A linear path smoothing method based on airthoid and arc

Abstract

The present invention relates to the field of motion planning technology, and specifically discloses a linear path smoothing method based on Airthoid and circular arc, which specifically includes: step one, according to the approximate error constraint and the no self-intersection constraint, determine the solution set of the curvature radius extreme value and the first tangent angle of the smooth curve; step two, calculate the solution set of the parameter combination that satisfies the tangent continuity, curvature continuity and position continuity constraints; step three, calculate the intersection of solution set one and solution set two to obtain solution set three; judge whether solution set three is empty, if so, run step five; if not, run step four; step four, calculate the optimal parameter combination in solution set three, and then run step six; step five: generate a smooth curve parameter combination that does not contain circular arcs according to the no self-intersection constraint, and then run step six; step six, analytically calculate the curvature extreme value and arc length of the smooth curve. The present invention has the advantages of being based on arc length parameter expression, controllable contour error, and analytical expression of curvature extreme value and arc length.

Description

Technical Field

[0001] The present invention relates to the technical field of motion planning, and in particular to a linear path smoothing method based on airthoid and circular arc. Background Art

[0002] During CNC machining, the linear path of the tool is tangentially discontinuous at the connection point, requiring the CNC machine tool to reduce its speed to zero at that point. This frequent start and stop significantly reduces machining efficiency and causes chatter, making it difficult to give full play to the advantages of high-speed and high-precision CNC machining of CNC machine tools. Inserting high-order continuous curves at corners to eliminate the tangential discontinuity of the connection point is expected to improve the movement speed and efficiency of the machine tool.

[0003] Reference 1 "Xiaoyong H, Fei Z, Tao T, et al. A newly developed corner smoothing methodology based on clothoid splines for high speed machine tools [J]. Robotics and Computer-Integrated Manufacturing, 2021, 70." discloses a spatial Airthoid curve based on arc length parameter expression and C3 continuity, which has the advantages of controllable contour error, curvature extreme value and arc length analytical expression, but on this basis, the curvature extreme value can be further reduced.

[0004] Reference 2 "Xu F, Sun YA circumscribed corner rounding method based on double cubic B-splines for a five-axis linear tool path [J]. The International Journal of Advanced Manufacturing Technology, 2018, 94 (1-4): 451-462." discloses a G2 continuous local smoothing method. This method uses a circumscribed cubic B-spline curve to smooth the corners formed by straight line segments. Although it effectively reduces the extreme curvature, it cannot analytically express the approximation error.

[0005] The typical characteristics of the above literature are: although the smoothing curve can achieve the smooth transition of continuous straight line segments, the smoothing method in the literature has a predefined fixed ratio constraint to determine the parameters of the transition spline curve. However, the curvature distribution of the transition spline has a direct impact on the transition speed and processing efficiency. A smaller curvature extreme value means that the tool can pass the corner at a faster speed and achieve high-efficiency processing. Therefore, it is of great significance to further reduce the curvature extreme value of the smoothing curve. The methods involved in the above literature do not consider the curvature optimization factor, and the obtained smoothing curve is difficult to further improve the processing efficiency. Summary of the invention

[0006] In order to solve the above technical problems, the present invention provides a linear path smoothing method based on Airthoid and circular arc.

[0007] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0008] A linear path smoothing method based on Airthoid and arc to construct a smooth curve To correct the straight line segment P in the tool path i-1 P i and P i P i+1 The angle ∠P i Smoothing is performed, and the smoothing curve is used as the actual machining path of the tool, 1≤i≤N, N is the total number of corners in the tool path; the smoothing curve Composed of two symmetrical Airthoid curves and and a i Angle bisector symmetric arc Points B0 and B1 are respectively the smoothing curves The entry and exit points are smoothed using curves Replace the straight line segment B0P i and the straight line segment P i B1, the length of the straight line segment replaced by the smooth curve is recorded as the transition length; C0 and C1 points are two Airthoid curves respectively. and The intersection of the arc and the tangent of the arc at the intersection C0 and the straight line P i-1 P i The angle between the tangent line at the midpoint of the arc and the straight line P is defined as the first tangent angle γ; i-1 P i The angle between is defined as the second tangent angle β; the linear path smoothing method specifically includes the following steps:

[0009] Step 1: According to the approximate error constraint and the no self-intersection constraint, determine the solution set of the extreme value r of the curvature radius of the smooth curve and the first tangent angle γ, which is recorded as solution set 1;

[0010] Step 2: According to the constraints of tangential continuity, curvature continuity and position continuity, calculate the solution set of the parameter combination (r, γ), which is recorded as solution set 2;

[0011] Step 3: Calculate the intersection of solution set 1 and solution set 2 to obtain solution set 3; determine whether solution set 3 is empty, if so, run step 5; if not, run step 4;

[0012] Step 4: Based on the principle of maximizing the speed extreme value, calculate the optimal parameter combination (r, γ) in solution set 3 and obtain the solution (r n ,γ n ), then run step six;

[0013] Step 5: Generate a smooth curve parameter combination that does not contain arcs according to the no self-intersection constraint, and get the solution (r n ,γ n ), then run step six;

[0014] Step 6: According to the solution in step 4 or step 5 (r n ,γ n ), and analytically calculate the extreme curvature and arc length of the smooth curve.

[0015] Furthermore, step one specifically includes:

[0016] For the corner ∠P i , according to the upper limit of the approximate error ε max A smooth curve containing only arc segments is constructed. To avoid the overlap of smooth curves corresponding to two adjacent corners, the transition length needs to be constrained, which is defined as no self-intersection constraint. The maximum transition length L under the geometric constraint consisting of the approximate error upper limit and the no self-intersection constraint is expressed as:

[0017]

[0018] where ε max is the predefined upper limit of the approximation error; α is the corner ∠P i Angle; ‖P i-1 P i ‖,‖P i P i+1 ‖ are straight line segments P i-1 P i , P i P i+1 The length of, when i=2,…,N, the parameter k1 is 0.5, otherwise k1 is 1; when i=1,…,N-1, k2 is 0.5, otherwise k2 is 1;

[0019] Based on the predefined geometric constraints, the upper limit R of the extreme value r of the curvature radius of the smooth curve is derived as:

[0020]

[0021] The upper limit β of the first tangent angle γ is expressed as:

[0022]

[0023] That is, we get the solution set 1, the mathematical form of which is: {(r,γ)|0 <r≤R,0<γ≤β}。

[0024] Furthermore, step 2 specifically includes:

[0025] In the scope of solution set 1, in order to satisfy the tangential continuity and curvature continuity of the connection between the Airthoid curve and the arc, the parameter combination (r, γ) must satisfy the following relationship:

[0026] Formula 1:

[0027] Where ε represents the actual approximation error, ε = L(csc0.5α-cot0.5α); α is the corner ∠P i Angle, intermediate variable: The boundary when the equation 1 takes the equal sign is recorded as curve C nsi ;

[0028] In order to make the connection between the airthoid curve and the arc have positional continuity, the parameter combination (r, γ) should satisfy the equation C g2 :

[0029]

[0030] Among them, the intermediate variable

[0031] That is, the second solution set is obtained, and the mathematical form of the second solution set is:

[0032] Furthermore, step three specifically includes:

[0033] The intersection of solution set 1 and solution set 2 is:

[0034]

[0035] D represents the feasible domain of (r,γ) under the upper limit of approximation error, no self-intersection constraint, tangent continuity and curvature continuity;

[0036] That is, we get solution set three.

[0037] Furthermore, the step 4 specifically includes:

[0038] The maximum velocity extreme value principle includes tangential kinematic constraints, normal kinematic constraints and target velocity; for different parameter combinations (r, γ), the function v corresponding to the maximum velocity extreme value principle is n (r,γ) is expressed as:

[0039]

[0040] Among them, I1 represents the straight line γ=0、r=R,and curve C i-ii The enclosed area, i.e.

[0041]

[0042] II is represented by curve C i-ii , C nsi , and the straight line r=R enclosed area, that is

[0043]

[0044] I2 is represented by the straight line γ=0, The closed area surrounded by r = 0, that is,

[0045]

[0046] III represents a straight line The closed area surrounded by γ=β and r=0, that is,

[0047]

[0048] Curve C i-ii : J n represents the predefined maximum normal jump, A n represents the predefined maximum normal acceleration;

[0049] Take solution set three as function v n The domain of (r,γ) gives the function v n The expression of (r,γ) is:

[0050]

[0051] Where θ1 is the curve C g2 The ordinate of the intersection with the straight line r=R; (r1,θ2) is the curve C g2 With curve C i-ii The intersection of r2 and c g2The abscissa of the point of intersection with the line γ = β;

[0052] From the above formula, we can know that the function v n (r,γ) has a maximum value v in each of the three intervals 1n 、v 2n 、v 3n , then the maximum value Get The corresponding solution (r n ,γ n ).

[0053] Furthermore, step five specifically includes:

[0054] If solution set 3 is empty, the first tangent angle of the smooth curve

[0055] According to the no self-intersection constraint, the actual approximation error ε is:

[0056]

[0057] Among them, the intermediate variable

[0058] On this basis, the curvature radius of the smooth curve is r n for:

[0059]

[0060] That is, we get the solution (r n ,γ n ).

[0061] Furthermore, step six specifically includes:

[0062] The curvature extreme value κ of the smooth curve max for:

[0063]

[0064] Where a is the scaling parameter of the Airthoid curve, which indicates the curvature of the curve.

[0065] Airthoid Curve The length of s1 is:

[0066]

[0067] Arc The length s2 is:

[0068]

[0069] The arc length l of the smooth curves for:

[0070]

[0071] Compared with the prior art, the beneficial technical effects of the present invention are:

[0072] The smoothing curve designed by the present invention is composed of Airthoid splines and arc segments, which significantly improves the problem of sharp increase in curvature compared to the method of corner smoothing using only Airthoid splines, which is beneficial to improving the speed of the machine tool during corner transition, thereby improving the machining efficiency of the machine tool. In addition, the parameters of the Airthoid splines and arcs are analytically determined by maximizing the allowed speed and predefined geometric constraints, and the analytical calculation method can improve the efficiency of calculating the parameters of the smoothing curve, which is beneficial to the online implementation of the smoothing algorithm. In addition, the arc length and curvature extreme value of the smoothing curve can be analytically expressed, which is more conducive to the real-time execution of smoothing motion by the numerical control system. The method realizes smooth motion speed by smoothing the machining path based on Airthoid and arcs, and has the advantages of expression based on arc length parameters, controllable contour error, and analytical expression of curvature extreme value and arc length. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] In order to more clearly illustrate the technical solutions in the specific implementation modes of the present invention, the drawings required for use in the description of the specific implementation modes will be briefly introduced below. Obviously, the drawings described below are some implementation modes of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0074] Figure 1 is a schematic diagram of locally smoothing an embodiment using the present invention;

[0075] Figure 2 It is a trajectory diagram of smoothing the embodiment using the Airthoid curve method and the smoothing method of the present invention.

[0076] Figure 3 It is a curvature distribution diagram of a smoothing curve obtained by using the Airthoid curve method and the smoothing method of the present invention. DETAILED DESCRIPTION

[0077] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0078] As a specific embodiment of the present invention, the present invention designs a smoothing curve that is a mixture of airthoid and arc and has the best time. Specifically, the smoothing curve is defined as follows: Composed of two symmetrical Airthoid curves and and an arc symmetric about the corner bisector Point B0 and point B1 are the entry and exit points of the transition curve, respectively, and are defined as transition points. Replace the straight line segment B0P i and P i B1, its length is recorded as the transition length. Points C0 and C1 are the intersection points of the Airthoid curve and the arc segment, defined as Airthoid-arc points. The first tangent angle at the intersection point C0 is recorded as γ. The second tangent angle at the midpoint of the transition curve is set to β.

[0079] Step 1: According to the approximate error constraint and the no self-intersection constraint, determine the solution set of the extreme value r of the curvature radius of the smooth curve and the first tangent angle γ, which is recorded as solution set 1.

[0080] Step 1 specifically includes:

[0081] For two adjacent straight line segments P i-1 P i and P i P i+1 The corners formed by the approximate error upper limit ε max A transition curve containing only arc segments can be constructed. In addition, in order to avoid the overlap of the smooth curves corresponding to two adjacent corners, the transition length should be constrained, which is defined as a no self-intersection constraint. The maximum transition length L under the geometric constraints consisting of the approximate error upper limit and the no self-intersection constraint is expressed as:

[0082]

[0083] where ε max is the predefined upper limit of the approximate error; a is the corner ∠P i Angle; ‖P i-1 P i ‖,‖P i P i+1 ‖ are straight line segments P i-1 P i , P i P i+1 The length of, when i=2,…,N, the parameter k1 is 0.5, otherwise k1 is 1; when i=1,…,N-1, k2 is 0.5, otherwise k2 is 1; N is the total number of corners.

[0084] Based on the predefined geometric constraints, the upper limit R of the extreme value r of the curvature radius of the smooth curve is derived as:

[0085]

[0086] The upper limit β of the first tangent angle γ is expressed as:

[0087]

[0088] That is, we get the solution set 1, the mathematical form of which is: {(r,β)|0 <r≤R,0<β≤β}。

[0089] Step 2: According to the constraints of tangential continuity, curvature continuity and position continuity, calculate the solution set of the parameter combination (r, γ), which is recorded as solution set 2.

[0090] Step 2 specifically includes:

[0091] Within the boundary obtained in step 1, in order to ensure that the connection between the Airthoid curve and the arc has tangent and curvature continuity, the parameter combination (r, γ) must satisfy the following relationship:

[0092] Formula 1:

[0093] Where ε represents the actual approximation error, ε = L(csc0.5α-cot0.5α), α is the corner ∠P i Angle, intermediate variable: The boundary when the equation 1 takes the equal sign is recorded as curve C nsi .

[0094] In order to make the connection between the airthoid curve and the arc have positional continuity, the parameter combination (r, γ) should satisfy the equation C g2 :

[0095]

[0096] Among them, the intermediate variable

[0097] That is, the second solution set is obtained, and the mathematical form of the second solution set is:

[0098] Step 3: Calculate the intersection of solution set 1 and solution set 2 to obtain solution set 3; determine whether solution set 3 is empty, if so, run step 5; if not, run step 4.

[0099] The intersection of the solution sets obtained in step 1 and step 2 is:

[0100]

[0101] D represents the feasible domain of (r,γ) under the upper limit of the approximation error, no self-intersection constraint, tangent continuity and curvature continuity.

[0102] Step 4: Based on the principle of maximizing the speed extreme value, calculate the optimal parameter combination (r, γ) in solution set 3 and obtain the solution (r n ,γ n ), then run step 6.

[0103] Step 4 specifically includes:

[0104] The maximum velocity extreme value principle includes tangential kinematic constraints, normal kinematic constraints and target velocity; for different parameter combinations (r, γ), the function v corresponding to the maximum velocity extreme value principle is n (r,γ) is expressed as:

[0105]

[0106] Among them, I1 represents the straight line γ=0、r=R,and curve C i-ii The enclosed area, i.e.

[0107]

[0108] II is represented by curve C i-ii , C nsi , and the straight line r=R enclosed area, that is

[0109]

[0110] I2 is represented by the straight line γ=0, The closed area surrounded by r = 0, that is,

[0111]

[0112] III represents a straight line The closed area surrounded by γ=β and r=0, that is,

[0113]

[0114] Curve C i-ii : J n represents the predefined maximum normal jump, A n Represents a predefined maximum normal acceleration.

[0115] Take solution set three as function v n The domain of (r,γ) gives the function v n The expression of (r,γ) is:

[0116]

[0117] Where θ1 is the curve C g2 The ordinate of the intersection with the straight line r=R; (r1,θ2) is the curve C g2 With curve C i-ii The intersection of r2 and c g2 The horizontal coordinate of the intersection with the straight line γ=β.

[0118] From the above formula, we can know that the function v n (r,γ) has a maximum value v in each of the three intervals 1n 、v 2n 、v 3n , then the maximum value Get The corresponding solution (r n ,γ n ).

[0119] Step 5: Generate a smooth curve parameter combination that does not contain arcs according to the no self-intersection constraint, and get the solution (r n ,γ n ), then run step 6.

[0120] Step 5 specifically includes:

[0121] If solution set 3 is empty, the first tangent angle of the smooth curve

[0122] According to the no self-intersection constraint, the actual approximation error ε is:

[0123]

[0124] Among them, the intermediate variable

[0125] On this basis, the curvature radius of the smooth curve is r n for:

[0126]

[0127] That is, we get the solution (r n ,γ n ).

[0128] Step 6: According to the solution in step 4 or step 5 (r n ,γ n ), and analytically calculate the extreme curvature and arc length of the smooth curve.

[0129] Step 6 specifically includes:

[0130] The curvature extreme value κ of the smooth curvemax for:

[0131]

[0132] Where a is the scaling parameter of the Airthoid curve, which indicates the curvature of the curve.

[0133] Airthoid Curve The length of s1 is:

[0134]

[0135] Arc The length s2 is:

[0136]

[0137] The arc length l of the smooth curve s for:

[0138]

[0139] The invention has the advantages of curvature continuity, formula control approximation error, analytical curvature extreme value and arc length. The analytical geometric properties greatly reduce the time consumption of calculation. The method is more meaningful for online smooth motion and is applied to commercial numerical control systems.

[0140] In order to compare the smoothing method in the present invention with the method of inscribed Airthoid curve, five coordinates are set for points A, B, C, D, and E, which are (0, 0), (0.525, 0), (1.050, 2.100), (-0.525, 1.050), and (-0.525, 0), respectively, and the approximate error ε is set max is 90 μm, then the smoothing curve obtained by the method of inscribing the Airthoid curve and the smoothing method of the present invention is as follows Figure 2 As shown, the curvature distribution of the smoothed curve obtained by the method of inscribed Airthoid curve and the smoothing method of the present invention is as follows Figure 3 shown.

[0141] It is obvious to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting, and the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that all changes falling within the meaning and scope of the equivalent elements of the claims are included in the present invention, and any reference numerals in the claims should not be regarded as limiting the claims involved.

[0142] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A linear path smoothing method based on Airthoid and arc, characterized in that: Constructing smooth curves To correct the straight line segment P in the tool path i-1 P i and P i P i+1 The angle ∠P i Smoothing is performed, and the smoothing curve is used as the actual machining path of the tool, 1≤i≤N, N is the total number of corners in the tool path; the smoothing curve Composed of two symmetrical Airthoid curves and and a i Angle bisector symmetric arc Points B0 and B1 are respectively the smoothing curves The entry and exit points are smoothed using curves Replace the straight line segment B0P i and the straight line segment P i B1, the length of the straight line segment replaced by the smooth curve is recorded as the transition length; C0 and C1 points are two Airthoid curves respectively. and The intersection point with the arc; The tangent line of the arc at the intersection point C0 and the straight line segment P i-1 P i The angle between the tangent line at the midpoint of the arc and the straight line P is defined as the first tangent angle γ; i-1 P i The angle is defined as the second tangent angle β; The linear path smoothing method specifically includes the following steps: Step 1: According to the approximate error constraint and the no self-intersection constraint, determine the solution set of the extreme value r of the curvature radius of the smooth curve and the first tangent angle γ, which is recorded as solution set 1; Step 2: According to the constraints of tangential continuity, curvature continuity and position continuity, calculate the solution set of the parameter combination (r, γ), which is recorded as solution set 2; Step 3: Calculate the intersection of solution set 1 and solution set 2 to obtain solution set 3; determine whether solution set 3 is empty, if so, run step 5; if not, run step 4; Step 4: Based on the principle of maximizing the speed extreme value, calculate the optimal parameter combination (r, γ) in solution set 3 and obtain the solution (r n ,γ n ), then run step six; Step 5: Generate a smooth curve parameter combination that does not contain arcs according to the no self-intersection constraint, and get the solution (r n ,γ n ), then run step six; Step 6: According to the solution in step 4 or step 5 (r n ,γ n ), analytically calculate the curvature extreme value and arc length of the smooth curve; Step 1 specifically includes: For the corner ∠P i , according to the upper limit of the approximate error ε max A smooth curve containing only arc segments is constructed. To avoid the overlap of smooth curves corresponding to two adjacent corners, the transition length needs to be constrained, which is defined as no self-intersection constraint. The maximum transition length L under the geometric constraint consisting of the approximate error upper limit and the no self-intersection constraint is expressed as: where ε max is the predefined upper limit of the approximation error; α is the corner ∠P i Angle; ‖P i-1 P i ‖,‖P i P i+1 ‖ are straight line segments P i-1 P i , P i P i+1 The length of, when i=2,···,N, the parameter k1 is 0.5, otherwise k1 is 1; when i=1,···,N-1, k2 is 0.5, otherwise k2 is 1; Based on the predefined geometric constraints, the upper limit R of the extreme value r of the curvature radius of the smooth curve is derived as: The upper limit of the first tangent angle γ is β: That is, we get the solution set 1, the mathematical form of which is: {(r,γ)|0 <r≤R,0<γ≤β}; Step 2 specifically includes: In the scope of solution set 1, in order to satisfy the tangential continuity and curvature continuity of the connection between the Airthoid curve and the arc, the parameter combination (r, γ) must satisfy the following relationship: Formula 1: Where ε represents the actual approximation error, ε = L(csc 0.5α - cot 0.5α); α is the corner ∠P i Angle, intermediate variable: The boundary when the equation 1 takes the equal sign is recorded as curve C nsi ; In order to make the connection between the airthoid curve and the arc have positional continuity, the parameter combination (r, γ) should satisfy the equation C g2 : Among them, the intermediate variable That is, the second solution set is obtained, and the mathematical form of the second solution set is:

2. The linear path smoothing method based on Airthoid and arc according to claim 1, characterized in that: Step three specifically includes: The intersection of solution set 1 and solution set 2 is: D represents the feasible domain of (r,γ) under the upper limit of approximation error, no self-intersection constraint, tangent continuity and curvature continuity; That is, we get solution set three.

3. The linear path smoothing method based on Airthoid and arc according to claim 2, characterized in that: The step 4 specifically includes: The maximum velocity extreme value principle includes tangential kinematic constraints, normal kinematic constraints and target velocity; for different parameter combinations (r, γ), the function v corresponding to the maximum velocity extreme value principle is n (r,γ) is expressed as: Among them, I1 represents the straight line γ=0、r=R,and curve C i-ii The enclosed area, i.e. II is represented by curve C i-ii , C nsi , and the straight line r=R enclosed area, that is I2 is represented by the straight line γ=0, The closed area surrounded by r = 0, that is, III represents a straight line The closed area surrounded by γ=β and r=0, that is, Curve C i-ii : J n represents the predefined maximum normal jump, A n represents the predefined maximum normal acceleration; Take solution set three as function v n The domain of (r,γ) gives the function v n The expression of (r,γ) is: Where θ1 is the curve C g2 The ordinate of the intersection with the straight line r=R; (r1,θ2) is the curve C g2 With curve C i-ii The intersection of r2 and c g2 The abscissa of the point of intersection with the straight line γ = β; From the above formula, we can know that the function v n (r,γ) has a maximum value v in each of the three intervals 1n 、v 2n 、v 3n , then the maximum value Get The corresponding solution (r n ,γ n ).

4. The linear path smoothing method based on Airthoid and arc according to claim 1, characterized in that: Step 5 specifically includes: If solution set 3 is empty, the first tangent angle of the smooth curve According to the no self-intersection constraint, the actual approximation error ε is: Among them, the intermediate variable On this basis, the curvature radius of the smooth curve is r n for: That is, we get the solution (r n ,γ n ).

5. The linear path smoothing method based on Airthoid and arc according to claim 1, characterized in that: Step 6 specifically includes: The curvature extreme value κ of the smooth curve max for: Where a is the scaling parameter of the Airthoid curve, which indicates the curvature of the curve. Airthoid Curve The length of s1 is: Arc The length s2 is: The arc length l of the smooth curve s for:

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