CN117371222B - Curve fairing optimization method based on discrete curvature and multistage optimization points - Google Patents
Curve fairing optimization method based on discrete curvature and multistage optimization points Download PDFInfo
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Abstract
The invention provides a curve fairing optimization method based on discrete curvature and multistage optimization points, which uses interpolation points on a curve as an optimization object, obtains defect points on the curve based on the discrete curvature of each interpolation point and uses curvature constraint values as judgment references, and replaces the defect points of the curve with multistage optimization points to realize correction optimization of the curve. And establishing a fairing criterion applicable to any curve from three dimensions of the fairing degree, the discrete curvature and the discrete deflection rate, and finally forming a fairing optimization method applicable to any curve. According to the method, interpolation points on the curve are used as optimization objects, so that the curve is subjected to finer fairing optimization; the multistage optimization point correction method is provided, and the regulation and control of the curve fairing optimization speed are realized through stage selection. And a fairing criterion for analyzing the curve fairing is established from three dimensions, so that the curve fairing connotation is enriched. The method has universality for the fairing optimization of any curve in space, and has important theoretical significance and use value for engineering practical application and computer graphic design.
Description
Technical Field
The invention relates to the technical field of curve fairing optimization, in particular to a curve fairing optimization method based on discrete curvature and multistage optimization points.
Background
The fields of computer graphic design, industrial design and manufacture and the like have certain requirements on the smoothness of curves. For example, during machine tool machining, the smoothness of the feed path profile affects the precision and surface finish of the formed workpiece and is related to the useful life of the tool; in the planning of a composite material fiber curve laying path, the influence of the smoothness of the laying path on a forming member is large, the poor smoothness of the laying path can cause overlarge local curvature of a laying filament bundle, and the fiber is easy to generate buckling deformation to influence the quality of the composite material member. Therefore, curve fairing optimization is widely used in a plurality of fields as an important technical means. With the progress of technology, the requirements of various fields on curve smoothness are higher and higher, and the types of the faced curves are more complex and various, so that a fairing optimization technology suitable for any curve is urgently needed. Curve fairing optimization currently has a technical bottleneck mainly in two aspects: a curve fairing criterion and a curve fairing processing method. The curve fairing criteria in the prior art are numerous in the aspect of the fairing criteria, but the fairing criteria applicable to any curve are not proposed, and the analysis dimension for curve fairing is too single. Most of the existing processing methods in the aspect of the fairing processing method focus on correcting curve control points, and the method is easy to cause the curve to generate larger deformation in the fairing optimization process so as to deviate from the change track of the original curve.
Disclosure of Invention
The invention aims to provide a curve fairing optimization method based on discrete curvature and multistage optimization points, which aims to solve the technical problems that the existing curve fairing optimization is poor in universality, the analysis dimension of the curve fairing is too single, and the deformation of a curve in the fairing optimization process is too large.
The technical problems solved by the invention can be realized by adopting the following scheme:
the curve fairing optimization method based on the discrete curvature and the multistage optimization points is characterized by comprising the following steps of: the method comprises the following steps:
s101, establishing a curve model before optimization, generating a curve before optimization, and solving an interpolation point set P 0＝{Qi (i=1, 2, …, n) on the curve;
s102, calculating curve light smoothness E before optimization, setting a discrete curvature constraint value [ K ], and selecting an optimization point level N;
S103, calculating the discrete curvature K i of interpolation points { Q i } (i=1, 2, …, n) on the curve, and obtaining a defect point set PQ and a defect point number set BH by using a defect point discriminant K i - [ K ] not less than 0;
s104, solving N-level optimization points to obtain optimization points to replace defect points Generating an optimized curve interpolation point set P 1;
S105, solving the discrete curvature of each point in the interpolation point set P 1 of the optimized curve, and obtaining a defect point set and a defect point number set of the optimized curve by using the defect point discriminant K i K not less than 0;
S106, judging whether the optimized finishing condition is met after the fairing optimization: the defect point number set is smaller than a set value; if not, repeating the steps S104 and S105 until the curve interpolation point set PUP is satisfied, and generating an optimized curve;
S107, solving the smooth degree and the discrete curvature of the optimized curve, and when the curve is a space curve, further requiring the discrete flexibility rate of the solution curve before and after optimization; and analyzing the light compliance before and after the curve optimization according to the light compliance before and after the light compliance optimization, the discrete curvature and the discrete flexibility rate.
Further: the solution formula of the curve light smoothness E is as follows:
Wherein, l i＝||Qi+1-Qi is the vector modular length of two adjacent points Q i,Qi+1 on the curve, e i is the unit polyline vector of two adjacent points Q i,Qi+1 on the curve, e i+1-ei is the variable quantity modular length of two adjacent unit polyline vectors, and the solution formula of e i is as follows:
Further: The interpolation point at the ith is the jth defect point, which is triangulated/>, with the two adjacent points Q i-1 and Q i+1 Mid-point of median line/>Is defect point/>Is parallel to the bottom edge Q i- 1Qi+1; the N-level optimization point is obtained by taking the N-1 level optimization point as a defect point and solving the midpoint/>, of the median againThe optimized point is used for directly cross-level replacement of the original defect point/>
Further: when the curve is a plane curve, the solution formula of the discrete curvature K i of the interpolation point Q i on the curve in step S103 is:
Wherein α i,βi is a first-order director and a second-order director of the curve at the point Q i, and a specific solution formula is as follows:
Where h is the defined step size.
Further: when the curve is a space curve, the solution formula of the discrete curvature K i of the interpolation point Q i on the curve in step S103 is:
Wherein α i,βi is a first-order director and a second-order director of the curve at the point Q i, and a specific solution formula is as follows:
Where h is the defined step size.
Further: in step S106, the termination condition is optimized: the defect point number set is smaller than a set value, which is at least 3.
Further: the spatial curve discrete flexural rate τ i is expressed as:
Where ρ i is the third order director of the curve at point Q i.
Further: in step S107, according to the discrete curvature and the discrete flexibility rate of the curve, a discrete curvature and a discrete flexibility rate change curve before and after the curve fairing optimization are obtained, the uniformity of the discrete curvature and the discrete flexibility rate change before and after the curve optimization is compared and analyzed, and the fairing before and after the curve optimization is qualitatively analyzed; and quantitatively analyzing the forward and backward light compliance of the curve optimization according to the light compliance value.
Further: in step S107, a solution curve fairing optimization rate η is also required, and the curve optimization degree is determined, where the expression is:
Where E is the smoothness of the curve before optimization and EUP is the smoothness of the curve after optimization.
The curve fairing optimization method based on discrete curvature and multistage optimization points uses interpolation points on any curve in space as an optimization object, acquires defect points on the curve based on the discrete curvature of each interpolation point and uses curvature constraint values as judgment references, and realizes correction optimization of the curve by replacing the defect points of the curve with the optimized points. And establishing a fairing criterion for evaluating the smoothness of the curve from three dimensions of the smoothness, the discrete curvature and the discrete flexibility rate, and finally forming a fairing optimization method suitable for any curve. According to the invention, interpolation points on the curve instead of control points are used as specific research objects, so that the curve is subjected to finer fairing optimization, and larger deformation of the curve after the curve is optimized by a curve optimization method based on the control points is avoided; and a multistage optimization point correction method is provided, and the regulation and control of the curve fairing optimization speed are realized through stage selection. In addition, a fairing criterion for analyzing the curve fairing is established from three dimensions, so that the curve fairing connotation is enriched. The method has universality and high efficiency for fairing optimization of any curve, and has important theoretical significance and use value for engineering practical application and computer graphic design.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a first embodiment and a second embodiment of a method for optimizing the fairing of a curve;
FIG. 2 is a flow chart of a planar curve model preprocessing in accordance with an embodiment of the present invention;
FIG. 3 is a graph showing a plane before a fairing optimization according to an embodiment of the invention;
FIG. 4 is a schematic diagram of a solution principle of N-level optimization points according to the first and second embodiments of the present invention;
FIG. 5 is a graph of a planar curve fairing optimization iterative process in accordance with an embodiment of the invention;
FIG. 6 is a graph of the discrete curvature change before smoothing optimization of a planar curve in accordance with an embodiment of the present invention;
FIG. 7 is a graph of the variation of discrete curvature after smoothing optimization of a planar curve in accordance with an embodiment of the present invention;
FIG. 8 is a graph of a plane after a fairing optimization in accordance with an embodiment of the invention;
FIG. 9 is a flow chart of a second space curve model preprocessing in accordance with an embodiment of the present invention;
FIG. 10 is a graph of space before fairing optimization in accordance with embodiments of the invention;
FIG. 11 is a diagram of a second space curve fairing optimization iteration process according to an embodiment of the invention;
FIG. 12 is a graph of the discrete curvature change before a second space curve fairing optimization in accordance with an embodiment of the invention;
FIG. 13 is a graph of discrete flex rate variation before fairing optimization of a second space curve in accordance with an embodiment of the invention;
FIG. 14 is a graph of the discrete curvature change after a second space curve fairing optimization in accordance with an embodiment of the invention;
FIG. 15 is a graph of discrete flex rate change after a two-dimensional curve fairing optimization in accordance with an embodiment of the invention;
FIG. 16 is a graph showing the space after the fairing optimization according to the second embodiment of the invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, the embodiments of the present invention will be described in further detail with reference to the accompanying drawings.
Example 1
Fig. 1 to 8 show a curve fairing optimization method based on discrete curvature and multistage optimization points according to the present embodiment, where the curve fairing optimization method is used for optimizing a plane curve, as shown in fig. 1, and specifically includes the following steps:
S101, preprocessing an optimized curve model, establishing a curve model before optimization, generating a curve before optimization, and solving an interpolation point set P 0＝{Qi (i=1, 2, …, n) on the curve.
Specifically, as shown in fig. 2, a plane curve model is built in CAD software, key node coordinate information on the curve is extracted, and the coordinate point information is used to generate an optimized front plane curve shown in fig. 3 in MATLAB.
Specifically, the interpolation point in step S101 is a specific study object of the fairing optimization in this embodiment, and is different from the conventional fairing algorithm in that the fairing optimization of the curve is implemented by modifying the control point on the curve. The control point is used for regulating and controlling the whole contour of the curve, the interpolation point not only can determine the contour of the local area of the curve, but also has a mesoscopic characteristic, and the local property of the curve can be reflected to the greatest extent, so that the mesoscopic correction and optimization of the defect part of the interpolation point on the curve can realize the improvement of the smoothness on the macro of the curve.
S102, calculating the light smoothness E of the curve of the plane before optimization, setting a discrete curvature constraint value [ K ], and selecting a proper optimization point level N according to the profile characteristics and the light smoothness of the curve.
Specifically, the solution formula of the curve light smoothness E is as follows:
Wherein, l i＝||Qi+1-Qi is the vector modular length of two adjacent points Q i,Qi+1 on the curve, e i is the unit polyline vector of two adjacent points Q i,Qi+1 on the curve, e i+1-ei is the variable quantity modular length of two adjacent unit polyline vectors, and the solution formula of e i is as follows:
The discrete curvature constraint value [ K ] is a judging standard of defect points on a curve, the value of [ K ] determines the height of a curve fairing optimization standard, and the smaller the discrete curvature constraint value [ K ] is selected, the number of marked defect points is increased, and the requirement on curve fairing is also improved. On the contrary, the value of [ K ] is larger, the number of marked defect points is also reduced, and the requirement on curve smoothness is also reduced.
Specifically, in this embodiment, a plane curve discrete curvature constraint value [ K ] =0.015 is set, and the selected optimization point level N is equal to 5.
S103, obtaining a discrete curvature K i of interpolation points { Q i } (i=1, 2, …, n) on the curve, and obtaining a defect point set PQ and a defect point number set BH by using a defect point discriminant K i - [ K ] gtoreq 0.
Specifically, the solution formula of the discrete curvature K i of the interpolation point Q i on the plane curve is as follows:
Wherein, α i,βi is a first-order director and a second-order director of the plane curve at the point Q i, which respectively represent a tangent vector and a normal vector of the plane curve at the point Q i, α i×βi has no special meaning in the plane curve, the curvature of the plane curve obtained by solving here has both positive and negative values, the positive and negative properties have no influence on the curve light compliance, therefore, the discrete curvature takes positive values during solving, and the specific solving formula of α i,βi in the plane curve is as follows:
Wherein the method comprises the steps of
In this embodiment, the step h=0.001 is set, and the first-order derivative and the second-order derivative of the point Q i on the curve are obtained by a calculation method of forward difference quotient.
S104, from the defect point that i is more than or equal to 2Starting correction optimization, and determining defect points/>Adjacent points Q i-1 and Q i+1 of the model (C) are subjected to N-level optimization point solving formulas to obtain an optimization point/>To replace defective points with the obtained optimized points/>And generating a primary optimization curve interpolation point set P 1, completing primary fairing optimization, and solving the curve light smoothness E 1.
Specifically, the embodiment gives a multistage optimization point correction method by utilizing the geometric characteristics of the defect points on the curve, the optimization point level in the embodiment is the correction point type adopted in the curve fairing optimization process,The interpolation point at the ith is the jth defect point, which is formed by adjacent two points Q i-1 and Q i+1/>Too small, curve at point/>The curvature at that point will be too great. As shown in FIG. 4, the three points are formed/>Mid-point of median line/>As defect points/>The median line is parallel to the bottom line Q i-1Qi+1, and the i-th interpolation point on the curve and the adjacent two points are formed/>Will increase significantly, effecting a reduction in curvature at that point. The second-level optimizing point is to consider the first-level optimizing point as a defect point and calculate the midpoint of the median again to directly cross-level replace the original defect point/>The N-level optimization point is taken as a defect point, and the midpoint/> -of the median is calculated againThe original defect point/> is directly replaced by the trans-grade(The principle of geometric solution of the first to third order optimization points is shown from left to right in fig. 4).
The N-level optimizing point is obtained by setting the optimizing point level N, the dropping speed of the curve curvature is greatly improved in a mode that the N-level optimizing point directly replaces the original curve defect point, and the N-level optimizing point is used by selecting and adjusting the optimizing point levelThe defect points are directly replaced, and the selective regulation and control of the curve fairing optimization speed can be realized.
Specifically, in this embodiment, the plane curve is first-order optimized pointAnd N-level optimization Point/>The solving formula is as follows:
specifically, in this embodiment, five-stage optimization point replacement curve defect points are solved according to an optimization point solving method and a formula.
S105, solving the discrete curvature of each point in the interpolation point set P 1 of the optimized curve, and obtaining a defect point set PQ 1 and a defect point number set BH of the primary optimized curve by utilizing the defect point discriminant K i K not less than 0 1
S106, judging whether the optimized termination condition is met after the fairing optimization, wherein the termination condition is as follows: the defect point number set is smaller than a set value; if not, repeating the steps S104 and S105 until the optimization is satisfied, and if the optimization is satisfied, obtaining an optimized curve interpolation point set PUP so as to generate an optimized curve;
specifically, the set value of the optimized termination condition in this embodiment is at least three.
The condition is not only the termination condition of the fairing optimization, but also the precondition of the optimization point solution. When the head end and the tail end of the curve are defect points, the left adjacent point or the right adjacent point is not corrected and optimized through the optimized point, and the curvature value of the curve is gradually reduced by the correction of the defect points in the neighborhood.
In the embodiment, the planar curve is subjected to loop optimization correction until the optimization termination condition is met, the smoothness of the curve is solved in the loop iteration process, and the smoothness in the loop optimization process is used for drawing the curve of the fairing optimization iteration process in the embodiment as shown in fig. 5.
And S107, solving the smoothness and the discrete curvature of the optimized curve, and evaluating the smoothness of the curve before and after optimization according to the smoothness and the discrete curvature before and after optimization.
Specifically, according to the discrete curvature of the plane curve, a discrete curvature change curve before the plane curve fairing optimization shown in fig. 6 and a discrete curvature change curve after the plane curve fairing optimization shown in fig. 7 are obtained, the uniformity of the discrete curvature change before and after the optimization is compared and analyzed, and the smoothness before and after the curve optimization is qualitatively analyzed; and quantitatively analyzing the forward and backward light smoothness of the curve according to the forward and backward light smoothness value of the curve.
The more uniform the discrete curvature and discrete flex rate change of the curve, the better the curve's smoothness can be assessed qualitatively, the less the curve's smoothness, and the better the curve's smoothness can be assessed quantitatively.
Further, in step S107, a solution curve fairing optimization rate η is also required, which indicates the degree of curve fairing optimization, and the larger η, the higher the degree of improving the fairing of the optimized curve, the expression is as follows:
Where E is the smoothness of the curve before optimization and EUP is the smoothness of the curve after optimization.
Comparing fig. 6 and fig. 7, it is obvious that the uniformity of the plane curve discrete curvature change curve of the present embodiment is greatly improved, and the smoothness of the curve in the present embodiment can be qualitatively evaluated to be optimized. As can be seen from fig. 5, the light smoothness of the curve in the fairing optimization process is in a decreasing trend, the light smoothness of the plane curve is decreased from e= 0.4736 to eup=0.0326, the curve fairing optimization rate is as high as η=93%, and the curve fairing in the embodiment can be quantitatively evaluated to be optimized.
In the embodiment, a curve fairing criterion is established from two dimensions of discrete curvature and fairing, and the curve fairing is rated from two layers of qualitative and quantitative mutually, so that the feasibility and the high efficiency of the curve fairing optimization method are verified.
Fig. 8 shows a schematic plan view of a fairing-optimized planar curve in this embodiment, and a curve is generated in MATLAB by using the interpolation point set PUP after the fairing optimization, so that it is obvious that the geometric appearance of the curve is greatly optimized, and the line is smoother.
According to the planar curve fairing optimization method based on the discrete curvature and the multistage optimization points, interpolation points on the curve are used as specific research objects, so that the curve is subjected to finer fairing optimization, the defect that the curve is excessively deformed after the optimization of the existing method is avoided, the geometric characteristics of the defect points on the curve are utilized to provide a solving and correcting method of the multistage optimization points, and the adjustment and control of the fairing optimization speed of the curve are realized through the stage selection of the optimization points. In the embodiment, a fairing criterion for evaluating the smoothness of the curve is established from two dimensions of the smoothness and the discrete curvature, and the smoothness of the optimized curve is evaluated in a complementary manner from a qualitative layer and a quantitative layer, so that a fairing optimization method for a plane curve is finally formed.
Example two
Fig. 1, fig. 4, and fig. 9-16 show a curve fairing optimization method based on discrete curvature and multistage optimization points according to the present embodiment, where the curve fairing optimization method is used for optimizing a space curve on a half cylinder, as shown in fig. 1, and specifically includes the following steps:
S101, preprocessing an optimized curve model, establishing a curve model before optimization, generating a curve before optimization, and solving an interpolation point set P 0＝{Qi (i=1, 2, …, n) on the curve;
Specifically, as shown in fig. 9, a space curve model is built in CAD software, key node coordinate information on a curve and curve key node information are extracted, and a space curve shown in fig. 10 is generated in MATLAB by using the coordinate point information. Aiming at the geometric characteristics of a space curve, a curve model arranged on the space curve is required to be selected to obtain the coordinate information of all key nodes of the curve, the curve is generated in MATLAB by a fitting method, and the curve are combined together.
S102, calculating the light smoothness E of the space curve before optimization, setting a discrete curvature constraint value [ K ], and selecting a proper optimization point level N according to the profile characteristics and the light smoothness of the curve;
specifically, the solution formula of the curve smoothness E is the same as that of the planar curve.
Specifically, in this embodiment, a space curve discrete curvature constraint value [ K ] =0.18 is set, and the selected optimization point level N is equal to 5.
S103, obtaining a discrete curvature K i of interpolation points { Q i } (i=1, 2, …, n) on the curve, and obtaining a defect point set PQ and a defect point number set BH by using a defect point discriminant K i - [ K ] gtoreq 0.
Specifically, the solution formula of the discrete curvature K i of the interpolation point Q i on the spatial curve is:
wherein, α i,βi is a first-order director and a second-order director of the space curve at the point Q i, α i,βi is a tangent vector and a main normal vector of the space curve at the point Q i, and α i×βi＝γi represents a unit auxiliary normal vector at the point Q i, which has the following specific solving formula:
Where h is a defined step size, and in this embodiment, the step size h=0.001 is set.
S104, from the defect point that i is more than or equal to 2Starting correction optimization, and determining defect points/>Adjacent points Q i-1 and Q i+1 of the model (C) are subjected to N-level optimization point solving formulas to obtain an optimization point/>To replace defective points with the obtained optimized points/>And generating a primary optimization curve interpolation point set P 1, completing primary fairing optimization, and solving the curve light smoothness E 1.
Specifically, as shown in fig. 4, the solution principle of the N-level optimization point in this embodiment is the same as that of the planar curve.
First order optimization point of space curve in this embodimentThe solving formula is as follows:
by recursion, the N-level optimization points of the space curve The coordinate solving formula is
Wherein (x i,yi,zi) is a defect pointCoordinates (x i-1,yi-1,zi-1) are Q i-1 coordinates (x i+1,yi+1,zi+1) are Q i+1 coordinates,/>For/>Is defined by the coordinates of (a).
Specifically, in this embodiment, five-stage optimization point replacement curve defect points are solved according to an optimization point solving method and a formula.
S105, solving the discrete curvature of each point in the interpolation point set P 1 of the optimized curve, and obtaining a defect point set PQ 1 and a defect point number set BH of the primary optimized curve by utilizing the defect point discriminant K i K not less than 0 1
S106, judging whether the optimized termination condition is met after the fairing optimization, wherein the termination condition is as follows: the defect point number set is smaller than a set value; if not, repeating the steps S104 and S105 until the optimization is satisfied, and if the optimization is satisfied, obtaining an optimized curve interpolation point set PUP so as to generate an optimized curve;
specifically, the set value of the optimized termination condition in this embodiment is at least three.
In the embodiment, the space curve is subjected to loop optimization correction until the optimization termination condition is met, the smoothness of the curve is solved in the loop iteration process, and the smoothness in the loop optimization process is used for drawing the curve of the fairing optimization iteration process in the embodiment as shown in fig. 11.
S107, solving the smooth degree and the discrete curvature of the optimized curve, and solving the discrete flexibility rate of the space curve before and after optimization; and evaluating the smoothness of the curve before and after the optimization according to the smoothness, the discrete curvature and the discrete flexibility rate of the curve before and after the optimization.
Further, the discrete flexibility ratio is a quantity for describing the torsion degree of the space curve, and the expression of the discrete flexibility ratio tau i is as follows:
Wherein, α i and β i have the same meaning and solving method as that described in step S103, and ρ i is the third-order vector of the curve at the point Q i.
Specifically, according to the discrete curvature and the discrete flexibility ratio of the space curve, a discrete curvature change curve before the space curve fairing optimization shown in fig. 12, a discrete curvature change curve after the space curve fairing optimization shown in fig. 14, a discrete flexibility ratio change curve before the space curve fairing optimization shown in fig. 13 and a discrete flexibility ratio after the space curve fairing optimization shown in fig. 15 are obtained, the discrete curvature and the uniformity of the discrete flexibility ratio change before and after the space curve fairing optimization are compared and analyzed, and the fairing before and after the space curve fairing optimization is qualitatively analyzed; and quantitatively analyzing the forward and backward fairing of the curve according to the forward and backward fairing value of the curve.
The more uniform the discrete curvature and discrete flex rate change of the curve, the better the curve's smoothness can be assessed qualitatively, the less the curve's smoothness, and the better the curve's smoothness can be assessed quantitatively.
The calculation method of the curve fairing optimization rate η in this embodiment is the same as that of the first embodiment.
Comparing fig. 12, 14 and fig. 13 and 15, it is apparent that the uniformity of the space curve discrete curvature and the discrete flexibility rate change curve in this embodiment is greatly improved, and the smoothness of the curve in this embodiment can be qualitatively evaluated to be optimized. As can be seen from fig. 11, the light smoothness of the curve in the fairing optimization process is in a decreasing trend, the light smoothness of the spatial curve is decreased from e= 21.4118 to eup= 1.5954, the curve fairing optimization rate is as high as η=92.5%, and the curve fairing in the embodiment can be quantitatively evaluated to be optimized.
In the embodiment, a curve fairing criterion is established from three dimensions of discrete curvature, discrete deflection and fairing, and the curve fairing is rated from two layers of qualitative and quantitative mutually to supplement, so that the feasibility and the high efficiency of the curve fairing optimization method are verified.
Fig. 16 shows a schematic diagram of a space curve after the fairing optimization in this embodiment, and a curve is generated in MATLAB by using the interpolation point set PUP after the fairing optimization, so that it is obvious that the geometric appearance of the curve is greatly optimized, and the line is smoother.
According to the space curve fairing optimization method based on the discrete curvature and the multistage optimization points, interpolation points on the curve are used as specific research objects, so that the curve is subjected to finer fairing optimization, the defect that the curve is excessively deformed due to processing of the existing method is avoided, the geometric characteristics of the defect positions on the curve are utilized, the method for correcting the fairing by the multistage optimization points is provided, and the adjustment and control of the fairing optimization speed of the curve are realized through the selection of the optimization points. In the embodiment, a curve fairing criterion is established from three dimensions of the fairing degree, the discrete curvature and the discrete flexibility, and the smoothness of the optimized curve is evaluated in a complementary mode from a qualitative layer and a quantitative layer, so that a fairing optimization method applicable to any curve is finally formed.
The foregoing is merely illustrative of the present invention, and the present invention is not limited thereto, and any person skilled in the art will readily recognize that variations or substitutions are within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (8)
1. The curve fairing optimization method based on the discrete curvature and the multistage optimization points is characterized by comprising the following steps of: the method comprises the following steps:
s101, establishing a curve model before optimization, generating a curve before optimization, and solving an interpolation point set P 0＝{Qi (i=1, 2, …, n) on the curve;
s102, calculating curve light smoothness E before optimization, setting a discrete curvature constraint value [ K ], and selecting an optimization point level N;
S103, calculating the discrete curvature K i of interpolation points { Q i } (i=1, 2, …, n) on the curve, and obtaining a defect point set PQ and a defect point number set BH by using a defect point discriminant K i - [ K ] not less than 0;
s104, solving N-level optimization points to obtain optimization points to replace defect points Generating an optimized curve interpolation point set P 1;
S105, solving the discrete curvature of each point in the interpolation point set P 1 of the optimized curve, and obtaining a defect point set and a defect point number set of the optimized curve by using the defect point discriminant K i K not less than 0;
S106, judging whether the optimized finishing condition is met after the fairing optimization: the defect point number set is smaller than a set value; if not, repeating the steps S104 and S105 until the curve interpolation point set PUP is satisfied, and generating an optimized curve;
S107, solving the smooth degree and the discrete curvature of the optimized curve, and when the curve is a space curve, further requiring the discrete flexibility rate of the solution curve before and after optimization; analyzing the light compliance before and after the curve optimization according to the light compliance before and after the light compliance optimization, the discrete curvature and the discrete flexibility rate;
the solution formula of the curve light smoothness E is as follows:
Wherein, l i＝||Qi+1-Qi is the vector modular length of two adjacent points Q i,Qi+1 on the curve, e i is the unit polyline vector of two adjacent points Q i,Qi+1 on the curve, e i+1-ei is the variable quantity modular length of two adjacent unit polyline vectors, and the solution formula of e i is as follows:
2. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points according to claim 1, wherein: The interpolation point at the ith is the jth defect point, which is triangulated with the adjacent two points Q i-1 and Q i+1 Mid-point of median line/>Is defect point/>Is parallel to the bottom edge Q i-1Qi+1; the N-level optimization point is obtained by taking the N-1 level optimization point as a defect point and solving the midpoint/>, of the median againThe original defect point is directly replaced by the cross-stage
3. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points according to claim 1, wherein: when the curve is a plane curve, the solution formula of the discrete curvature K i of the interpolation point Q i on the curve in step S103 is:
Wherein α i,βi is a first-order director and a second-order director of the curve at the point Q i, and a specific solution formula is as follows:
Where h is the defined step size.
4. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points according to claim 1, wherein: when the curve is a space curve, the solution formula of the discrete curvature K i of the interpolation point Q i on the curve in step S103 is:
Wherein α i,βi is a first-order director and a second-order director of the curve at the point Q i, and a specific solution formula is as follows:
Where h is the defined step size.
5. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points according to claim 1, wherein: in step S106, the termination condition is optimized: the defect point number set is smaller than a set value, which is at least 3.
6. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points as recited in claim 4, wherein: the spatial curve discrete flexural rate τ i is expressed as:
Where ρ i is the third order director of the curve at point Q i.
7. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points as recited in claim 6, further comprising: in step S107, according to the discrete curvature and the discrete flexibility rate of the curve, a discrete curvature and a discrete flexibility rate change curve before and after the curve fairing optimization are obtained, the uniformity of the discrete curvature and the discrete flexibility rate change before and after the curve optimization is compared and analyzed, and the fairing before and after the curve optimization is qualitatively analyzed; and quantitatively analyzing the forward and backward light compliance of the curve optimization according to the light compliance value.
8. The method for optimizing curve fairing based on discrete curvature and multi-stage optimization points according to claim 1, wherein: in step S107, a solution curve fairing optimization rate η is also required, and the curve optimization degree is determined, where the expression is:
Where E is the smoothness of the curve before optimization and EUP is the smoothness of the curve after optimization.
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