CN108898656A - A kind of free curve in three-dimensional modeling efficiently produces method - Google Patents

A kind of free curve in three-dimensional modeling efficiently produces method Download PDF

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CN108898656A
CN108898656A CN201810454782.4A CN201810454782A CN108898656A CN 108898656 A CN108898656 A CN 108898656A CN 201810454782 A CN201810454782 A CN 201810454782A CN 108898656 A CN108898656 A CN 108898656A
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free curve
curvature
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赵慧卿
王慧
黄佳敏
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Abstract

The invention discloses the free curves in a kind of three-dimensional modeling to efficiently produce method, including：The generation for the continuous free curve of G1 for determining endpoint according to two；Generation in core business according to the continuous free curve of G2 of two free curves；The generation of the open continuous free curve of G2；The generation of the continuous free curve of closed G2 in key structure Holistic modeling.The present invention can be realized the model that free curve is generated on the basis of minimum computer resource occupancy, have resources occupation rate low, calculating speed is fast, the high advantage of modeling efficiency.

Description

Efficient free curve generation method in three-dimensional modeling
Technical Field
The invention belongs to the field of computer graphic processing, and particularly relates to a free curve efficient generation method in three-dimensional modeling.
Background
When a three-dimensional modeling task is carried out, if a free curve of the object appearance is close to an actual radian and a smooth degree as much as possible, computer resources are additionally consumed, and a large amount of resources are occupied by accumulation of a plurality of curve curves, so that the operation of a modeling core service is influenced. In the prior art, a broken line approximation method is adopted to save as many resources as possible to complete the core service, which causes distortion of a free curve model and influences the authenticity of the core service.
Disclosure of Invention
In order to solve the problems, the invention provides a method for efficiently generating a free curve in three-dimensional modeling, which not only ensures the authenticity of free curve modeling, but also solves the problem of resource occupation.
The technical scheme adopted by the invention for solving the problems is as follows:
a method for efficiently generating a free curve in three-dimensional modeling is characterized by comprising the following steps: the method comprises the generation methods of four curves, namely a G1 continuous free curve, a G2 continuous free curve, an open G2 continuous free curve and a closed G2 continuous free curve:
the generation method of the G1 continuous free curve according to two determined end points comprises the following steps:
suppose P1，P2Is two feature points on the target free curve, the method for generating G1 continuous free curve according to these two feature points is as follows:
(1) over characteristic point P1And P2Auxiliary straight line L by pointo：lo(x,y)＝aox+boy+co＝0；
(2) Over characteristic point P1As an auxiliary straight line Lm：lm(x,y)＝amx+bmy+cm＝0；
(3) Over characteristic point P2As an auxiliary straight line Ln：ln(x,y)＝anx+bny+cn＝0；
(4) On the basis of the above-mentioned auxiliary line, a controllable function curve C: F (x, y) is inserted, which
In (1),namely, it is
Mu is a control parameter of the free curve C, w is more than 1 and is an integer, and represents the element number of the function curve;
in the configuration of the target free curve, by adjusting the controlThe flatness of the function curve C can be adjusted by the size of the parameter mu, and the curve is more gentle when | mu | is larger; mu.s>When 0, the curve is convex, otherwise, the curve is concave; by adjusting LmAnd lnThe slope of the curve can control the bias degree of the curve;
after the G1 continuous free curve with controllable parameters is generated, the free curve can be continuously controlled by the following parameters until the free curve is closest to the target curve:
wherein k ism，knIs an auxiliary straight line LmAnd LnSlope of (2)
At this point, the generation of a continuous free curve according to the G1 with two determined end points is completed;
(III) a method for generating a G2 continuous free curve according to the two free curves:
suppose P1，P2A free curve C which is two characteristic points on the target free curve and passes through the two points1:F1(x, y) ═ 0 and C2:F2If (x, y) is 0, then the method for generating G2 continuous free curves according to these two feature points is as follows:
(1) per P1，P2As an auxiliary straight line Lo：lo(x,y)＝aox+boy+co＝0；
(2) Per P1，P2F (x, y) ═ 0:
wherein,namely, it is
F(x,y)＝F1(x,y)·F2(x,y)+μ·(aox+boy+co)30; mu is a control parameter of the free curve C;
① the partial derivatives of the above equation are obtained:
because of the free curve F1(x, y) and an auxiliary line LoAll pass through the feature point P1Thus at P1The dotted equation above can be simplified as:
the compound can be obtained by the formula,
i.e. at the characteristic point P1Free curve C1:F1(x, y) 0 is normal to the interpolation function curve C: F (x, y) 0, and so, at P1Free curve C1A common tangent to the interpolation function curve C;
② calculating the characteristic point P of the interpolation function curve C, where F (x, y) is 0, according to the formula for calculating curvature1Curvature ρ:
the partial derivative and the second partial derivative are calculated for F (x, y) equal to 0, and the following formula is obtained:
substituting the above formula into curvature calculation formula to obtain interpolation function curve C at characteristic point P1Curvature ρ ofcThe calculation is as follows:
i.e. the interpolation function curve C and the free curve C1At a characteristic point P1The same curvature, the same C and C2At a characteristic point P2The curvature is also the same;
(4) for a given feature point P1And P2Recording the controllable tangent lines at two points as LnAnd LnCurvature is ρmAnd ρnRespectively making a circle Cr through the feature points1And Cr2So that their tangent is LmAnd LnRadius ofThe two circular function expressions resulting from the above constraints can be expressed as a circle Cr1:Fr1(x, y) is 0; circular Cr2:Fr2(x, y) ═ 0, and as described in steps (1) and (2), an interpolation function curve C can be used: the G2 free curve which is used as the determined two characteristic points and has controllable offset and curvature is used for adjusting the concavity and convexity of the controllable target free curve by the positive and negative of the control parameter mu, adjusting the offset of the controllable target curve by the slope of the tangent line, adjusting the curvature parameter and adjusting the smoothness of the connection of the free curve and the front and rear sections;
(III) a method for generating a continuous free curve of the open G2:
suppose that a series of feature points sampled in the modeling is P1,P2,P3…PnThen, the method for generating an open free curve passing through the series of feature points includes the following steps:
(1) determining a first edge feature point P1Tangent line Tl of1：
① when P1The tangent at (B) is predetermined in the modeling, and the predetermined tangent is the edge feature point P1Tangent line Tl of1；
② when P1The tangent line of (b) has no preset value in the modeling, it is determined by the following method:
per P1And P2Make a straight line intersect with point N1So that △ P1N1P2Is an isosceles triangle, then tangent line Tl1Is a straight line P1N1According to the constraint condition, the method realizes the use of P by using the characteristic that one-dimensional constraint condition is needed to determine the unique isosceles triangle1Tangent line Tl of1This constraint continuously adjusts the target free curve;
(2) determining P in sequence2,P3…PnTangent at point, the method is as follows:
in step (1), if the line Tl is cut1Case ① is met, P2Tangent line Tl of2The generation method comprises the following steps:
per P2Make a straight line P2N1So that P is2N1||P1N1(Tl1) Then, the tangent is obtained; tl2＝P2N1；
Subsequent sequential passing of Pi(i e (2, n)) is taken as a straight line PiNi-1So that P isiNi-1||Pi-1Pi+1Cross line tangent Tli-1In Ni-1Then, the tangent Tl is obtainedi＝PiNi-1；
At the termination feature point PnAt tangent line TlnThe generation method comprises the following steps:
per PnMake a straight line PnNn-1So that △ PnNn-1Pn-1Is an isosceles triangle, then Tln＝PnNn-1。
In conclusion, the tangent lines at all the characteristic points are determined;
(3) and (3) calculating the curvature radius of each characteristic point by the following method:
① edge feature point P1Radius of curvature ρ1The calculation method comprises the following steps:
auxiliary circle ⊙ P1N1P2The auxiliary circle is unique, assuming that its radius is r1Then at the feature point P1Radius of curvature ρ of1＝r1；
② middle feature point P2,P3,…,Pn-1Radius of curvature ρ of2,ρ3,…,ρn-1Calculating by adopting an adjacent average value method:
at a characteristic point Pi(i∈[2,n-1]) Here, make an auxiliary circle ⊙ PiNiPi+1Let its radius be riThen PiRadius of curvature of
③ terminating characteristic point PnRadius of curvature ρnThe calculating method of (2):
auxiliary circle ⊙ Pn-1Nn_1PnLet its radius be rnThen ρn＝rn；
In conclusion, the curvature radius of each characteristic point is calculated;
(4) adopting the method described in the second step, and adopting an interpolation function curve C through each characteristic point, wherein F (x, y) is equal to F1(x,y)·F2(x,y)+μ·(aox+boy+co)3When the value is equal to 0, the generation of the open G2 continuous free curve can be completed;
(IV) method for generating closed G2 continuous free curve:
assume a series of feature points P1,P2,P3…Pn(Pn＝P1) The whole process is the same as the generation method (III), but at the end point Pn(or P)1) Tangent line Tl ofi(i 1or n) and radius of curvature ρiThe following processes are required:
the invention has the beneficial effects that:
the method for efficiently generating the free curve in the three-dimensional modeling can generate a model of the free curve on the basis of extremely small resource occupancy rate of a computer, has the advantages of low resource occupancy rate, high calculation speed and high modeling efficiency, and most importantly, can realize the free control of the global curve at an end point and the local control at an intermediate characteristic point, and has high adaptability and flexibility.
Drawings
FIG. 1 is a general flow chart of a method for efficiently generating a free curve in three-dimensional modeling;
FIG. 2 is the generation and control effect of a G1 continuous free curve passing through two feature points;
FIG. 3 is the generation and control effect of a G2 continuous free curve passing through two feature points;
FIG. 4 is the effect of the generation of a continuous free curve of open G2;
fig. 5 shows the effect of the closed G2 continuous free curve.
Detailed Description
Referring to fig. 1, the invention relates to a method for efficiently generating a free curve in three-dimensional modeling, which comprises four curve generation methods of a G1 continuous free curve, a G2 continuous free curve, an open G2 continuous free curve and a closed G2 continuous free curve:
according to the generation of the G1 continuous free curve of two determined end points, in modeling, for the occasion that the continuity requirement of the free curve is not high or only the outline needs to be approximately constructed, a free curve generation method of determining the end points of the curve and the tangent lines thereof is adopted:
as shown in FIG. 2(a), assume P1，P2Is two feature points on the target free curve, the method for generating G1 continuous free curve according to these two feature points is as follows:
(1) over characteristic point P1And P2Auxiliary straight line L by pointo：lo(x,y)＝aox+boy+co＝0；
(2) Over characteristic point P1As an auxiliary straight line Lm：lm(x,y)＝amx+bmy+cm＝0；
(3) Over characteristic point P2As an auxiliary straight line Ln：ln(x,y)＝anx+bny+cn＝0；
(4) On the basis of the above-mentioned auxiliary line, a controllable function curve C: F (x, y) is inserted, which
In (1),namely, it is
Mu is a control parameter of the free curve C, w > 1 and is an integer, representing the number of elements of the function curve.
In the configuration of the target free curve, by adjusting the control parameter muThe flatness of the function curve C can be adjusted by the size, and the curve is more gentle when | mu | is larger, as shown in FIG. 2 (b); mu.s>At 0, the curve is convex, whereas the curve is concave, as shown in fig. 2 (c); by adjusting LmAnd lnThe slope of (c) can control the degree of bias of the curve.
Therefore, if two characteristic points can be determined in the modeling process, the continuous free curve of G1 with controllable parameters can be generated by adopting the steps. In an example application, let w be 2 (binary function curve), after generating the free curve, the free curve can be continuously controlled until it is closest to the target curve by the following parameters:
wherein k ism，knIs an auxiliary straight line LmAnd LnSlope of (2)
At this point, the generation of a continuous free curve according to the two end-determined G1 is completed.
And secondly, generating a G2 continuous free curve according to the two characteristic points, and adopting a free curve generating method according to the two free curves to obtain the continuity of more than G2 when the requirement on the continuity of the core service part or the curve is high. The specific process is as follows:
as shown in FIG. 3, assume P1，P2A free curve C which is two characteristic points on the target free curve and passes through the two points1:F1(x, y) ═ 0 and C2:F2If (x, y) is 0, then the method for generating G2 continuous free curves according to these two feature points is as follows:
(1) per P1，P2As an auxiliary straight line Lo：lo(x,y)＝aox+boy+co＝0；
(2) Per P1，P2F (x, y) ═ 0:
wherein,namely, it is
F(x,y)＝F1(x,y)·F2(x,y)+μ·(aox+boy+co)30; μ is the control parameter of the free curve C.
① the partial derivatives of the above equation are obtained:
because of the free curve F1(x, y) and an auxiliary line LoAll pass through the feature point P1Thus at P1The dotted equation above can be simplified as:
the compound can be obtained by the formula,
i.e. at the characteristic point P1Free curve C1:F1(x, y) 0 is normal to the interpolation function curve C: F (x, y) 0, and so, at P1Free curve C1Having a common tangent with the interpolation function curve C.
② calculating the characteristic point P of the interpolation function curve C, where F (x, y) is 0, according to the formula for calculating curvature1Curvature ρ:
the partial derivative and the second partial derivative are calculated for F (x, y) equal to 0, and the following formula is obtained:
at a characteristic point P1A subsidiary line LoAnd free curve C1Expression F of1Are both 0, so the above formula can be simplified as:
substituting the above formula into curvature calculation formula to obtain interpolation function curve C at characteristic point P1Curvature ρ ofcThe calculation is as follows:
i.e. the interpolation function curve C and the free curve C1At a characteristic point P1The same curvature, the same C and C2At a characteristic point P2The curvature is also the same.
(5) For a given feature point P1And P2Recording the controllable tangent lines at two points as LmAnd LnCurvature is ρmAnd ρn. Respectively making a circle Cr through the characteristic points1And Cr2So that their tangent is LmAnd LnRadius ofThe two circular function expressions resulting from the above constraints can be expressed as a circle Cr1:Fr1(x, y) is 0; circular Cr2:Fr2(x, y) ═ 0, interpolation function curves can be used as described in steps (1) and (2) The G2 free curve is used as two determined characteristic points and has controllable bias and curvature. As shown in fig. 3(a), the concavity and convexity of the target free curve can be controlled by adjusting the positive and negative of the control parameter μ, the bias of the target curve can be controlled by adjusting the slope of the tangent as shown in fig. 3(b), and the smoothness of the connection between the free curve and the front and rear segments as shown in fig. 3(c) can be adjusted by adjusting the curvature parameter.
(III) generation of open G2 continuous free curve, which is suitable for generation method of open G2 continuous free curve such as marking line;
suppose that a series of feature points sampled in the modeling is P1,P2,P3…PnThen, the method for generating an open free curve through the series of feature points, as shown in fig. 4, includes the following steps:
(1) determining a first edge feature point P1Tangent line Tl of1：
① when P1The tangent at (B) is predetermined in the modeling, and the predetermined tangent is the edge feature point P1Tangent line Tl of1；
② when P1The tangent line of (b) has no preset value in the modeling, it is determined by the following method:
per P1And P2Make a straight line intersect with point N1So that △ P1N1P2Is an isosceles triangle, then tangent line Tl1Is a straight line P1N1According to the constraint condition, the method realizes the use of P by using the characteristic that one-dimensional constraint condition is needed to determine the unique isosceles triangle1Tangent line Tl of1This constraint continuously adjusts the target free curve.
(2) Determining P in sequence2,P3…PnTangent at point, the method is as follows:
in the step of(1) In, if tangent line Tl1Case ① is met, P2Tangent line Tl of2The generation method comprises the following steps:
per P2Make a straight line P2N1So that P is2N1||P1N1(Tl1) Then, the tangent is obtained; tl2＝P2N1；
Subsequent sequential passing of Pi(i e (2, n)) is taken as a straight line PiNi-1So that P isiNi-1||Pi-1Pi+1Cross line tangent Tli-1In Ni-1Then, the tangent Tl is obtainedi＝PiNi-1；
At the termination feature point PnAt tangent line TlnThe generation method comprises the following steps:
per PnMake a straight line PnNn-1So that △ PnNn-1Pn-1Is an isosceles triangle, then Tln＝PnNn-1。
In conclusion, the tangents at all feature points are determined.
(3) On the basis of the steps, the curvature radius of each characteristic point is calculated by the following method:
① edge feature point P1Radius of curvature ρ1The calculation method comprises the following steps:
auxiliary circle ⊙ P1N1P2The auxiliary circle is unique, assuming that its radius is r1Then at the feature point P1Radius of curvature ρ of1＝r1。
② middle feature point P2,P3,…,Pn-1Radius of curvature ρ of2,ρ3,…,ρn-1Calculating by adopting an adjacent average value method:
at a characteristic point Pi(i∈[2,n-1]) Here, make an auxiliary circle ⊙ PiNiPi+1Let its radius be riThen PiRadius of curvature of
③ terminating characteristic point PnRadius of curvature ρnThe calculating method of (2):
auxiliary circle ⊙ Pn-1Nn-1PnLet its radius be rnThen ρn＝rn。
In conclusion, the curvature radius at each feature point is calculated.
(4) The method is used for generating a continuous free curve by a method (II) G2, and an interpolation function curve C is adopted by passing through each characteristic point, wherein F (x, y) is equal to F1(x,y)·F2(x,y)+μ·(aox+boy+co)3When the value is 0, the generation of the open G2 continuous free curve is completed.
And (IV) the generation of the closed G2 continuous free curve is suitable for the generation method of the closed G2 continuous free curve of special equipment, facilities, marks or models and the like.
For the overall construction scene of the key equipment shape, the generation of a closed G2 continuous free curve is needed, and the method is as follows:
assume a series of feature points P1,P2,P3…Pn(Pn＝P1) The whole process is the same as the generation method of the (three) open G2 continuous free curve, but at the end point Pn(or P)1) Tangent line Tl ofi(i 1or n) and radius of curvature ρiThe following processes are required:
the resulting effect of the closed G2 continuous free curve is shown in fig. 5.
In conclusion, the method for efficiently generating the free curve in the three-dimensional modeling according to the present invention is completed. The invention can generate the model of the free curve on the basis of the minimum computer resource occupancy rate, has the advantages of low resource occupancy rate, high calculation speed and high modeling efficiency, can realize the global curve free control at an end point, can realize the local control at an intermediate characteristic point, and has high adaptability and flexibility.

Claims (1)

1. A method for efficiently generating a free curve in three-dimensional modeling is characterized by comprising the following steps: the method comprises the generation methods of four curves, namely a G1 continuous free curve, a G2 continuous free curve, an open G2 continuous free curve and a closed G2 continuous free curve:
the generation method of the G1 continuous free curve according to two determined end points comprises the following steps:
suppose P1，P2Is two feature points on the target free curve, the method for generating G1 continuous free curve according to these two feature points is as follows:
(1) over characteristic point P1And P2Auxiliary straight line L by pointo：lo(x,y)＝aox+boy+co＝0；
(2) Over characteristic point P1As an auxiliary straight line Lm：lm(x,y)＝amx+bmy+cm＝0；
(3) Over characteristic point P2As an auxiliary straight line Ln：ln(x,y)＝anx+bny+cn＝0；
(4) Inserting controllable function curve C, F (x, y),
wherein,namely, it is
Mu is a control parameter of the free curve C, w is more than 1 and is an integer, and represents the element number of the function curve;
in the configuration of the target free curve, the flatness of the function curve C can be adjusted by adjusting the size of the control parameter mu, and the curve is more gentle when | mu | is larger; mu.s>When 0, the curve is convex, otherwise, the curve is concave; by adjusting LmAnd lnThe slope of the curve can control the bias degree of the curve;
after the G1 continuous free curve with controllable parameters is generated, the free curve can be continuously controlled by the following parameters until the free curve is closest to the target curve:
wherein k ism，knIs an auxiliary straight line LmAnd LnSo far, the generation of a continuous free curve according to the G1 with two determined end points is completed;
(II) a method for generating a G2 continuous free curve according to the two free curves:
suppose P1，P2Is two on the target free curveCharacteristic point and free curve C passing through the two points1:F1(x, y) ═ 0 and C2:F2If (x, y) is 0, then the method for generating G2 continuous free curves according to these two feature points is as follows:
(1) per P1，P2As an auxiliary straight line Lo：lo(x,y)＝aox+boy+co＝0；
(2) Per P1，P2F (x, y) ═ 0:
wherein,namely, it is
F(x,y)＝F1(x,y)·F2(x,y)+μ·(aox+boy+co)30; mu is a control parameter of the free curve C;
① the partial derivatives of the above equation are obtained:
because of the free curve F1(x, y) and an auxiliary line LoAll pass through the feature point P1Thus at P1The dotted equation above can be simplified as:
from the above formula, (F)x(x,y)，
I.e. at the characteristic point P1Free curve C1:F1(x, y) 0 is normal to the interpolation function curve C: F (x, y) 0, and so, at P1Free curve C1In common with the interpolation function curve CThe tangent line of (a);
② calculating the characteristic point P of the interpolation function curve C, where F (x, y) is 0, according to the formula for calculating curvature1Curvature ρ:
the partial derivative and the second partial derivative are calculated for F (x, y) equal to 0, and the following formula is obtained:
substituting the above formula into curvature calculation formula to obtain interpolation function curve C at characteristic point P1Curvature ρ ofcThe calculation is as follows:
i.e. the interpolation function curve C and the free curve C1At a characteristic point P1The same curvature, the same C and C2At a characteristic point P2The curvature is also the same;
(3) for a given feature point P1And P2Recording the controllable tangent lines at two points as LmAnd LnCurvature is ρmAnd ρnRespectively making a circle Cr through the feature points1And Cr2So that their tangent is LmAnd LnRadius ofThe two circular function expressions resulting from the above constraints can be expressed as a circle Cr1:Fr1(x, y) is 0; circular Cr2:Fr2(x, y) ═ 0, interpolation function curves can be used as described in steps (1) and (2) As the G2 free curve with two determined characteristic points and controllable bias and curvature, the positive and negative of the control parameter mu are adjusted to control the target free curveThe concave-convex characteristic of the curve, the offset of the target curve can be controlled by adjusting the slope of the tangent line, the curvature parameter can be adjusted, and the smoothness of the connection of the free curve and the front and rear sections can be adjusted;
(III) a method for generating a continuous free curve of the open G2:
suppose that a series of feature points sampled in the modeling is P1,P2,P3…PnThen, the method for generating an open free curve passing through the series of feature points includes the following steps:
(1) determining a first edge feature point P1Tangent line Tl of1：
① when P1The tangent at (B) is predetermined in the modeling, and the predetermined tangent is the edge feature point P1Tangent line Tl of1；
② when P1The tangent line of (b) has no preset value in the modeling, it is determined by the following method:
per P1And P2Make a straight line intersect with point N1So that △ P1N1P2Is an isosceles triangle, then tangent line Tl1Is a straight line P1N1According to the constraint condition, the method realizes the use of P by using the characteristic that one-dimensional constraint condition is needed to determine the unique isosceles triangle1Tangent line Tl of1This constraint continuously adjusts the target free curve;
(2) determining P in sequence2,P3…PnTangent at point, the method is as follows:
in step (1), if the line Tl is cut1Case ① is met, P2Tangent line Tl of2The generation method comprises the following steps:
per P2Make a straight line P2N1So that P is2N1||P1N1(Tl1) Then, the tangent is obtained; tl2＝P2N1；
Subsequent sequential passing of Pi(i e (2, n)) is taken as a straight line PiNi-1So that P isiNi-1||Pi-iPi+1Cross line tangent Tli-1In Ni-1Then, the tangent Tl is obtainedi＝PiNi-1；
At the termination feature point PnAt tangent line TlnThe generation method comprises the following steps:
per PnMake a straight line PnNn-1So that △ PnNn-1Pn-1Is an isosceles triangle, then Tln＝PnNn-1。
In conclusion, the tangent lines at all the characteristic points are determined;
(3) and (3) calculating the curvature radius of each characteristic point by the following method:
① edge feature point P1Radius of curvature ρ1The calculation method comprises the following steps:
auxiliary circle ⊙ P1N1P2The auxiliary circle is unique, assuming that its radius is r1Then at the feature point P1Radius of curvature ρ of1＝r1；
② middle feature point P2,P3,…,Pn-1Radius of curvature ρ of2,ρ3,…,ρn-1Calculating by adopting an adjacent average value method:
at a characteristic point Pi(i∈[2,n-1]) Here, make an auxiliary circle ⊙ PiNiPi+1Let its radius be riThen PiRadius of curvature of
③ terminating characteristic point PnRadius of curvature ρnThe calculating method of (2):
auxiliary circle ⊙ Pn-1Nn-1PnLet its radius be rnThen ρn＝rn；
In conclusion, the curvature radius of each characteristic point is calculated;
(4) adopting the method described in the second step, and adopting an interpolation function curve C through each characteristic point, wherein F (x, y) is equal to F1(x,y)·F2(x,y)+μ·(aox+boy+co)3Finish when equal to 0Generation of a continuous free curve in the open G2 form;
(IV) method for generating closed G2 continuous free curve:
assume a series of feature points P1,P2,P3…Pn(Pn＝P1) The whole process is the same as the generation method (III), but at the end point Pn(or P)1) Tangent line Tl ofi(i 1orn) and radius of curvature ρiThe following processes are required:
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113419409A (en) * 2021-07-16 2021-09-21 中国科学院长春光学精密机械与物理研究所 Method and device for controlling free-form surface

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113419409A (en) * 2021-07-16 2021-09-21 中国科学院长春光学精密机械与物理研究所 Method and device for controlling free-form surface
CN113419409B (en) * 2021-07-16 2022-09-20 中国科学院长春光学精密机械与物理研究所 Method and device for controlling free-form surface

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