CN114936389B - Mean camber line structure and geometric feature segmentation method for section line of blade - Google Patents
Mean camber line structure and geometric feature segmentation method for section line of blade Download PDFInfo
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Abstract
The invention provides a mean camber line structure and geometric feature segmentation method for a blade section line, which solves the problems that the existing mean camber line structure method cannot judge whether an inscribed circle exists or not and cannot process iteration interval selection under the condition of severe curvature change of an area near a blade edge head, and a method adopted by geometric feature segmentation cannot be suitable for manufacturing states of large blade edge head errors, irregular profile distribution and severe curvature change of the blade edge head. According to the method, a mean camber line is obtained by traversing the measuring blade, the circle center is extracted under the constraint of the residual arc length and the increment of a mean camber line construction point based on the geometric information of a design blade edge, and then the measuring section line is divided based on the circle center projected on the mean camber line and the arc angle of the design blade edge. The method is not influenced by errors introduced in the processes of blade manufacturing, blade measurement and the like, the feature segmentation of the blade in the front edge and the rear edge regions has higher precision, and the method is suitable for more blade types and provides more effective reference for the reconstruction of the blade geometric model.
Description
Technical Field
The invention belongs to the technical field of geometric modeling of an aero-engine blade, and particularly relates to a camber line structure and geometric feature segmentation method of a blade section line.
Background
The blade is used as a key part of the aero-engine, has the characteristics of complex appearance, high precision requirement, high manufacturing difficulty and the like, and the thrust performance of the whole engine is greatly influenced by the blade which does not meet the design precision requirement. Therefore, the geometric model reconstruction of the manufactured blade is an effective means for evaluating the manufacturing precision of the blade and improving the process level.
The blade geometric model reconstruction is generally completed by constructing a blade section line and performing sweep on the basis of the measured data. Among them, how to construct the section line based on the measurement data is the key of the blade geometric model reconstruction technique. The section line of the blade consists of curves of a blade basin, a blade back, a front edge and a rear edge, the front edge and the rear edge are used as edge head parts of the section line and are the most important geometrical characteristics, and the front edge and the rear edge are divided from the curves of the blade basin and the blade back by separating points. The mean camber line is used as a fitting line of the centers of inscribed circles between the curves of the leaf basin and the leaf back, is a skeleton of a section line of the leaf, is a basis for constructing a geometric model of the leaf, and particularly has a direct relation with geometric feature segmentation points through the inscribed circles at the endpoints of the mean camber line.
The conventional camber line construction method cannot completely solve the problem of existence of an inscribed circle on one hand, and cannot solve the problem of selection of an iteration interval under the condition of severe curvature change of an area near the edge head on the other hand. In addition, in the aspect of geometric feature segmentation, both the curvature change analysis method and the edge head included angle judgment method are judged by an empirical threshold value, and the method cannot be applied to real manufacturing states of large blade edge head errors, irregular profile distribution, severe edge head curvature change and the like.
Disclosure of Invention
The invention provides a camber line structure and geometric feature segmentation method for a blade section line, and aims to solve the technical problems that an existing camber line construction method cannot judge whether an inscribed circle exists or not and cannot process iteration interval selection under the condition that curvature of an area near a blade edge head changes violently, and a method adopted by geometric feature segmentation cannot be suitable for manufacturing states of blades with large blade edge head errors, irregular profile distribution and violent and real changes in the curvature of the blade edge head.
In order to achieve the purpose, the technical scheme provided by the invention is as follows:
a camber line structure and geometric feature segmentation method of a blade section line is characterized by comprising the following steps:
Step 2, calculating all discrete points of the discrete point concentration edge head approximate area in the section line curve s c Average value of the above arc length parameter and using the value to cross-section the line curve s c Upper corresponding points, cross-sectional linesCurve s c Is divided into curves s 1 And s 2 ；
if the curve s can be obtained 2 The tangent circle is then marked as the inscribed circle c 1 And with the inscribed circle c 1 The center of the circle is taken as the initial construction point of the mean camber line, and the inscribed circle c is respectively taken as 1 And curve s 1 And s 2 The intersection of (A) is denoted as p 1 And q is 1 ；
If not, curve s is used 2 Discrete point q having the furthest distance from the edge head b For one point on the circle to draw a curve s 1 Tangent circle, which is designated as inscribed circle c 1 And with the inscribed circle c 1 The center of the circle is used as the initial construction point of the mean camber line, and an inscribed circle c is respectively formed 1 And curve s 1 And s 2 Is denoted as p 1 And q is 1 ；
Step 4, from curve s 1 Upper and point p 1 The curves s are sequentially obtained from adjacent discrete points along the direction close to the edge head 1 Upper discrete point and curve s 2 Inscribed circle c of 2 Inscribed circle c 3 8230, inscribed circle c 2 The center of the circle is used as a second construction point, a third construction point and a third construction point of a mean camber line (8230), and the inscribed circle c is obtained 2 Respectively associated with curve s 1 And s 2 Cross point p of 2 And q is 2 Inscribed circle c 3 Respectively with curve s 1 And s 2 Cross point p of 3 And q is 3 、…；
Up to the point of intersection p i And q is i At curve s c The arc length of the upper blade is less than or equal to the arc length of the designed blade edge head, and step 5 is executed;
or, until | c is satisfied i -c i-1 |＜|c i-1 -c i-2 I/2 or (c) i -c i-1 )·(c i-1 -c i-2 ) If not less than 0, executing step 6;
wherein i =2,3,4.. A, a is the total number of inscribed circles in step 4;
step 5, removing the inscribed circle c i And extracting the curve s c The middle arc length parameter is at point p i And q is i Points in between, and performing circle fitting to obtain a fitting circle c f And the circle center thereof, skipping to the step 7;
step 6, removing the inscribed circle c i And c is i The center of the circle is used as a fitting circle c f The center of the circle;
step 7, fitting the initial construction point, the second construction point, the third construction point, \8230; (i-1) th construction point into a mean camber line m, and fitting a circle c f The center of the circle of the arc line is projected along the normal direction of the mean arc line m to obtain a projection point c e Taking the point as the terminal point of the mean camber line to finish the mean camber line structure of the section line of the blade;
step 8, obtaining c e Rotating the vector of tangent direction t on the mean camber line clockwise and counterclockwise by half of arc angle theta at the design edge and corresponding to the section line curve s c Intersecting to obtain two separation points a for measuring the geometric characteristics of the edge head of the section 1 And a 2 According to the point of separation a 1 And a 2 And cutting the section line of the blade.
Further, in step 3), the curve s 1 Discrete point with upper edge most distant from position of rim head
For a point on the circle, draw the curve s 2 The tangent circle is specifically:
defining a curve s 1 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 2 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the crossing point p being plotted as a curve s s Normal l, calculating normal l and target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curve s t And its corresponding point on the normal line l is the point q t ；
The second step is that: point p and point q are pointed to t Dispersing the line segments between the points to obtain a discrete point d, and iteratively calculating the point d and the curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the point d, a normal n of the point q and a vector are determined
Whether the directions of the two are consistent; when the two are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent circle; when the two are not consistent, the position of the test point p is matched with the target curve s t The tangent circle acquisition fails.
Further, in step 3), the curve s is used 2 Discrete point with upper edge most distant from the position of the rim head
For one point on the circle to draw a curve s 1 The tangent circle is specifically:
defining a curve s 2 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 1 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the crossing point p being plotted as a curve s s Normal l, calculating normal l and target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curves t And its corresponding point on the normal line l is the point q t ；
The second step is that: point p and q are opposite t Dispersing the line segments between the points to obtain a discrete point d, and iteratively calculating the point d and the curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the point d, a normal n to the point q and a vector
If the directions of the test points p are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent circle.
Further, in step 4, the curve s is obtained 1 Upper discrete point and curve s 2 The inscribed circle of (1) is specifically as follows:
defining a curve s 1 Is a source curve s s Curve s 1 Discrete points on it are the source curve s s Test point p, curve s of 2 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the crossing point p being plotted as a curve s s Calculating the normal l and the target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curve s t And its corresponding point, the corresponding point on the normal l being the point q t ；
The second step: point p and point q are pointed to t Dispersing line segments between the points to obtain a discrete point d, and iteratively calculating the point d and a curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the point d, a normal n of the point q and a vector are determined
Whether the directions of the two are consistent;when the two are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent circle; when the two are not consistent, the calculation of the inscribed circle c at the test point p fails.
Compared with the prior art, the invention has the advantages that:
1. according to the method, a camber line is obtained by traversing the front edge or the rear edge area of the measuring blade, the circle center is extracted under the constraint of the residual arc length and the increment of the camber line construction point based on the geometric information of the design blade edge head, and the measured section line is divided based on the circle center projected on the camber line and the arc angle at the design blade edge head. The method is not influenced by errors introduced by the processes of manufacturing and measuring the blade and the like, the characteristic segmentation of the blade in the front edge and the rear edge area has higher precision, and the method is suitable for more blade types and provides more effective reference for the reconstruction of the geometric model of the blade.
2. According to the invention, by improving the iterative computation method of the inscribed circle, the extraction success rate and accuracy of the inscribed circle in the area near the edge head are improved, the accurate extraction of the fitting discrete points of the edge head and the high-precision segmentation of the geometric characteristics are realized, and meanwhile, the method has higher algorithm stability and is suitable for the diversified error distribution forms after the blade is manufactured.
Drawings
FIG. 1 is a schematic view of a camber line structure and geometric feature segmentation method for a section line of a blade according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a single curve divided into two curves in an embodiment of the camber line configuration and geometric feature division method of the cross-sectional line of the blade of the present invention;
FIG. 3 shows a method for dividing the camber line and geometric feature of a blade section line according to an embodiment of the present invention 1 Successfully calculating an inscribed circle schematic diagram;
FIG. 4 shows a method for dividing the camber line and geometric feature of a blade section line according to an embodiment of the present invention 2 Successfully calculating an inscribed circle schematic diagram;
FIG. 5 is a schematic diagram of a geometric feature separation point obtained in an embodiment of the camber line configuration and geometric feature segmentation method for a blade section line according to the present invention;
FIG. 6 is a schematic diagram showing that 2 intersection points exist between the normal of the test point and the target curve in the method for constructing the mean camber line and dividing the geometric features of the blade section line according to the present invention;
FIG. 7 is a schematic diagram of the successful calculation of inscribed circles and the absence of intersection points between the normals of the test points and the target curves in the method for constructing the mean camber line and dividing the geometric features of the section lines of the blade according to the present invention;
FIG. 8 is a failure diagram of the method for constructing the mean camber line and dividing the geometric features of the blade section line according to the present invention, wherein there is no intersection point between the test point normal and the target curve and the inscribed circle is calculated.
Detailed Description
The invention is described in further detail below with reference to the figures and specific embodiments.
The embodiment of the invention provides a mean camber line structure and geometric feature segmentation method of a blade section line, wherein a blade measurement data point is from a blue light scanner device, and the method comprises the following steps:
step 2, calculating all discrete points of the discrete point concentration edge head approximate area in the section line curve s c Average value of the above arc length parameter and using the value to cross-section the line curve s c At the corresponding point a, the section line curve s c Is divided into curves s 1 And s 2 See, fig. 2;
if the curve s can be obtained 2 The tangent circle is then designated as the inscribed circle c 1 And with the inner sideTangent circle c 1 The center of the circle is used as the initial construction point of the mean camber line, and an inscribed circle c is respectively formed 1 And curve s 1 And s 2 The intersection of (A) is denoted as p 1 And q is 1 See, fig. 3;
if not, show, curve s 1 End point p of b And curve s 2 Fails to form the curve s 2 Discrete point q having the furthest distance from the edge head b For a point on the circle, draw the curve s 1 Tangent circle, which is designated as inscribed circle c 1 And with the inscribed circle c 1 The center of the circle is used as the initial construction point of the mean camber line, and an inscribed circle c is respectively formed 1 And curve s 1 And s 2 The intersection of (A) is denoted as p 1 And q is 1 See, fig. 4;
step 4, continuously traversing the curve s 1 At the next discrete point of time, and the curve s is simultaneously obtained 2 The center of the inscribed circle c is used as a construction point of the mean camber line; in particular, from curve s 1 Upper and point p 1 The curves s are sequentially obtained from adjacent discrete points along the direction close to the edge head 1 Upper discrete point and curve s 2 Inscribed circle c of 2 Inscribed circle c 3 8230, inscribed circle c 2 Inscribed circle c 3 The center of the circle is used as a second construction point, a third construction point and a construction point of '8230' \ of a mean camber line, and an inscribed circle c is obtained respectively 2 Respectively with curve s 1 And s 2 Cross point p of 2 And q is 2 Inscribed circle c 3 Respectively with curve s 1 And s 2 Cross point p of 3 And q is 3 、…；
Until the point is traversed to the ith point, an inscribed circle c i Respectively with curve s 1 And s 2 Cross point p of i And q is i At curve s c Stopping traversing when the arc length of the blade edge is less than or equal to the arc length of the designed blade edge, and jumping to the step 5;
or until the point i is traversed, the vector formed by the centers of adjacent inscribed circles meets | c i -c i-1 |＜|c i-1 -c i-2 I/2 or (c) i -c i-1 )·(c i-1 -c i-2 ) If the value is less than 0, stopping traversing and jumping to the step 6;
step 5, removing the inscribed circle c i And extracting the curve s c Medium arc length parameter at point p i And q is i Points in between, and performing circle fitting to obtain a fitting circle c f And the circle center thereof, skipping to the step 7;
step 6, removing the inscribed circle c i And c is i The center of the circle is used as a fitting circle c f The center of the circle;
step 7, fitting the starting construction point, the second construction point, the third construction point, \8230 \ (i-1) th construction point (namely the centers of the other inscribed circles, namely all the centers in the dotted line frame D in the figure 5) into a mean camber line m, and fitting a fitting circle c f The center of the circle of the projection lens is projected along the normal direction of the mean camber line m to obtain a projection point c e Taking the point as the terminal point of the mean camber line to finish the mean camber line structure of the section line of the blade;
step 8, obtaining c e The tangent direction t on the mean camber line is rotated clockwise and anticlockwise to design half of the arc angle theta at the edge head and to the section line curve s c Intersecting to obtain two separated points a for measuring the geometric characteristics of the section edge head 1 And a 2 See FIG. 5 for a separation point a 1 And a 2 And cutting the section line of the blade.
Further, in step 3, the curve s 1 Discrete point with upper edge most distant from the position of the rim head
For one point on the circle to draw a curve s 2 When the circles are tangent, define a curve s 1 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 2 Is a target curve s t ；
By curve s 2 Discrete point with upper edge most distant from position of rim head
For a point on the circle, draw the curve s 1 Tangent circle time, defining curve s 2 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 1 Is a target curve s t ；
In step 4, curve s is obtained 1 Upper discrete point and curve s 2 When inscribing a circle, a curve s is defined 1 Is a source curve s s Curve s, curve s 1 The discrete point on is the source curve s s Test point p, curve s of 2 Is a target curve s t
The iterative calculation method of the inscribed circle mainly comprises the following three steps:
the first step is as follows: for the source curve s s And a test point p on the curve, the crossing point p being plotted as a curve s s Normal l, calculating normal l and target curve s t The number of intersections; there are 3 cases: when there are 1 intersection, the point is denoted as q t (ii) a When there are 2 intersections, the point closest to the point p is q t Points, see FIG. 6; when there is no intersection point, then the normal l and the target curve s are calculated t And its corresponding point, the corresponding point on the normal l being the point q t See, fig. 7;
the second step: point p and point q are pointed to t Dispersing line segments between the points to obtain a discrete point d, and iteratively calculating the point d and a curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the point d, a normal n of the point q and a vector are determined
Whether the directions of (A) and (B) are consistent; when the two are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent inscribed circle; when the two are not consistent, the position of the test point p is matched with the target curve s t The tangent inscribed circle failed to acquire, as shown in fig. 8.
The method comprises the steps of combining design parameters of the blade, obtaining a mean camber line by traversing a front edge or a rear edge area of the measured blade, extracting the center of a front edge or a rear edge under the constraint of residual arc length and the increment of a mean camber line construction point based on geometric information of a design edge head, and dividing a measurement section line based on the center of a circle projected on the mean camber line and an arc angle at the design blade edge head. The method is not influenced by errors introduced by the processes of manufacturing and measuring the blade and the like, the characteristic segmentation of the blade in the front edge and the rear edge area has higher precision, and the method is suitable for more blade types and provides more effective reference for the reconstruction of the geometric model of the blade.
The above description is only for the preferred embodiment of the present invention and does not limit the technical solution of the present invention, and any modifications made by those skilled in the art based on the main technical idea of the present invention belong to the technical scope of the present invention.
Claims (4)
1. A method for dividing camber line structure and geometric features of section lines of blades is characterized by comprising the following steps:
step 1, obtaining a marginal head approximate area of a cross section and discrete points of an area near the marginal head from blade measurement data to form a discrete point set, and obtaining a section line curve s through fitting c ；
Step 2, calculating all discrete points of the discrete point concentration edge head approximate area in the section line curve s c Average value of the above arc length parameter and using the value to cross-section the line curve s c At corresponding points, cross-sectional line curve s c Is divided into curves s 1 And s 2 ；
Step 3, curve s 1 Discrete point p having the greatest distance from the edge position b For a point on the circle, draw the curve s 2 A tangent circle;
if the curve s can be obtained 2 The tangent circle is then designated as the inscribed circle c 1 And with the inscribed circle c 1 The center of the circle of (a) is used as the start of the mean camber lineStructural points, respectively inscribed the circle c 1 And curve s 1 And s 2 The intersection of (A) is denoted as p 1 And q is 1 ；
If not, curve s 2 Discrete point q having the furthest distance from the edge head b For one point on the circle to draw a curve s 1 Tangent circle, which is designated as inscribed circle c 1 And with the inscribed circle c 1 The center of the circle is used as the initial construction point of the mean camber line, and an inscribed circle c is respectively formed 1 And curve s 1 And s 2 Is denoted as p 1 And q is 1 ；
Step 4, from curve s 1 Upper and point p 1 The curves s are sequentially obtained from adjacent discrete points along the direction close to the edge head 1 Upper discrete point and curve s 2 Inscribed circle c of 2 Inscribed circle c 3 823060, and is inscribed with a circle c 2 Inscribed circle c 3 The circle center of the arc line is used as a second construction point, a third construction point and a structure point of 82309 of the mean arc line, and an inscribed circle c is obtained respectively 2 Respectively with curve s 1 And s 2 Cross point p of 2 And q is 2 Inscribed circle c 3 Respectively associated with curve s 1 And s 2 Cross point p of 3 And q is 3 、…；
Up to the point of intersection p i And q is i At curve s c The arc length of the upper blade is less than or equal to the arc length of the designed blade edge head, and step 5 is executed;
or, until | c is satisfied i -c i-1 |＜|c i-1 -c i-2 I/2 or (c) i -c i-1 )·(c i-1 -c i-2 ) If not less than 0, executing step 6;
wherein i =2,3,4.. A, a is the total number of inscribed circles in step 4;
step 5, removing the inscribed circle c i And extracting the curve s c The middle arc length parameter is at point p i And q is i Points in between, and performing circle fitting to obtain a fitting circle c f And the circle center thereof, skipping to the step 7;
step 6, removing the inscribed circle c i And c is i The center of the circle is used as a fitting circle c f The center of the circle;
step 7, fitting the initial construction point, the second construction point, the third construction point, \8230 \ 8230;. The (i-1) th construction point into a mean arc line m, and fitting a circle c f The center of the circle of the arc line is projected along the normal direction of the mean arc line m to obtain a projection point c e Taking the point as the terminal point of the mean camber line to finish the mean camber line structure of the section line of the blade;
step 8, obtaining c e Rotating the vector of tangent direction t on the mean camber line clockwise and counterclockwise by half of arc angle theta at the design edge and corresponding to the section line curve s c Intersecting to obtain two separated points a for measuring the geometric characteristics of the section edge head 1 And a 2 According to the point of separation a 1 And a 2 And cutting the section line of the blade.
2. The method for constructing the mean camber line and dividing the geometric characteristics of the section line of the blade according to claim 1, wherein: in step 3, the curve s 1 Discrete point with upper edge most distant from the position of the rim head
For a point on the circle, draw the curve s 2 The tangent circle is specifically:
defining a curve s 1 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 2 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the curve s being drawn by passing the point p s Normal l, calculating normal l and target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curve s t And its corresponding point, the corresponding point on the normal l being the point q t ；
The second step is that: point p and q are opposite t Dispersing line segments between the points to obtain a discrete point d, and iteratively calculating the point d and a curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the point d, a normal n of the point q and a vector are determined
Whether the directions of (A) and (B) are consistent; when the two are consistent, the point d is the point of the test point p and the target curve s t The center of the tangent circle; when the two are not consistent, the position of the test point p is matched with the target curve s t The tangent circle acquisition fails.
3. The method for constructing the mean camber line and dividing the geometric characteristics of the section line of the blade according to claim 1, wherein: in step 3, the curve s 2 Discrete point with upper edge most distant from position of rim head
For one point on the circle to draw a curve s 1 The tangent circle is specifically:
defining a curve s 2 Is a source curve s s Discrete points
Is a source curve s s Test point p, curve s of 1 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the curve s being drawn by passing the point p s Calculating the normal l and the target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curve s t And its corresponding point on the normal line l is the point q t ；
The second step is that: point p and point q are pointed to t Dispersing the line segments between the points to obtain a discrete point d, and iteratively calculating the point d and the curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a point q nearest to the point d in the curve st, a normal n of the point q and a vector
If the directions of the test points p are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent circle.
4. The method for constructing the mean camber line and dividing the geometric characteristics of the section line of the blade according to claim 1, wherein: in step 4, the curve s is obtained 1 Upper discrete point and curve s 2 The inscribed circle specifically is as follows:
defining a curve s 1 Is a source curve s s Curve s, curve s 1 Discrete points on it are the source curve s s Test point p, curve s of 2 Is a target curve s t ；
The first step is as follows: for the source curve s s And a test point p on the curve, the curve s being drawn by passing the point p s Calculating the normal l and the target curve s t The number of intersections; when there are 1 intersection, the point is denoted as q t When there are 2 intersections, the point closest to the point p is q t Point, when there is no intersection point, then calculate the normal l and the target curve s t And its corresponding point, the corresponding point on the normal l being the point q t ；
The second step is that: point p and point q are pointed to t Dispersing the line segments between the points to obtain a discrete point d, and iteratively calculating the point d and the curve s t Until the distance between the point d and the point p is equal to the distance between the point d and the point p, ending the iteration;
the third step: obtaining a curve s t A point q closest to the intermediate distance point d, and a normal n and a vector of the point q are determined
Whether the directions of (A) and (B) are consistent; when the two are consistent, the point d is the point where the test point p is located and the target curve s t The center of the tangent circle; when the two are not consistent, the calculation of the inscribed circle c at the test point p fails.
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