Railway track calculation method and system based on spatial curve automatic generation algorithm

Abstract

The application discloses a kind of track calculation method and system based on spatial curve automatic generation algorithm, comprising: the starting point coordinate of acquisition spatial curve, terminal point coordinate and the curve parameter of space track;Based on the starting point coordinate of spatial curve and the curve parameter of space track, obtain discrete point set;In the discrete point set obtained, the starting point coordinate of spatial curve is re-acquired, based on the starting point coordinate of re-acquired spatial curve and the curve parameter of space track, obtain next discrete point set;Constantly generate discrete point set, until terminal point coordinate in last discrete point set;Based on the coordinate of each starting point, construct spatial curve.Judge whether the transition of last spatial curve and terminal point connected curve and previous curve is smooth;If not, adjust the curve parameter of space track, until transition is smooth;If yes, then output spatial curve.The present application can be used in the scene that only start and end points are determined without intermediate points, can reduce the cost of proofing, improve efficiency.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of space curve calculation, and relates to a track calculation method and system based on a space curve automatic generation algorithm. BACKGROUND

[0002] Differential geometry has a wide range of applications in various aspects of social life such as civil, military, astronomy, etc. Currently, in the problem of space curve generation, the traditional detection, machine vision and other detection methods are mainly used to detect existing points and fit to generate space curves, which cannot be applied to scenarios where only the start and end points are determined without intermediate points. In particular, in some traditional industries, the actual space curve track is usually first manually punched out, and then a detection method is used to detect and fit the manually created track to generate a space curve. This process usually needs to be repeated multiple times because it cannot be guaranteed to achieve the goal at one time. SUMMARY

[0003] The present application aims to solve the problem that the existing technology cannot be applied to scenarios where only the start and end points are determined without intermediate points in the generation of space curves, and provides a track calculation method and system based on a space curve automatic generation algorithm.

[0004] To achieve the above-mentioned purpose, the present application adopts the following technical solutions:

[0005] The track calculation method based on the space curve automatic generation algorithm comprises:

[0006] Step 1: Collect the start point coordinates, end point coordinates and curve parameters of the space track of the space curve;

[0007] Step 2: Based on the start point coordinates of the space curve and the curve parameters of the space track, obtain a discrete point set;

[0008] Step 3: In the obtained discrete point set, re-collect the start point coordinates of the space curve, and based on the re-collected start point coordinates of the space curve and the curve parameters of the space track, obtain a next discrete point set;

[0009] Step 4: Repeat step 3 until the end point coordinates are in the last discrete point set;

[0010] Step 5: Based on the coordinates of each start point, construct a space curve;

[0011] Step 6: Determine whether the transition of the last space curve connected with the end point and the previous curve is smooth; if not, adjust the curve parameters of the space track, repeat steps 1-5 until the transition is smooth; if yes, output the space curve.

[0012] Further improvements of the present application are as follows:

[0013] Further, the curve parameters of the spatial orbit include: curvature, torsion, smoothness and arc length.

[0014] Further, the smoothness is set by human.

[0015] Further, the arc length s step is fixed. step The distance between the starting point and the adjacent discrete point set.

[0016] Further, the curvature k(s) and the torsion τ(s) of the spatial curve are respectively shown in formula (1) and formula (2):

[0017]

[0018]

[0019] Wherein, α(s) is a unit tangent vector, β(s) is a principal normal vector, and γ(s) is a secondary normal vector.

[0020] Further, the principal normal vector β(s) of the spatial curve is:

[0021]

[0022] Further, the secondary normal vector γ(s) of the spatial curve is:

[0023]

[0024] Further, the unit tangent vector α(s) of the spatial curve is

[0025]

[0026] Wherein, r(s) is the equation of the spatial curve, and s is the arc length parameter of the curve; |α(s)|=1, and α(s)⊥α′(s).

[0027] Further, based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit, the discrete point set is obtained, specifically:

[0028] If the discrete point set is [P0,P1,P2,...,P n ], wherein P0 is the starting point S i , P n is the end point E i ; the point P k+1 is determined by P k , the arc length s step , the curvature c k and the torsion τ k , and the coordinates of the point P k are recorded as (p xkp yk p zk ), the relationship is calculated as:

[0029]

[0030] Further, based on the coordinates of each starting point, a space curve is constructed; specifically, the starting points in the discrete points are fitted to obtain a space curve, specifically:

[0031] The discrete point coordinates are [P0, P1, P2,..., P n ], wherein P i The space curve at the point P i-1 , P i , P i+1 , P i+2 , the boundary condition is that the starting point and the ending point of the curve are natural boundaries, the second derivative of the starting point and the ending point is 0, and the other places are non-kink boundaries, the third derivative of the two end points is equal to the third derivative of the adjacent points of the two end points, then the curve equation can be obtained according to the piecewise cubic spline interpolation;

[0032] P i (u)=a i u 3 +b i u 2 +c i u+d i (7)

[0033] wherein,

[0034] The orbit calculation system based on the space curve automatic generation algorithm comprises:

[0035] A first acquisition module, the first acquisition module is used for acquiring the starting point coordinates and the ending point coordinates of the space curve and the curve parameters of the space orbit;

[0036] An acquisition module, the acquisition module is used for acquiring a discrete point set based on the starting point coordinates of the space curve and the curve parameters of the space orbit;

[0037] A second acquisition module, the second acquisition module is used for reacquiring the starting point coordinates of the space curve in the acquired discrete point set, and acquiring a next discrete point set based on the reacquired starting point coordinates of the space curve and the curve parameters of the space orbit;

[0038] A termination module, the termination module is used for determining that the ending point coordinates are in the last discrete point set;

[0039] A construction module, the construction module is used for constructing a space curve based on the coordinates of each starting point;

[0040] A judging module is configured to judge whether the transition of the last spatial curve connected with the end point and the previous curve is smooth or not, adjust the curve parameters of the spatial track until the transition is smooth if the transition is not smooth, and output the spatial curve if the transition is smooth.

[0041] Compared with the prior art, the present application has the following beneficial effects:

[0042] The present application obtains the discrete point set through the start point coordinate, the end point coordinate and the curve parameters of the spatial track, re-collects the start point coordinate of the spatial curve in the obtained discrete point set, obtains the next discrete point set based on the re-collected start point coordinate of the spatial curve and the curve parameters of the spatial track, until the end point coordinate is in the last discrete point set, and generates the spatial curve through fitting based on the coordinates of the start points. BRIEF DESCRIPTION OF DRAWINGS

[0043] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the following will briefly introduce the drawings needed to be used in the embodiments, and it should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as a limitation to the scope, and for those skilled in the art, other related drawings can also be obtained without creative labor on the basis of these drawings.

[0044] Figure 1 The flow chart of the track calculation method based on the spatial curve automatic generation algorithm of the present application;

[0045] Figure 2 The structure diagram of the track calculation system based on the spatial curve automatic generation algorithm of the present application;

[0046] Figure 3 The start point in each discrete point set is constructed into a spatial curve graph. DETAILED DESCRIPTION

[0047] In order to make the purpose, technical solutions and advantages of the embodiments of the present application more clear, the following will combine the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application, and obviously, the described embodiments are some embodiments of the present application, but not all the embodiments. The components of the embodiments of the present application described and shown in the drawings here can be arranged and designed in various different configurations.

[0048] Therefore, the following detailed description of the embodiments of the application provided in the drawings is not intended to limit the scope of the application claimed, but merely represents selected embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by those of ordinary skill in the art without creative labor fall within the scope of protection of the application.

[0049] It should be noted that similar reference numbers and letters represent similar items in the following drawings, so once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings.

[0050] In the description of the embodiments of the application, it should be noted that if the terms "upper", "lower", "horizontal", "inner" and the like indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, or the orientation or positional relationship in which the product of the application is usually placed, only for the convenience of describing the application and simplifying the description, and it is not intended to indicate or imply that the device or element referred to must have a particular orientation, be constructed and operated in a particular orientation, and therefore cannot be understood as limiting the application. In addition, the terms "first", "second", etc. are only used to distinguish the description and cannot be understood as indicating or implying relative importance.

[0051] In addition, if the term "horizontal" appears, it does not mean that the component must be absolutely horizontal, but can be slightly inclined. For example, "horizontal" only means that its direction is relatively more horizontal than "vertical", and does not mean that the structure must be completely horizontal, but can be slightly inclined.

[0052] In the description of the embodiments of the application, it should also be noted that unless otherwise explicitly specified and limited, if the terms "set", "mount", "connected", "connected" appear, they should be understood in a broad sense, for example, they can be fixedly connected, or can be detachably connected, or integrally connected; can be mechanically connected, or can be electrically connected; can be directly connected, or can be indirectly connected through an intermediate medium; can be the communication between the two elements inside. For those of ordinary skill in the art, the specific meaning of the above terms in the application can be understood according to the specific circumstances.

[0053] The application will be described in further detail below in conjunction with the accompanying drawings:

[0054] Referring to Figure 1 The application discloses a track calculation method based on a spatial curve automatic generation algorithm, comprising:

[0055] S101: Collecting the starting point coordinates, the terminal point coordinates and the curve parameters of the spatial track of the spatial curve.

[0056] The curve parameters of the spatial track include: curvature, torsion, smoothness and arc length. The smoothness is set by human.

[0057] The curvature k(s) and torsion τ(s) of the spatial curve are respectively shown in formula (1) and formula (2):

[0058]

[0059]

[0060] Wherein, α(s) is a unit tangent vector, β(s) is a principal normal vector, and γ(s) is a binormal vector.

[0061] The principal normal vector β(s) of the spatial curve is:

[0062]

[0063] The binormal vector γ(s) of the spatial curve is:

[0064]

[0065] The unit tangent vector α(s) of the spatial curve is

[0066]

[0067] Wherein, r(s) is an equation of the spatial curve, and s is an arc length parameter of the curve; |α(s)| = 1, and α(s) is perpendicular to α'(s).

[0068] S102: Based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit, a discrete point set is obtained.

[0069] If the discrete point set is [P0, P1, P2,..., Pn], wherein P0 is the starting point S, and Pn is the terminal point E; the point Pn-1 is determined by Pn, the arc length s, the curvature c, and the torsion τ, and the coordinates of the point Pn-1 are (p, p, p), and the calculation relationship is: n i n i k+1 k step k k k xk yk zk

[0070]

[0071] S103: In the obtained discrete point set, the starting point coordinates of the spatial curve are re-acquired, and based on the re-acquired starting point coordinates of the spatial curve and the curve parameters of the spatial orbit, a next discrete point set is obtained. ​​​​​​​​​​​​​

[0072] Arc length s step The distance between the starting points of two adjacent discrete point sets is s, where s is the arc length. step The length is fixed.

[0073] S104: Repeat S103 until the coordinates of the termination point are in the last discrete point set.

[0074] S105: Construct a spatial curve based on the coordinates of each starting point.

[0075] Based on the coordinates of each starting point, a spatial curve is constructed; specifically, the starting points among the discrete points are fitted to obtain the spatial curve.

[0076] The coordinates of the discrete points are [P0, P1, P2, ..., P... n ], where P i The spatial curve at point P passes through point P. i-1 ,P i ,P i+1 ,P i+2 The boundary conditions are defined as follows: the starting and ending points of the curve are natural boundaries, the second derivative of the starting and ending points is 0, the rest are non-knotted boundaries, and the third derivative of the two endpoints is equal to the third derivative of the points adjacent to the two endpoints. Then the curve equation can be obtained by piecewise cubic spline interpolation.

[0077] P i (u)=a i u 3 +b i u 2 +c i u+d i (7)

[0078] in,

[0079] S106: Determine whether the transition between the last spatial curve connected to the termination point and the previous curve is smooth; if not, adjust the curve parameters of the spatial track and repeat S101-S105 until the transition is smooth; if yes, output the spatial curve.

[0080] See Figure 2 This invention discloses a trajectory calculation system based on an automatic space curve generation algorithm, comprising:

[0081] The first acquisition module is used to acquire the starting point coordinates, ending point coordinates, and curve parameters of the spatial curve.

[0082] The acquisition module acquires a discrete point set based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit;

[0083] The second acquisition module reacquires the starting point coordinates of the space curve from the acquired discrete point set, and obtains the next discrete point set based on the reacquired starting point coordinates of the space curve and the curve parameters of the space track.

[0084] Termination module, the termination module is used to determine the coordinates of the termination point in the last discrete point set;

[0085] The construction module constructs a spatial curve based on the coordinates of each starting point;

[0086] The judgment module is used to determine whether the transition between the last spatial curve connected to the termination point and the previous curve is smooth; if not, the curve parameters of the spatial track are adjusted until the transition is smooth; if yes, the spatial curve is output.

[0087] Example:

[0088] This invention discloses a trajectory calculation method based on an automatic space curve generation algorithm, comprising:

[0089] Step 1: The user specifies the curve parameters of the space orbit, such as curvature, torsion, and smoothness; and specifies the output method of the space orbit, such as parametric equation output or discrete point output. The maximum and minimum curvature are denoted as k. max ,k min The maximum and minimum torsion are denoted as τ. max ,τ min Curve smoothness requirements.

[0090] Step 2: Obtain the start and end points of the space curve using contact or non-contact measuring equipment, and label them as {S1, S2, ..., S...} m ,E1,E2,...,E m}, where m represents the number of points whose positions need to be considered, and S i E represents the spatial coordinates of the starting point of the i-th position. i Represents the spatial coordinates of the endpoint at position i.

[0091] Step 3: Based on the starting point coordinates, ending point coordinates, and curve parameters of the spatial orbit, calculate and generate a discrete point set that meets the parameter requirements, and fit the discrete point set into m interval spatial curves that meet the parameter requirements.

[0092] For any interval space curve C:f(x,y,z)=0 in space, it can be represented by a vector function, as shown in formula (8):

[0093] r(t)=(x(t),y(t),z(t)) (8)

[0094] where t is an arbitrary parameter, in particular, t can be the arc length parameter s of the interval space curve.

[0095] Let the equation of the interval space curve be r(s), s is the arc length parameter of the interval space curve. The unit tangent vector of the interval space curve α(s), that is,

[0096]

[0097] Naturally, |α(s)| = 1, and α(s) α'(s).

[0098] Definition: k(s) is called the curvature of the curve, where k(s) is:

[0099]

[0100] β(s) is the principal normal vector of the curve, β(s) is:

[0101]

[0102] The tangent vector and the principal normal vector of the curve determine the binormal vector of the curve

[0103]

[0104] Definition τ(s) is the torsion of the curve, τ(s) is:

[0105] τ(s) = -γ'(s) · β(s) (13)

[0106] Using the unit tangent vector α(s), the principal normal vector β(s), the binormal vector γ(s), the analytical expressions of the curvature k(s) and the torsion τ(s) are given as follows:

[0107]

[0108]

[0109] The geometric meaning of the curvature is the rotational speed of the tangent vector of the curve with respect to the arc length. The greater the curvature, the greater the degree of bending of the curve, so it reflects the degree of bending of the curve. The absolute value of the torsion is the rotational speed of the binormal vector of the curve with respect to the arc length. The curve with constant torsion is a planar curve. According to the basic theorem of space curve theory, the curvature and the torsion can completely determine the space curve.

[0110] According to the curve parameters of the space orbit: maximum curvature / torsion, minimum curvature / torsion, and the coordinates of the starting position point and the ending position point measured, a series of discrete points of the space curve can be calculated and generated, and then a space curve meeting the conditions can be fitted and generated.

[0111] Referring toFigure 3 Let the starting and ending position coordinates be S i E i Taking the i-th spatial curve as an example, starting from the starting position S... i Let's start from S i Starting from point S, the arc length is s. step The space curve, which needs to satisfy the curvature range of [c], is a space curve that has a curvature value of [c]. min ,c max ], and the range of torsion values ​​is [τ min ,τ max If the coordinates of the endpoint are within a certain range, then the range is defined by the starting point S. i Arc length s step The range of curvature and the range of torsion are jointly determined. This range is considered a candidate region, and the starting point of the next curve segment is selected from this region, based on the starting point and the arc length s. step The range of curvature and the range of torsion together determine the endpoint range of the next segment of the space curve. This continues until the endpoint E. i .

[0112] Starting from the initial point, there are infinitely many endpoints for each segment of a spatial curve with a defined arc length. One of these points is selected as the starting point for the next segment of the spatial curve, and the calculation continues until the endpoint is reached.

[0113] In the process of calculating the discrete points of a space curve, three parameters need to be set for each small space curve segment: curvature, torsion, and arc length. These three parameters, along with the starting point, together determine the space curve and its endpoint. These will be described in detail below.

[0114] To make the space curve smooth, the curvature change of each small space curve segment should be uniform and the transition smooth, and the endpoint of each space curve segment should gradually move towards the termination point E. i Approaching. At this point, the curvature should first be kept within the range of curvature variation, then gradually increase and enter a stable value, and then gradually decrease to ensure that the natural boundary is satisfied at the starting and ending points, with the second derivative being 0. The curvature values ​​of each small spatial curve segment constitute a curvature sequence.

[0115] Similar to curvature, the same requirements apply to deflection. However, it's advisable to consider that deflection primarily occurs within a portion of the space curve, such as the beginning, end, or middle section. This way, once the deflection reaches the termination point, it can become zero, allowing this section of the space curve with zero deflection to be transformed into a planar curve, thus facilitating processing. Likewise, the deflection values ​​of each small space curve segment constitute a curvature sequence.

[0116] The selection of the arc length should also be noted, the arc length cannot be too large, too large may affect the accuracy of the curve. The arc length largely determines the number of discrete points, thus determining the accuracy of the final space curve fitting. At the same time, the arc length cannot be too small, too small will increase the amount of calculation. Generally for each small space curve segment arc length selected uniform value, denoted as s step .

[0117] Consider the S i as the starting point, the arc length of the space curve s step , the starting point S i , determine the parameter arc length s step , curvature sequence, torsion sequence, through the following formula to calculate the series of discrete points. Denoted as [P0, P1, P2,..., P n ], wherein P0 is S i , P n is E i . Then the point P k+1 is determined by P k , the arc length s step , the curvature c k , the torsion τ k , denoted as the coordinates of the point P k (p xk , p yk , p zk ), then the calculation relationship is:

[0118]

[0119] After getting the discrete points, the last space curve connected with the end point needs to be analyzed, to judge whether the curvature and torsion are smooth with the transition before, if not meet the requirements, the curvature sequence, torsion sequence or arc length needs to be adjusted to meet the requirements of the last interval space curve.

[0120] Through the previous calculation, the discrete point set of the ith space curve is obtained. The previous segmented space curve can be directly used to represent the space curve, or the curve fitting method can be applied to the discrete points to obtain the space curve. Note the relationship between the discrete points and the fitting accuracy: the more the discrete points, the higher the accuracy of the fitted space curve; the step size can be appropriately reduced where needed to ensure the accuracy of the curve. Taking the piecewise cubic spline as an example of the curve fitting method, suppose the discrete points have n, the coordinates are [P1, P2,..., P n ], wherein P i point of the space curve through the point P i-1 , P i , P i+1 , P i+2The boundary condition is defined as the natural boundary at the start and end points of the curve, the second derivative of the start and end points is 0, the other points are non-kinked boundaries, the third derivative of the two end points is equal to the third derivative of the adjacent points of the two end points, and then the curve equation can be obtained according to the piecewise cubic spline interpolation.

[0121] P i (u)=a i u 3 +b i u 2 +c i u+d i (17)

[0122] wherein,

[0123] P i =[P xi ,P yi ,P zi ]',a i =[a xi ,a yi ,a zi ]',b i =[b xi ,b yi ,b zi ]',c i =[c xi ,c yi ,c zi ]',d i =[d xi ,d yi ,d zi ]';

[0124] Step four: according to the calculation results of the spatial curve, including the discrete points on the curve and the expression of the fitting curve, output the data of the spatial track in the specified output mode.

[0125] The above is only the preferred embodiment of the present application, and is not used to limit the present application. For those skilled in the art, the present application can have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A trajectory calculation method based on an automatic spatial curve generation algorithm, characterized in that, include: Step 1: Collect the starting point coordinates, ending point coordinates, and curve parameters of the spatial curve. The spatial curve is generated by fitting the spatial curve track produced by manual sheet metal prototyping. Step 2: Obtain a discrete point set based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit; Step 3: In the acquired set of discrete points, re-acquire the coordinates of the starting point of the space curve, and based on the re-acquired coordinates of the starting point of the space curve and the curve parameters of the space orbit, obtain the next set of discrete points. Step 4: Repeat step 3 until the coordinates of the termination point are in the last discrete point set; Step 5: Construct a space curve based on the coordinates of each starting point; Step 6: Determine whether the transition between the last spatial curve connected to the termination point and the previous curve is smooth; if not, adjust the curve parameters of the spatial track and repeat steps 1-5 until the transition is smooth; if yes, output the spatial curve. The discrete point set is obtained based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit, specifically as follows: If the discrete point set is ,in Starting point , End point ;point Depend on Arc length curvature torsion Decision, note the points The coordinates are The calculation formula is: The construction of a spatial curve based on the coordinates of each starting point involves: fitting the starting points among the discrete points to obtain the spatial curve. The coordinates of the discrete point are ,in The spatial curve at point A passes through point B. The boundary conditions are defined as follows: the starting and ending points of the curve are natural boundaries, the second derivative of the starting and ending points is 0, the rest are non-knotted boundaries, and the third derivative of the two endpoints is equal to the third derivative of the points adjacent to the two endpoints. Then the curve equation can be obtained by piecewise cubic spline interpolation. in, .

2. The orbit calculation method based on the automatic spatial curve generation algorithm according to claim 1, characterized in that, The curve parameters of the space track include: curvature, torsion, smoothness, and arc length.

3. The orbit calculation method based on the automatic spatial curve generation algorithm according to claim 2, characterized in that, The smoothness is set manually.

4. The orbit calculation method based on the automatic spatial curve generation algorithm according to claim 3, characterized in that, Arc length The distance between the starting points of two adjacent discrete point sets, and the arc length. The length is fixed.

5. The orbit calculation method based on the automatic spatial curve generation algorithm according to claim 4, characterized in that, The curvature c(s) and torsion of the space curve As shown in formulas (1) and (2) respectively: in, As the unit tangent vector, Principal normal vector This is the normal vector.

6. The orbit calculation method based on the automatic spatial curve generation algorithm according to claim 5, characterized in that, The principal normal vector of the space curve for: The normal vector of a space curve for: Unit tangent vector of a space curve for in, Let be the equation of the space curve. It is the arc length parameter of the curve; ,and .

7. A trajectory calculation system based on an automatic space curve generation algorithm, characterized in that, include: The first acquisition module is used to acquire the starting point coordinates, ending point coordinates, and curve parameters of the spatial curve. The spatial curve is generated by fitting the spatial curve track based on the manual sheet metal prototyping. The acquisition module acquires a discrete point set based on the starting point coordinates of the spatial curve and the curve parameters of the spatial orbit; The second acquisition module reacquires the starting point coordinates of the space curve from the acquired discrete point set, and obtains the next discrete point set based on the reacquired starting point coordinates of the space curve and the curve parameters of the space track. Termination module, the termination module is used to determine the coordinates of the termination point in the last discrete point set; The construction module constructs a spatial curve based on the coordinates of each starting point; The judgment module is used to determine whether the transition between the last spatial curve connected to the termination point and the previous curve is smooth; if not, the curve parameters of the spatial track are adjusted until the transition is smooth; if yes, the spatial curve is output. The acquisition module is used if the discrete point set is ,in Starting point , End point ;point Depend on Arc length curvature torsion Decision, note the points The coordinates are The calculation formula is: The construction module is used to fit the starting point among the discrete points to obtain a spatial curve, specifically: The coordinates of the discrete point are ,in The spatial curve at point A passes through point B. The boundary conditions are defined as follows: the starting and ending points of the curve are natural boundaries, the second derivative of the starting and ending points is 0, the rest are non-knotted boundaries, and the third derivative of the two endpoints is equal to the third derivative of the points adjacent to the two endpoints. Then the curve equation can be obtained by piecewise cubic spline interpolation. in, .

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