SDL parametric modeling method and system for arc
Through the SDL parameterization modeling method, the problems of low parameterization efficiency and poor uniformity of arc quadratic curves are solved, faster calculation speed and better uniformity are achieved, and the real-time requirements of CAD design are met.
Patent Information
- Application Number
- CN202510249807.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-03-04
- Publication Date
- 2025-06-20
AI Technical Summary
In the prior art, the parameterization efficiency of arc quadratic curves is low and the uniformity is poor, which cannot meet the real-time requirements of CAD design.
The SDL parameterization modeling method is adopted to derive multiple parametric equations through different parametric methods, and the SDL parameterization equation is constructed and normalized. The parameterized expression composed of four segmented functions is used to calculate the parameters of the previous point of the arc by only two addition/subtraction operations.
SDL parameterization is faster than arc length parameterization and chord length parameterization, has global continuity, can meet the application needs of CAD, and has better performance in weighing calculation speed and uniformity, which can improve modeling efficiency and machining accuracy.
Smart Images
Figure CN120180714A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of curved surface arc processing, and particularly relates to a method and system for SDL parametric modeling of an arc. Background Art
[0002] Curve and surface parameterization plays an important role in industries such as CAD, especially the parameterization of conic sections (quadratic curves) and circles, which is widely used in fields such as architecture, automotive, shipbuilding, and PCB design. The mutual conversion between points and parameters (i.e., curve evaluation and point inversion) is a basic and high-frequency task of the CAD kernel, and its effectiveness greatly affects subsequent geometric algorithms. The arc length parameterization of a curve is the optimal parameterization in the sense of uniformity, but most curves do not have polynomial or rational polynomial arc length parameterization. In fact, the only quadratic curve that can be accurately arc length parameterized is a circular arc, but it involves time-consuming trigonometric operations in curve evaluation and point inversion operations, which cannot meet the real-time requirements of CAD design. Chord length parameterization is often used to replace arc length parameterization. Its parameters have geometric meanings, but the parameterization effect is poor. The parameters of rational Bezier have no specific geometric meanings, the parameterization quality is poor, and it is difficult to invert the parameters. In fact, it is very difficult to balance both computational efficiency and uniformity in curve evaluation and point inversion operations for these parameterization methods.
[0003] There is a common problem in PCB design: given a circular arc and a point on the circular arc , calculate whether it is on the circular arc . Transforming it into a mathematical problem is: calculate the parameters of three points, and judge whether it is located within the interval, as shown in Figure 2 . Figure 2 For the query of points on a circular arc, point is between points and on the circular arc.
[0004] Currently, the strategies of Parasolid, Acis, and OCC geometric engines are to calculate the inverse trigonometric function (arc length parameterization) of the angle between and the positive direction of the X-axis, that is, to calculate Currently, the calculation strategy of inverse trigonometric functions adopted by the Windows standard library is to convert trigonometric functions into polynomials through Taylor expansion for approximate calculation (retaining some terms). The hardware parameters of the machine used in the experiment are 64GB of memory and an Intel i7-14700KF CPU. The Microsoft-provided math library Windows SDK, version 10.0.22621.0, is used. The results are as follows: Executing 100,000 times of sin and cos operations takes 3.7 times the time of the same-order double operations; while executing 100,000 times of arcsin and arccos operations takes 15 times and 25 times the time of the same-order double operations. This shows that the calculation of inverse trigonometric functions is very time-consuming.
[0005] Currently, the most commonly used parameterization methods in engineering are arc length parameterization, chord length parameterization, and rational Bezier parameterization. Among them, arc length parameterization is the optimal parameterization in the sense of uniformity. In fact, Farouki has proved that the conic sections that can accurately achieve arc length parameterization can only be circular arcs, and the ones that can achieve arc length parameterization of polynomials or rational polynomials can only be straight lines. Chinese scholar Shi Fazhong further proved that high-degree parametric polynomial curves cannot take their own arc length as a parameter. In the sense of uniformity, arc length parameterization is the most ideal parameterization. The uniformity of chord length parameterization is poor, and it does not necessarily satisfy continuity at the segmentation points. Rational Bezier parameterization is a commonly used parameterization method in engineering. It has a polynomial expression and satisfies continuity, but there are also some disadvantages: the parameters have no specific geometric meaning, the parameterization quality is poor, it is difficult to inverse the parameters, and it is not conducive to specific modeling and machining. Summary of the Invention
[0006] The purpose of the present invention is to provide a method and system for SDL parameterization modeling of circular arcs to overcome the problems of low efficiency and poor uniformity in the parameterization of circular arc conic sections in the prior art.
[0007] To achieve the above purpose, the technical solution adopted by the present invention is as follows: A method for SDL parameterization modeling of circular arcs, comprising the following steps: S1, according to the set unit circular arc degree, use different parameterization methods to derive a variety of parameterization equations; S2, construct an SDL parameterization equation according to the unit circular arc degree, perform normalization processing on the constructed SDL parameterization equation to obtain a normalization result, use different parameterization methods and the derived variety of parameterization equations to perform parameter normalization on the unit circle to obtain a parameter equation, and use the obtained parameter equation to obtain circular arc construction parameters for circular arc construction.
[0008] Preferably, according to the set unit circular arc degree, a variety of parametric equations are derived by using different parameterization methods, and the different parameterization methods include arc length parameterization and chord length parameterization.
[0009] Preferably, given a unit circle with the center at the origin, taking the point as the point with a radian of 0, and the counterclockwise direction as the direction of increasing radian. For the points on the circle as the radian increases from 0 to , the length is introduced to represent the increase with the radian .
[0010] Preferably, the arc length parameterization expression can be expressed as: (1).
[0011] Preferably, the chord length parameterization expression can be expressed as: (2).
[0012] Preferably, based on the chord length parameterization, chord length square parameterization, X-parameterization, and Y-parameterization are derived: The X-parameterization expression can be expressed as: (3).
[0013] Preferably, the chord length square parameterization expression can be expressed as: (4).
[0014] Preferably, the Y-parameterization expression can be expressed as: (5).
[0015] Preferably, the parametric equations after parameter normalization of the above different parameterization methods for the unit circle are established : .
[0016] An SDL parametric modeling method for a circular arc includes a parameterization module and a modeling module: The parameterization module, according to the set unit circular arc degree, uses different parameterization methods to derive a variety of parametric equations; A modeling module constructs an SDL parametric equation based on a unit circular arc degree, normalizes the constructed SDL parametric equation to obtain a normalization result, uses different parametric methods and various derived parametric equations to perform parametric normalization on a unit circle to obtain a parametric equation, and uses the obtained parametric equation to obtain arc construction parameters for arc construction.
[0017] Compared with the prior art, the present invention has the following beneficial technical effects: The present invention provides an SDL parametric modeling method for an arc. According to a set unit circular arc degree, different parametric methods are used to derive various parametric equations. An SDL parametric equation is constructed based on the unit circular arc degree, the constructed SDL parametric equation is normalized to obtain a normalization result, different parametric methods and various derived parametric equations are used to perform parametric normalization on a unit circle to obtain a parametric equation, and the obtained parametric equation is used to obtain arc construction parameters for arc construction. It is composed of four piecewise functions and only involves two addition / subtraction operations to calculate the parameters of a point on the arc. SDL parameterization is faster than arc length parameterization and chord length parameterization. SDL parameterization is global continuous and can meet the applications of CAD. At the same time, SDL parameterization is closer to arc length parameterization than chord length parameterization. SDL parameterization can be reparameterized into a quadratic rational Bézier representation through a suitable quadratic transformation, and two addition and subtraction operations are used for point inverse calculation. It has good performance in weighing calculation speed and uniformity, and can greatly improve the modeling efficiency and machining accuracy. BRIEF DESCRIPTION OF THE DRAWINGS
[0018] Figure 1 It is a schematic flow diagram of an SDL parametric modeling method for an arc in an embodiment of the present invention.
[0019] Figure 2 It is a schematic diagram of the arc mathematical conversion structure in an embodiment of the present invention.
[0020] Figure 3 It is a schematic diagram of the reparameterization structure of a unit circle in a rectangular coordinate system in an embodiment of the present invention.
[0021] Figure 4 It is a schematic diagram of the time comparison between SDL parameterization and other parameterizations in point inversion in an embodiment of the present invention.
[0022] Figure 5 It is a schematic diagram of the time comparison between SDL parameterization and other parameterizations in curve evaluation in an embodiment of the present invention.
[0023] Figure 6This is a schematic circuit diagram of a circuit board in the interference check of applying SDL parameterization to the circuit board in the embodiments of the present invention. Detailed implementation manners
[0024] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.
[0025] It should be noted that the terms "first", "second", etc. in the specification and claims of the present invention and the above drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged under appropriate circumstances so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device comprising a series of steps or units is not necessarily limited to those clearly listed steps or units, but may include other steps or units not clearly listed or inherent to these processes, methods, products or devices.
[0026] In the specific implementation manner of the present invention, as Figure 1 shown, a method for SDL parameterization modeling of an arc is provided, which involves two addition / subtraction operations to calculate the parameters of a point on the arc. SDL parameterization is faster than arc length parameterization and chord length parameterization. SDL parameterization is globally continuous and can meet the applications of CAD. At the same time, SDL parameterization is closer to arc length parameterization than chord length parameterization. SDL parameterization can be re-parameterized into a quadratic rational Bezier representation through appropriate quadratic variation, and two addition and subtraction operations are used for point inverse calculation. It has good performance in weighing calculation speed and uniformity, and can effectively improve the efficiency of mutual conversion between points and parameters. The specific steps are as follows: S1. According to the set unit circular arc degree, derive multiple parameterization equations by using different parameterization methods; S2. Construct an SDL parameterization equation according to the unit circular arc degree, perform normalization processing on the constructed SDL parameterization equation to obtain a normalization result, use different parameterization methods and the derived multiple parameterization equations to perform parameter normalization on the unit circle to obtain a parameter equation, and use the obtained parameter equation to obtain arc construction parameters for arc construction.
[0027] In the specific implementation of this application, according to the set unit circular arc degree, a variety of parametric equations are derived using different parameterization methods. The different parameterization methods include arc length parameterization and chord length parameterization, and specifically include the following steps: Given a unit circle with the center at the origin, with the point as the radian being the point where the radian is 0, and the counterclockwise direction being the direction of increasing radian. For the point on the circle that varies with the radian increasing from 0 to , the length that varies with the radian is introduced for representation.
[0028] The arc length parameterization expression can be expressed as: (1) The chord length parameterization expression can be expressed as: (2) Based on the chord length parameterization, chord length squared parameterization, X - parameterization, and Y - parameterization are derived: The X - parameterization expression can be expressed as: (3) The chord length squared parameterization expression can be expressed as: (4) The Y - parameterization expression can be expressed as: (5) According to the unit circular arc degree, an SDL parameterization equation is constructed, specifically including the following steps: Establish the SDL parameterization expression of the present invention: As the radian increases, the sum of the projection of the point on the unit circle on the y - axis plus 1 minus the projection of the point on the x - axis is also increasing, increasing from 0 to 8. Therefore, the SDL parameterization expression can be established: (6) After the above re - parameterization, at the break point, it satisfies continuity.
[0029] Establish the parametric equations after parameter normalization of the above different parameterization methods for the unit circle : Normalize the arc length parameterization expression, and let , then the unit circle parameterized by arc length can be expressed as: (7) Normalize the chord length parameterization expression, and let , then the unit circle parameterized by chord length can be expressed as: (8) Normalize the X-parameterization expression, and let , then the unit circle parameterized by X can be expressed as: (9) Normalize the chord length squared parameterization expression, and let , then the unit circle parameterized by chord length squared can be expressed as: (10) Normalize the Y-parameterization expression, and let , then the unit circle parameterized by Y can be expressed as: (11) Normalize the SDL parameterization expression, and let , then the unit circle parameterized by SDL can be expressed as: (12) Use different parameterization methods and a variety of derived parametric equations to perform parameter normalization on the unit circle to obtain parametric equations, and obtain the solutions of the above equation (12) according to the characteristics of the unit circle: (12) In the specific implementation of the present invention, the numerical experiment part uses Matlab R2019a in Windows 10 as the development platform. Specifically as follows: To measure the gap between the parametric equation of the above unit circle and the arc length parameterization, randomly select within values as the parameter , determine the coordinates of the points on the circle according to the parametric equation of the unit circle under different parameterizations , and take the arc length distance between the points obtained by other parameterizations and the arc length parameterization as the energy function (within ). The smaller the fluctuation of the distance, the more uniform the parameterization. When , the experimental results are as follows Figure 3 shown.
[0030] Figure 3It is the re-parameterized representation of the unit circle in the rectangular coordinate system, namely arc length parameterization, chord length parameterization, X parameterization, chord length squared parameterization, Y parameterization, and SDL parameterization respectively.
[0031] above Figure 3 It represents the unit circle represented by different parameterizations. It can be seen that the distribution of points on the circle corresponding to SDL parameterization is more uniform, and the energy of each point is closer to 0, which means that the uniformity of SDL parameterization is closer to that of arc length parameterization.
[0032] In order to quantitatively characterize this feature of uniformity mathematically, randomly select inside values as parameters, and determine the coordinates of points on the circle according to different parameter methods , and then calculate the normalized radian values corresponding to these coordinates , then the uniformity is equal to the number of divided by . When , the calculation results are shown in Table 1 below.
[0033] Table 1 Uniformity of different parameterizations
[0034] The results show that the uniformity of chord length parameterization is the lowest, which is 0.6615. The uniformity of SDL parameterization is 0.9887, which is the closest to 1. Generally speaking, the uniformity of SDL parameterization is better and is relatively close to that of arc length parameterization.
[0035] Furthermore, analyze the efficiency of parameterization from the aspects of curve evaluation and point inverse calculation operations. Compare the time required for SDL parameterization of the unit circle with the above other parameterizations in curve evaluation and point inverse calculation operations. Specifically, randomly select points on the unit circle, . Calculate the time required for point inverse calculation and curve evaluation operations using different parameterizations, and take the average value of the above 100 experiments, as shown in Figure 4 .
[0036] Figure 4 It is the time comparison between SDL parameterization and other parameterizations in point inversion.
[0037] According to the above Figure 4 The results show that when the number of executions is large, for the point inverse calculation operation, arc length parameterization requires inverse trigonometric function operations, so it is the most time-consuming, while the SDL parameterization of the present invention only requires two addition and subtraction operations, so the efficiency is relatively the highest and the time spent is the least.
[0038] Figure 5Time comparison between SDL parameterization and other parameterizations in curve evaluation. According to the above Figure 5 The results show that when the number of executions is large, for curve evaluation operations, SDL parameterization takes the least time and has the highest efficiency. The chord length parameterization expression is complex, so it takes the most time relatively. Apply the SDL parameterization method of the present invention to the interference check of the circuit board. The circuit diagram of the circuit board is as follows Figure 6 shown. It has 40,630 vertices, 9,805 line segments, and 993 arcs.
[0039] The experimental results of using a computer with 64GB of memory and a CPU model of Intel i7-12700H to perform interference check (intersection) on the circuit board are as follows in the table: Table 2 Total time spent on interference check of the circuit board
[0040] It can be seen from Table 2 above that the point inverse calculation operation was called 9,928,422 times in total. The time spent by ACIS is 3.1524 times that of SDL parameterization, and the time spent by OCC is 2.0351 times that of SDL parameterization.
Claims
1. A SDL parametric modeling method for arcs, characterized in that: The following steps are involved: S1, according to the set unit arc, different parameterization methods are used to derive multiple parameterized equations; S2, construct an SDL parametric equation according to the unit circle arc, normalize the constructed SDL parametric equation to obtain a normalized result, use different parameterization methods and derived multiple parameterization equations to normalize the parameters of the unit circle to obtain a parametric equation, and use the obtained parametric equation to obtain arc construction parameters to construct the arc.
2. The SDL parametric modeling method of an arc according to claim 1, characterized in that: According to the set unit arc, different parameterization methods are used to derive a variety of parameterization equations, and the different parameterization methods include arc length parameterization and chord length parameterization.
3. The SDL parametric modeling method of an arc according to claim 2, characterized in that: Given a unit circle centered at the origin, Points as radians For points on the circle, the counterclockwise direction is the direction in which the arc increases. Follow the arc Increment from 0 to , introduction length Follow the arc Incremental representation of .
4. The SDL parametric modeling method of an arc according to claim 3, characterized in that: Arc length parameterized expression It can be expressed as: （1）。 5. The SDL parametric modeling method of an arc according to claim 3, characterized in that: Chord length parameterization expression It can be expressed as: （2）。 6. The SDL parametric modeling method of an arc according to claim 5, characterized in that: Chord length square parameterization, X parameterization and Y parameterization are derived from chord length parameterization: X - parameterized expression It can be expressed as: （3）。 7. The SDL parametric modeling method of an arc according to claim 6, characterized in that: Parameterized expression of the square of the chord length It can be expressed as: （4）。 8. The SDL parametric modeling method of an arc according to claim 5, characterized in that: Y-parameterized expression It can be expressed as: （5）。 9. The SDL parametric modeling method of an arc according to claim 5, characterized in that: Establish the parametric equations after the above different parameterization methods are normalized for the unit circle : 。 10. A SDL parametric modeling method for arcs, characterized in that: Includes parameterization modules and modeling modules: The parameterization module uses different parameterization methods to derive various parameterization equations according to the set unit arc; The modeling module constructs an SDL parametric equation according to the unit circle arc, normalizes the constructed SDL parametric equation to obtain a normalized result, uses different parameterization methods and a variety of derived parameterized equations to normalize the parameters of the unit circle to obtain a parametric equation, and uses the obtained parametric equation to obtain arc construction parameters for arc construction.