters: target belly length w and width h, upper and lower bow heights aw, left and right bow heights ah, and transition zone parameter c. If the user provides substrate dimensions, also enter substrate length and width (X, Y), long bow height h1, wide bow height h2, and transition zone length t. Calculate chord length using the formulas s1=X-2*t-2*h2 and s2=Y-2*t-2*h1.

[0031] 2. Calculation steps for length and width arc 1. Calculate the coordinates of point B based on the input w, h, ah, aw, and c: Bx=-(w / 2-ah), which means point B is in the negative direction of the x-axis and is (w / 2-ah) away from the origin.

[0032] By=h / 2-aw-c, which means point B is in the positive direction of the y-axis and is (h / 2-aw-c) away from the origin.

[0033] 2. Calculate the coordinates of point F: Fx=-(w / 2-ah-c), the x-axis coordinate is c more than point B, and moves in the negative direction.

[0034] Fy=h / 2-aw, the y-axis coordinate is c higher than point B and moves in the positive direction.

[0035] 3. The coordinates of point C are obtained by the symmetry of point B about the x-axis and y-axis: (w / 2-ah, h / 2-aw-c); the coordinates of point E are obtained by the symmetry of point F: (w / 2-ah-c, h / 2-aw).

[0036] 4. Select points B and C, and a third point on the corresponding arc (e.g., a point symmetrical to the origin). Using the principle that a line perpendicular to a point on a circle must pass through its center, draw perpendicular lines to the arc at points B and C. The intersection of these two perpendicular lines is the arc's center. Similarly, determine the centers of the other three arcs. Mirroring the image reveals the complete parameters of the four large arcs.

[0037] 3. Construction of Bezier Curve Transition Segment 1. Taking the arc at point B as an example, calculate the slope of the tangent line of the arc at point B: Based on the coordinates of the arc center and the coordinates of point B, find the slope of the connecting line. The slope of the tangent line is the negative reciprocal of the slope.

[0038] 2. Obtain the tangent expression f through the point-slope equation y-By=k(x-Bx) (k is the slope of the tangent). Similarly, obtain the tangent expression at point F.

[0039] 3. Combine the two tangent expressions to find the coordinates of the intersection point M. Use M as the common control point of the Bézier curve (the control point can be repeated) to construct a cubic Bézier curve with endpoints B and F and a tangent direction consistent with the adjacent arcs.

[0040] 4. The user can set the parameter rw (e.g. 0.5) to make point B' located at the midpoint of the BM segment. Similarly, the position of point F' can be adjusted by rh to achieve fine control of the smoothness of the transition segment.

[0041] 5. With the x-axis and y-axis as the axes of symmetry, mirror the Bezier curve of a single transition segment to obtain curves for the upper left, lower left, upper right, and lower right transition zones.

[0042] 4. Double arc fitting implementation 1. Using a Bezier curve with endpoints A and D and control points B and C as an example, determine the center F of the inscribed circle of triangle AED (that is, the intersection point E of the two tangents).

[0043] 2. Find the midpoint E of the chord AD, calculate the slope k1 of the chord AD, and the slope of its perpendicular line r is -1 / k1. Use the coordinates of point E and the slope to obtain the point-slope equation of the perpendicular line r.

[0044] 3. Calculate the slope k2 of the tangent at point A. The slope of its perpendicular line (i.e., the diameter direction) is -1 / k2, and the point-slope equation of the diameter at point A is obtained.

[0045] 4. Combine the equations of the perpendicular line r and the diameter of point A to obtain the coordinates of the center of the circle G. Similarly, obtain the coordinates of the center of the circle H and determine the parameters of the two arcs AF and FD.

[0046] 5. Through the mirroring operation, the double arc fitting result of a single transition segment is used to obtain 8 arcs of all four transition segments, plus the original 4 long and wide side arcs, and finally 12 arcs are output as the cutting trajectory.

[0047] Through the above steps, this method realizes the accurate calculation of the horse belly edge contour based on the Bezier curve, solves the problems of insufficient feasible solutions and low smoothness in the existing technology, and provides a better technical solution for stone cutting.

[0048] Parameter calculation rules for this solution: Given the base material's length and width, X and Y, these values ​​determine the maximum length and width of the finished flat. This solution requires two additional inputs: bow heights h1 and h2, representing the bow heights of the long and wide sides, respectively, and the length of the transition interval, t. It's easy to see that the long side chord length, s1, = X - 2 * t - 2 * h2, and the wide side chord length, s2, = Y - 2 * t - 2 * h1. This solution uses these values ​​as input for calculation.

[0049] The result sample is Figure 5 As shown in the figure, the dashed segments selected from left to right are the wide side arc, the long side arc, and the two arcs used to fit the Bezier curve of the transition segment. The four long and wide sides are symmetrical in pairs, and the four transition segments, upper left, lower left, upper right, and lower right, are symmetrical around the center.

[0050] The above description is only for explaining the embodiments of the present invention and is not intended to limit the present invention. For those skilled in the art, any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for calculating the contour of a horse's belly based on a Bezier curve, characterized in that: include: Step (1) Preset the coordinate system so that the constructed figure is centered on the origin of the coordinate system, the chords of the upper and lower arcs are perpendicular to the y-axis, and the chords of the left and right arcs are perpendicular to the x-axis, ensuring that the starting point and end point of the arc are symmetrical about the origin; Step (2) obtaining the length, width, bow height and transition zone range parameters input by the user; Step (3) Calculate the endpoint coordinates of the length and width arcs according to the input parameters, construct a circle by three points to get the center, and mirror to obtain four large arcs; Step (4) Calculate the Bezier curves of the four transition segments, and determine the control points of the Bezier curves under the condition that the tangent directions at the two end points are consistent with the tangent directions of the adjacent arcs; Step (5) uses the double arc fitting method to convert the Bezier curve into an arc, and then obtains all transition segment trajectories after mirroring, and finally outputs 12 arc segments.

2. The animation simulation method for sampling different motion attributes of an object according to claim 1, characterized in that: The endpoint coordinates of the length and width arcs are calculated based on the input parameters in step (3), specifically: Set the length and width entered by the user in step (2) as w and h, the upper and lower parts of the bow height as aw, the left and right parts of the bow height as ah, the transition zone range parameter as c, and set the left long side transition endpoint as B and the upper left transition starting point as F. The coordinates of B are as follows: Bx=-(w / 2-ah) By=h / 2-aw-c The coordinates of F are as follows: Fx=-(w / 2-ah-c) Fy=h / 2-aw, And through the symmetry of B and F, we can obtain the transition endpoint C of the long side on the right and the starting point E of the upper right transition section.

3. The animation simulation method for sampling different motion attributes of an object according to claim 2, characterized in that: Construct a circle using three points in B, C, F, and E. Using the theorem that a line perpendicular to a point on a circle must pass through the center of the circle, find the intersection of the two perpendicular lines as the center of the arc.

4. The animation simulation method for sampling different motion attributes of an object according to claim 3, characterized in that: The step (4) is specifically as follows: taking B and F as endpoints, calculating the slope of the tangent of the arc at the endpoints, obtaining the tangent expression through the point-slope equation, and taking the intersection point M of the two tangents as the control point of the Bezier curve.

5. The animation simulation method for sampling different motion attributes of an object according to claim 4, characterized in that: By setting the parameters rw and rh (value range 0 to 1), the percentage of control points B' and F' on the line segments BM and FM can be controlled to achieve fine adjustment of the smoothness of the transition segment.

6. The animation simulation method for sampling different motion attributes of an object according to claim 5, characterized in that: When fitting the double arc to the Bezier curve in step (5), the center of the inscribed circle of the triangle is selected as the middle point, and the center of the arc is determined by finding the intersection of the perpendicular line of the chord midpoint and the perpendicular line of the tangent line of the endpoint.

7. The animation simulation method for sampling different motion attributes of an object according to claim 5, characterized in that: The step (5) is specifically as follows: for a Bezier curve with endpoints A and D and control points B and C, find the midpoint E of the chord AD, calculate the slope of the perpendicular line r based on the slope of the chord, and solve the point-slope equation of the diameter at the endpoint A to obtain the center G.

8. The animation simulation method for sampling different motion attributes of an object according to claim 1, characterized in that: The Bezier curves of the four transition segments in step (4) and the double arc fitting trajectory in step (5) are obtained by performing a mirror operation with the x and y axes as the symmetry axes, and are obtained from the calculation results of a single transition segment.

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