# Cubic Trigonometric Hermite Interpolation Curve: Construction, Properties, and Shape Optimization

Authors: Juncheng Li; Chengzhi Liu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Cubic Hermite interpolation curve plays a very important role in interpolation curves modeling, but it has three shortcomings including low continuity, difficult shape adjustment, and the inability to accurately represent some common engineering curves. We construct a cubic trigonometric Hermite interpolation curve to make up the three shortcomings of cubic Hermite interpolation curve once and for all. The cubic trigonometric Hermite interpolation curve not only inherits the features of cubic Hermite interpolation curve but also achievesC2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In addition, we give the schemes for optimizing the shape of the cubic trigonometric Hermite interpolation curve based on internal energy minimization. The schemes include optimizing the shape of planar curve and spatial curve. Some modeling examples show that the proposed schemes are effective and the cubic trigonometric Hermite interpolation curve is more practical than cubic Hermite interpolation curve.


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## Body

## 1. Introduction

It is well known that cubic Hermite interpolation curve [1] is a common model to construct interpolation curves in engineering, and it has been widely applied in practical engineering problems [2–5]. However, cubic Hermite interpolation curve has three shortcomings: the first is that it only achieves C1 continuity, so it cannot meet some engineering problems with better requirements for smoothness; the second is that its shape would be fixed once the interpolation conditions are given, so it cannot meet some engineering problems requiring high flexibility in shape adjustment; the last is that it cannot accurately represent some common engineering curves. Some generalized Hermite interpolation curves have been presented to make up for these shortcomings, such as the rational Hermite interpolation curves with free parameters [6], the trigonometric Hermite interpolation curves with free parameters [7], the higher degree Hermite interpolation curves with free parameters [8–10], etc. Most of these curves can make up for one or two shortcomings of cubic Hermite interpolation curve.In the recent years, trigonometric polynomials have received much attention within geometric modeling. Such as the trigonometric Bézier curves [11–14], the trigonometric B-spline curves [15–17], the trigonometric interpolation curves [18, 19], etc. The curves constructed in trigonometric polynomial space have broad application prospects in engineering. To make up the three shortcomings of cubic Hermite interpolation curve once and for all, the first purpose of this paper is to construct a generalized Hermite interpolation curve in cubic trigonometric polynomial space. After the new Hermite interpolation curve is constructed, a further problem arises: how to optimize the shape of the curve to meet some specific requirements? Indeed, shape optimization of parametric curves has attracted more and more attention in recent years. Various objective functions have been proposed to optimize the shape of parametric curves, among which the internal energy of curves is a widely used objective function. Generally, the internal energy of curves includes stretch energy, strain energy (also called bending energy), and curvature variation energy (often replaced by Jerk’s energy), see [20–28]. This paper’s second purpose is to optimize the shape of the new Hermite interpolation curve by minimizing the internal energy.The main contributions of this paper are as follows:(a)
We construct the generalized Hermite interpolation curve named CTHI curve in cubic trigonometric polynomial space. CTHI curve not only interpolates the points and the corresponding tangent vectors but also achievesC2 continuity, has local and global adjustability, and can accurately represent elliptical arc, circular arc, quadratic parabolic arc, cubic parabolic arc, and astroid arc that often appear in engineering. In other words, CTHI curve not only inherits the features of cubic Hermite interpolation curve but also makes up for the three shortcomings of cubic Hermite interpolation curve once and for all(b)
We give the detailed construction process of CTHI curve, which can provide a valuable reference for the construction of other generalized Hermite interpolation curves(c)
We adopt the internal energy minimization to determine the optimal values of the free parameters contained in CTHI curve. The internal energy minimization methods include the cases of planar CTHI curve and spatial CTHI curve. The CTHI curves generated by minimizing the internal energy are satisfactory compared with the curves with inappropriate free parametersThe rest of this paper is organized as follows. In Section2, we briefly review cubic Hermite interpolation curve. In Section 3, we give the specific construction process of CTHI curve. In Section 4, we present the properties of CTHI curve. In Section 5, we provide shape optimization methods of CTHI curve based on internal energy minimization. Finally, we give a brief conclusion in Section 6.


## 2. Review of Cubic Hermite Interpolation Curve

Given a series of pointspj together with the corresponding tangent vectors mjj=0,1,⋯,n, cubic Hermite interpolation curve can be expressed by [1]
(1)Hit=f0tpi+f1tpi+1+g0tmi+g1tmi+1,i=0,1,⋯,n−1,where 0≤t≤1, fjt, and gjtj=0,1 are blending functions which can be described as follows,
(2)f0t=1−3t2+2t3,f1t=3t2−2t3,g0t=t−2t2+t3,g1t=−t2+t3.The blending functions expressed in Equation (2) have the following characteristics at the endpoints,
(3)f00f10g00g10f01f11g01g11f0′0f1′0g0′0g1′0f0′1f1′1g0′1g1′1=1000010000100001.Benefit from Equation (3), cubic Hermite interpolation curve expressed in Equation (1) satisfies that
(4)Hi0=pi,Hi1=pi+1,Hi′1=mi+1,Hi′0=mi,i=0,1,⋯,n−1.Equation (4) shows that cubic Hermite interpolation curve interpolates the points pj and the tangent vectors mjj=0,1,⋯,n. In addition, Equation (4) also shows that Hik1=Hi+1k0, k=0,1, i=0,1,⋯,n−1, which means cubic Hermite interpolation curve satisfies C1 continuity.As mentioned in Introduction, cubic Hermite interpolation curve has three shortcomings. To make up for these shortcomings once and for all, we construct a kind of generalized cubic Hermite interpolation curve in cubic trigonometric polynomial space in this paper.


## 3. Construction of Cubic Trigonometric Hermite Interpolation Curve

Cubic Hermite interpolation curve is defined in the polynomial space1,t,t2,t3. To make the constructed generalized Hermite interpolation curve can make up for the three shortcomings of cubic Hermite interpolation curve once and for all, we consider using the cubic trigonometric polynomial space 1,sint,cost,sin2t,sin3t,cos3t to instead of the polynomial space 1,t,t2,t3. It should be noted that cos2t is left out in the cubic trigonometric polynomial space because sin2t+cos2t≡1, which can ensure that the base functions are linearly independent.We need to construct cubic trigonometric blending functions first, denoted byFjt and Gjtj=1,2, and then can express the corresponding curve as follows:
(5)THit=F0tpi+F1tpi+1+G0tmi+G1tmi+1,i=0,1,⋯,n−1,where pj and mjj=0,1,⋯,n are given points and corresponding tangent vectors, respectively.From Equation (5), we have
(6)THi+1″0=F0″0pi+1+F1″0pi+2+G0″0mi+1+G1″0mi+2,(7)THi″π2=F0″π2pi+F1″π2pi+1+G0″π2mi+G1″π2mi+1.Because we hope the curve expressed in (5) can achieve C2 continuity, it must satisfy that THi″π/2=THi+1″0. Then, from Equations (6) and (7), we obtain
(8)F0″π2=0,F1″0=0,G0″π2=0,G1″0=0,F0″0=F1″π2,G0″0=G1″π2.Thus, the blending functionsFjt and Gjtj=1,2 need to have the characteristics expressed in Equation (8) in addition to the similar characteristics given by Equation (3).We expressFjt and Gjtj=1,2 as follows:
(9)F0tF1tG0tG1t=1sintcostsin2tsin3tcos3tM,where M is an undetermined matrix expressed as
(10)M=a11a12a13a14a21a22a23a24a31a32a33a34a41a42a43a44a51a52a53a54a61a62a63a64.From Equation (9), we have
(11)F00F10G00G10=101001M,(12)F0π/2F1π/2G0π/2G1π/2=110110M,(13)F0′0F1′0G0′0G1′0=010000M,(14)F0′π/2F1′π/2G0′π/2G1′π/2=00−1000M,(15)F0″0F1″0G0″0G1″0=00−120−3M,(16)F0″π/2F1″π/2G0″π/2G1″π/2=0−10−2−30M.BecauseFjt and Gjtj=1,2 should have the similar characteristics given by Equation (3), then from Equations (11), (12), (13), and (14) we obtain
(17)a11+a61=1,a12+a62=0,a13+a63=0,a14+a64=1,a11+a41+a51=0,a12+a42+a52=1,a13+a43+a53=−1,a14+a44+a54=0,a21=0,a22=0,a23=1,a24=0,a31=0,a32=0,a33=0,a34=−1.Recall thatFjt and Gjtj=1,2 should have the characteristics expressed in Equation (8), then from Equations (15) and (16) we obtain
(18)−a21−2a41−3a51=0,−a32+2a42−3a62=0,−a23−2a43−3a53=0,−a34+2a44−3a64=0,−a31+2a41−3a61=−a22−2a42−3a52,−a33+2a43−3a63=−a24−2a44−3a54.Note thatM has twenty-four undetermined numbers, it is necessary to set two free parameters in these undetermined numbers because Equations (17) and (18) only have twenty-two equations in total. If we set a61=α and a63=β as free parameters, α,β∈R, then from Equations (17) and (18) we could obtain
(19)M=1−α2α−1−β6β−1030010000−13α−131−α3β−232−β21−αα1−2β3β−83α21−αβ13−6β3.By substituting Equation (19) into Equation (9), we can obtain the cubic trigonometric blending functions expressed as
(20)F0t=1−α+3α−1S2+21−αS3+αC3,F1t=2α−1+31−αS2+αS3+21−αC3,G0t=−β+S+3β−2S2+1−2βS3+βC3,G1t=1323β−5−3C+92−βS2+3β−8S3+13−6βC3,where S≔sint, C≔cost, t∈0,π/2, α and β are free parameters, α,β∈R. Then, the cubic trigonometric Hermite interpolation curve expressed in Equation (5) is gotten naturally.To make the free parametersα and β local, we rewrite Equation (20) as follows:
(21)Fi,0t=1−αi+3αi−1S2+21−αiS3+αiC3,Fi,1t=2αi+1−1+31−αi+1S2+αi+1S3+21−αi+1C3,Gi,0t=−βi+S+3βi−2S2+1−2βiS3+βiC3,Gi,1t=1323βi+1−5−3C+92−βi+1S2+3βi+1−8S3+13−6βi+1C3,where S≔sint, C≔cost, t∈0,π/2, αi, αi+1, βi and βi+1 are free parameters. Then, we get the definition of the cubic trigonometric Hermite interpolation curve.Definition 1.
Given a series of pointspj together with the corresponding tangent vectors mjj=0,1,⋯,n, the cubic trigonometric Hermite interpolation curve (CTHI curve for short) is defined by
(22)THit=Fi,0tpi+Fi,1tpi+1+Gi,0tmi+Gi,1tmi+1,i=0,1,⋯,n−1,where 0≤t≤1, Fi,jt and Gi,jtj=0,1 are the cubic trigonometric blending functions expressed in Equation (21).Given the same endpoints and the corresponding tangent vectors, CTHI curve segments with different free parameters are shown in Figure1.Figure 1
CTHI curve segments with different free parameters. (a)αi,αi+1,βi,βi+1=−1,−0.5,0,0.5. (b) αi,αi+1,βi,βi+1=−0.5,0,1,0.5. (c) αi,αi+1,βi,βi+1=0.5,0,−0.5,−1. (d) αi,αi+1,βi,βi+1=0.5,1,0,−0.5.
(a)(b)(c)(d)


## 4. Properties of CTHI Curve

### 4.1. Interpolation and Continuity

Theorem 2.
CTHI curve interpolatespj and mjj=0,1,⋯,n, and achieves C2 continuity.Proof.
By computing from Equations (21) and (22), we have
(23)THi0=pi,THiπ/2=pi+1,THi′0=mi,THi′π/2=mi+1,THi″0=3αi−6pi+3βi−4mi,THi″π/2=3αi+1−6pi+1+3βi+1−4mi+1.
Equation (23) shows that CTHI curve interpolates the points pj and the tangent vectors mjj=0,1,⋯,n. In addition, Equation (23) also shows that THikπ/2=THi+1k0, k=0,1,2, i=0,1,⋯,n−1, which means CTHI curve achieves C2 continuity.


### 4.2. Shape Adjustability

Theorem 3.
For givenpj and mjj=0,1,⋯,n, the shape of CTHI curve can be adjusted locally or globally.Proof.
Equation (22) shows that CTHI curve is composed of n segments, and each segment contains four free parameters αi, αi+1, βi, and βi+1. When the points pj and the tangent vectors mjj=0,1,⋯,n are kept unchanged, we can adjust the shape of the curve locally by altering the values of the four free parameters contained in each segment. It is easy to find that the αi, βii=0,n−1 affect the shape of one segment, and αi, βii=1,2,⋯,n−2 affect the shape of two segments.
If we setαi=α, βi=β, i=0,1,⋯,n, CTHI curve only contains two free parameters α and β. When the points pj and the tangent vectors mjj=0,1,⋯,n are fixed, we can adjust the shape of the curve globally by changing the values of α and β.Example 4.
Givenp0=0,0, p1=1,0, p2=2,0, p3=3,0, p4=4,0, m0=0,1, m1=0,−1, m2=0,1, m3=0,−1, m4=0,1.
The corresponding CTHI curve is composed of four segments. Figures2 and 3 show the local and global adjustment of CTHI curve by modifying the values of the free parameters, respectively.Example 5.
Givenp0=−1/2,0,5/3, p1=−2,0,1, p2=−2,−5,0, p3=2,−5,0, p4=9/2,0,1, p5=2,0,5/3, p6=0,0,6, p7=−1/2,0,5/3, m0=−1,0,−1/2, m1=−3/4,−1/2,−1/2, m2=2,−5/2,−1/2, m3=13/4,5/2,1/2, m4=0,5/2,5/6, m5=−9/4,−1/2,0, m6=−5/4,0,0, m7=−1,0,−5/2.
The corresponding closed CTHI curve is composed of seven segments. Figures4 and 5 show the local and global adjustment of CTHI curve by modifying the values of the free parameters, respectively.Figure 2
Local adjustment of planar CTHI curve. (a)αi,βi=1,1i=1,2,3,4, but α0,β0=−0.5,0.5 (green line), α0,β0=0,1 (blue line), α0,β0=1.5,0 (red line). (b) αi,βi=1,1i=0,1,3,4, but α2,β2=−0.5,0.5 (green line), α2,β2=0,1 (blue line), α2,β2=1.5,0 (red line).
(a)(b)Figure 3
Global adjustment of planar CTHI curve. (a)α,β=0,1. (b) α,β=1.5,0.
(a)(b)Figure 4
Local adjustment of spatial CTHI curve. (a)αi,βi=1.5,1.5i=0,1,⋯,5,7, but α6,β6=0.5,2.5. (b) αi,βi=1.5,1.5i=0,1,⋯,5,7, but α6,β6=1.5,0.5.
(a)(b)Figure 5
Global adjustment of spatial CTHI curve. (a)α,β=1,1.5. (b) α,β=1.5,1.
(a)(b)


### 4.3. Accurate Representation of Some Engineering Curves

We consider using one segment of CTHI curve to represent some engineering curves.


#### 4.3.1. Accurate Representation of Elliptical Arc and Circular Arc

Theorem 6.
Letpi=0,b, pi+1=a,0, mi=a,0, mi+1=0,−b, ab≠0, and αi=αi+1=5/3, βi=βi+1=4/3, the CTHI curve segment THit accurately represents an elliptical arc as long as a≠b and a circular arc as long as a=b.Proof.
By substituting the given information into Equation (22), we have
(24)THit=asint,bcost,
wheret∈0,π/2. Equation (24) is the exact expression of an elliptical arc as long as a≠b and is the exact expression of a circular arc as long as a=b, see Figure 6.Figure 6
Accurate representation of an elliptical arc, wherea=1 and b=2.


#### 4.3.2. Accurate Representation of Quadratic Parabolic Arc

Theorem 7.
Letpi=−3a/4,−b/2, pi+1=a/4,b/2, mi=a,0, mi+1=0,0, ab≠0, and αi=αi+1=2/3, βi=βi+1=1/3, the CTHI curve segment THit accurately represents a quadratic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(25)THit=−3a4+asint,−b2+bsin2t,
wheret∈0,π/2. Equation (25) is the exact expression of a quadratic parabolic arc, see Figure 7.Figure 7
Accurate representation of a quadratic parabolic arc, wherea=1 and b=2.


#### 4.3.3. Accurate Representation of Cubic Parabolic Arc

Theorem 8.
Letpi=−a/3,b, pi+1=2a/3,0, mi=0,0, mi+1=a,0, ab≠0, and αi=αi+1=1, βi=βi+1=2, the CTHI curve segment THit accurately represents a cubic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(26)THit=2a3−acost,bcos3t,
wheret∈0,π/2. Equation (26) is the exact expression of a cubic parabolic arc, see Figure 8.Figure 8
Accurate representation of a cubic parabolic arc, wherea=1 and b=2.


#### 4.3.4. Accurate Representation of Astroid Arc

Theorem 9.
Letpi=0,a, pi+1=a,0, mi=0,0, mi+1=a,0, a≠0, and αi=αi+1=1, βi,βi+1∈R, the CTHI curve segment THit accurately represents an astroid arc.Proof.
By substituting the given information into Equation (22), we have
(27)THit=asin3t,acos3t,
wheret∈0,π/2. It is clear that Equation (27) is the exact expression of an astroid arc, see Figure 9.Figure 9
Accurate representation of an astroid arc, wherea=1.


## 4.1. Interpolation and Continuity

Theorem 2.
CTHI curve interpolatespj and mjj=0,1,⋯,n, and achieves C2 continuity.Proof.
By computing from Equations (21) and (22), we have
(23)THi0=pi,THiπ/2=pi+1,THi′0=mi,THi′π/2=mi+1,THi″0=3αi−6pi+3βi−4mi,THi″π/2=3αi+1−6pi+1+3βi+1−4mi+1.
Equation (23) shows that CTHI curve interpolates the points pj and the tangent vectors mjj=0,1,⋯,n. In addition, Equation (23) also shows that THikπ/2=THi+1k0, k=0,1,2, i=0,1,⋯,n−1, which means CTHI curve achieves C2 continuity.


## 4.2. Shape Adjustability

Theorem 3.
For givenpj and mjj=0,1,⋯,n, the shape of CTHI curve can be adjusted locally or globally.Proof.
Equation (22) shows that CTHI curve is composed of n segments, and each segment contains four free parameters αi, αi+1, βi, and βi+1. When the points pj and the tangent vectors mjj=0,1,⋯,n are kept unchanged, we can adjust the shape of the curve locally by altering the values of the four free parameters contained in each segment. It is easy to find that the αi, βii=0,n−1 affect the shape of one segment, and αi, βii=1,2,⋯,n−2 affect the shape of two segments.
If we setαi=α, βi=β, i=0,1,⋯,n, CTHI curve only contains two free parameters α and β. When the points pj and the tangent vectors mjj=0,1,⋯,n are fixed, we can adjust the shape of the curve globally by changing the values of α and β.Example 4.
Givenp0=0,0, p1=1,0, p2=2,0, p3=3,0, p4=4,0, m0=0,1, m1=0,−1, m2=0,1, m3=0,−1, m4=0,1.
The corresponding CTHI curve is composed of four segments. Figures2 and 3 show the local and global adjustment of CTHI curve by modifying the values of the free parameters, respectively.Example 5.
Givenp0=−1/2,0,5/3, p1=−2,0,1, p2=−2,−5,0, p3=2,−5,0, p4=9/2,0,1, p5=2,0,5/3, p6=0,0,6, p7=−1/2,0,5/3, m0=−1,0,−1/2, m1=−3/4,−1/2,−1/2, m2=2,−5/2,−1/2, m3=13/4,5/2,1/2, m4=0,5/2,5/6, m5=−9/4,−1/2,0, m6=−5/4,0,0, m7=−1,0,−5/2.
The corresponding closed CTHI curve is composed of seven segments. Figures4 and 5 show the local and global adjustment of CTHI curve by modifying the values of the free parameters, respectively.Figure 2
Local adjustment of planar CTHI curve. (a)αi,βi=1,1i=1,2,3,4, but α0,β0=−0.5,0.5 (green line), α0,β0=0,1 (blue line), α0,β0=1.5,0 (red line). (b) αi,βi=1,1i=0,1,3,4, but α2,β2=−0.5,0.5 (green line), α2,β2=0,1 (blue line), α2,β2=1.5,0 (red line).
(a)(b)Figure 3
Global adjustment of planar CTHI curve. (a)α,β=0,1. (b) α,β=1.5,0.
(a)(b)Figure 4
Local adjustment of spatial CTHI curve. (a)αi,βi=1.5,1.5i=0,1,⋯,5,7, but α6,β6=0.5,2.5. (b) αi,βi=1.5,1.5i=0,1,⋯,5,7, but α6,β6=1.5,0.5.
(a)(b)Figure 5
Global adjustment of spatial CTHI curve. (a)α,β=1,1.5. (b) α,β=1.5,1.
(a)(b)


## 4.3. Accurate Representation of Some Engineering Curves

We consider using one segment of CTHI curve to represent some engineering curves.


### 4.3.1. Accurate Representation of Elliptical Arc and Circular Arc

Theorem 6.
Letpi=0,b, pi+1=a,0, mi=a,0, mi+1=0,−b, ab≠0, and αi=αi+1=5/3, βi=βi+1=4/3, the CTHI curve segment THit accurately represents an elliptical arc as long as a≠b and a circular arc as long as a=b.Proof.
By substituting the given information into Equation (22), we have
(24)THit=asint,bcost,
wheret∈0,π/2. Equation (24) is the exact expression of an elliptical arc as long as a≠b and is the exact expression of a circular arc as long as a=b, see Figure 6.Figure 6
Accurate representation of an elliptical arc, wherea=1 and b=2.


### 4.3.2. Accurate Representation of Quadratic Parabolic Arc

Theorem 7.
Letpi=−3a/4,−b/2, pi+1=a/4,b/2, mi=a,0, mi+1=0,0, ab≠0, and αi=αi+1=2/3, βi=βi+1=1/3, the CTHI curve segment THit accurately represents a quadratic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(25)THit=−3a4+asint,−b2+bsin2t,
wheret∈0,π/2. Equation (25) is the exact expression of a quadratic parabolic arc, see Figure 7.Figure 7
Accurate representation of a quadratic parabolic arc, wherea=1 and b=2.


### 4.3.3. Accurate Representation of Cubic Parabolic Arc

Theorem 8.
Letpi=−a/3,b, pi+1=2a/3,0, mi=0,0, mi+1=a,0, ab≠0, and αi=αi+1=1, βi=βi+1=2, the CTHI curve segment THit accurately represents a cubic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(26)THit=2a3−acost,bcos3t,
wheret∈0,π/2. Equation (26) is the exact expression of a cubic parabolic arc, see Figure 8.Figure 8
Accurate representation of a cubic parabolic arc, wherea=1 and b=2.


### 4.3.4. Accurate Representation of Astroid Arc

Theorem 9.
Letpi=0,a, pi+1=a,0, mi=0,0, mi+1=a,0, a≠0, and αi=αi+1=1, βi,βi+1∈R, the CTHI curve segment THit accurately represents an astroid arc.Proof.
By substituting the given information into Equation (22), we have
(27)THit=asin3t,acos3t,
wheret∈0,π/2. It is clear that Equation (27) is the exact expression of an astroid arc, see Figure 9.Figure 9
Accurate representation of an astroid arc, wherea=1.


## 4.3.1. Accurate Representation of Elliptical Arc and Circular Arc

Theorem 6.
Letpi=0,b, pi+1=a,0, mi=a,0, mi+1=0,−b, ab≠0, and αi=αi+1=5/3, βi=βi+1=4/3, the CTHI curve segment THit accurately represents an elliptical arc as long as a≠b and a circular arc as long as a=b.Proof.
By substituting the given information into Equation (22), we have
(24)THit=asint,bcost,
wheret∈0,π/2. Equation (24) is the exact expression of an elliptical arc as long as a≠b and is the exact expression of a circular arc as long as a=b, see Figure 6.Figure 6
Accurate representation of an elliptical arc, wherea=1 and b=2.


## 4.3.2. Accurate Representation of Quadratic Parabolic Arc

Theorem 7.
Letpi=−3a/4,−b/2, pi+1=a/4,b/2, mi=a,0, mi+1=0,0, ab≠0, and αi=αi+1=2/3, βi=βi+1=1/3, the CTHI curve segment THit accurately represents a quadratic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(25)THit=−3a4+asint,−b2+bsin2t,
wheret∈0,π/2. Equation (25) is the exact expression of a quadratic parabolic arc, see Figure 7.Figure 7
Accurate representation of a quadratic parabolic arc, wherea=1 and b=2.


## 4.3.3. Accurate Representation of Cubic Parabolic Arc

Theorem 8.
Letpi=−a/3,b, pi+1=2a/3,0, mi=0,0, mi+1=a,0, ab≠0, and αi=αi+1=1, βi=βi+1=2, the CTHI curve segment THit accurately represents a cubic parabolic arc.Proof.
By substituting the given information into Equation (22), we have
(26)THit=2a3−acost,bcos3t,
wheret∈0,π/2. Equation (26) is the exact expression of a cubic parabolic arc, see Figure 8.Figure 8
Accurate representation of a cubic parabolic arc, wherea=1 and b=2.


## 4.3.4. Accurate Representation of Astroid Arc

Theorem 9.
Letpi=0,a, pi+1=a,0, mi=0,0, mi+1=a,0, a≠0, and αi=αi+1=1, βi,βi+1∈R, the CTHI curve segment THit accurately represents an astroid arc.Proof.
By substituting the given information into Equation (22), we have
(27)THit=asin3t,acos3t,
wheret∈0,π/2. It is clear that Equation (27) is the exact expression of an astroid arc, see Figure 9.Figure 9
Accurate representation of an astroid arc, wherea=1.


## 5. Shape Optimization of CTHI Curve Based on Internal Energy Minimization

As mentioned above, each segment of CTHI curve contains four free parametersαi, αi+1, βi, and βi+1. Although this will make the curve more flexible, it will also make the geometric operation more complex. We suggest taking αi=βi and αi+1=βi+1 in general, which cannot only maintain a certain degree of flexibility but also not make the operation too complicated. Then, the expression of CTHI curve can be described by
(28)THiαi,αi+1;t=Fi,0αi;tpi+Fi,1αi+1;tpi+1+Gi,0αi;tmi+Gi,1αi+1;tmi+1,i=0,1,⋯,n−1,where 0≤t≤1, pj, and mjj=0,1,⋯,n are given points and corresponding tangent vectors, respectively, Fi,jαi+j;t and Gi,jαi+j;tj=0,1 are expressed as
(29)Fi,0αi;t=1−αi+3αi−1S2+21−αiS3+αiC3,Fi,1αi+1;t=2αi+1−1+31−αi+1S2+αi+1S3+21−αi+1C3,Gi,0αi;t=−αi+S+3αi−2S2+1−2αiS3+αiC3,Gi,1αi+1;t=1323αi+1−5−3C+92−αi+1S2+3αi+1−8S3+13−6αi+1C3,and S≔sint, C≔cost, t∈0,π/2, and αi,αi+1∈R.Of course, we can adjust the shape of CTHI curve locally or globally by altering the values ofαi and αi+1 contained in each segment. However, it should also be noted that the shape of CTHI curve would be unsatisfactory if the free parameters are not selected properly.Example 10.
Given the same data as in Examples4 and 5. Figure 10 shows CTHI curves with inappropriate free parameters.Figure 10
CTHI curves with inappropriate free parameters. (a)α0,α1,⋯,α4=−2,1,−1,0,−3. (b) αi=0, i=0,1,⋯,7.
(a)(b)In addition, sometimes people may need to determine the free parameters to make the shape of CTHI curve meet certain specific geometric requirements. Hence, we can give schemes for optimizing the free parameters according to some objectives. Because the internal energy minimization has been successfully used to optimize the shape of some parametric curves, we adopt it to optimize the shape of CTHI curve.


### 5.1. Shape Optimization of Planar CTHI Curve

Three types of commonly used internal energy of a planar curve are stretch energy, strain energy (also called bending energy), and curvature variation energy (often replaced by Jerk’s energy). For a planar curvebta≤t≤b, its stretch energy, strain energy, and Jerk’s energy are usually approximately described by ∫abb′t2dt, ∫abb″t2dt and ∫abb‴t2dt, respectively, see [20, 21]. Since CTHI curve expressed in Equation (28) is composed of n segments, we can define the internal energy of planar CTHI curve as follows.Definition 11.
Three types of internal energy of planar CTHI curve are approximately expressed as(30)Ek=∑i=0n−1∫0THikαi,αi+1;t2dt,k=1,2,3,where THikαi,αi+1;t represents the kth derivative of THiαi,αi+1;t about t. Specifically, E1, E2 and E3 is the stretch energy, the strain (or bend) energy, and the curvature variation (or Jerk’s) energy of the curve, respectively.From Equations (29), we can rewrite Equation (28) as
(31)THiαi,αi+1;t=Aitαi+Bitαi+1+Cit,i=0,1,⋯,n−1,where
(32)Ait≔−1+3S2−2S3+C3pi+miBit≔2−3S2+S3−2C3pi+1+mi+1Cit≔1−3S2+2S3pi+−2+3S2+2C3pi+1+S−2S2+S3mi+13−10−3C+18S2−8S3+13C3mi+1,S≔sint,C≔cost,αi,αi+1∈R.By substituting Equation (31) into Equation (30) and by derivation, we obtain
(33)Ek=∑i=0n−1ak,iαi2+bk,iαi+12+ck,i+2dk,iαiαi+1+2ek,iαi+2fk,iαi+1,whereak,i≔∫0π/2Aikt2dt, bk,i≔∫0π/2Bikt2dt, ck,i≔∫0π/2Cikt2dt,dk,i≔∫0π/2Aikt⋅Biktdt, ek,i≔∫0π/2Aikt⋅Ciktdt, fk,i≔∫0π/2Bikt⋅Ciktdt.Becauseak,i, bk,i, ck,i, dk,i, ek,i, fk,i are constants once pi+j, mi+jj=0,1 and k are selected, the internal energy expressed in Equation (33) are quadratic functions of αii=0,1,⋯,n. Then, the following model can be gotten for generating planar CTHI curve with minimal internal energy,
(34)minα0,α1,⋯,αn∈REkα0,α1,⋯,αn,where k=1,2,3.From Equation (34), we have
(35)∂Ek∂α0=2ak,0α0+2dk,0α1+2ek,0,∂Ek∂αi=2dk,i−1αi−1+2ak,i+bk,i−1αi+2dk,iαi+1+2ek,i+fk,i−1,i=1,2,⋯,n−1,∂Ek∂αn=2dk,n−1αn−1+2bk,n−1αn+2fk,n−1.Since the solution of Equation (34) can be obtained by solving ∂Ek/∂αi=0, i=0,1,⋯,n, from Equation (35), we get the following equation system,
(36)ak,0α0+dk,0α1=−ek,0,dk,i−1αi−1+ak,i+bk,i−1αi+dk,iαi+1=−ek,i+fk,i−1,i=1,2,⋯,n−1,dk,n−1αn−1+bk,n−1αn=−fk,n−1,where k=1,2,3.LetAk be the coefficient matrix of Equation (36). It is clear that Ak is tridiagonal; then, the solution of Equation (36) can be easily obtained by using the LU method as long as Ak−1 exists for the given pj and mjj=0,1,⋯,n. Then, we can get the following theorem without proving.Theorem 12.
For selectedkk=1,2,3, suppose Ak−1 exists for the given pj and mjj=0,1,⋯,n, let hk,0≔−ek,0, hk,j≔−ek,j+fk,j−1, j=1,2,⋯,n−1, hk,n≔−fk,n−1, gi,jk≔Ak−1; the solution of Equation (34), denoted by αk,i, i=0,1,2,⋯,n, is expressed as
(37)αk,i=∑j=0ngi,jkhk,j,i=0,1,2,⋯,n.Example 13.
Given
p0=1,2, p1=3,4, p2=5,2, p3=7,4, p4=9,2,
m0=1,2, m1=−4,0, m2=2,0, m3=−4,0, m4=1,2.The corresponding CTHI curve is composed of four segments. By computing from Equation (37), the optimal free parameters can be obtained as follows:
(a)
Stretch energy minimization (viz.k=1),(38)α1,0,α1,1,⋯,α1,4=1.3955,1.3959,2.1150,1.7660,2.3859(b)
Strain energy minimization (viz.k=2),(39)α2,0,α2,1,⋯,α2,4=1.8858,1.6040,1.9430,2.1402,1.8706(c)
Curvature variation energy minimization (viz.k=3),(40)α3,0,α3,1,⋯,α3,4=1.8262,1.5796,1.7821,1.9774,1.5931The CTHI curves generated by taking inappropriate free parameters and minimizing the internal energy are shown in Figure11.Example 14.
Givenp0=0,0, p1=3,1, p2=5,2, p3=6,4, p4=8,5, p5=11,6, m0=0.8,1.5, m1=3.5,0.9, m2=0.9,0.6, m3=1.6,1.0, m4=1.3,1.3, and m5=4.9,1.3.
The corresponding CTHI curve is composed of five segments. By computing from Equation (37), the optimal free parameters can be obtained as follows:
(a)
Stretch energy minimization (viz.k=1),(41)α1,0,α1,1,⋯,α1,5=1.4738,1.6062,1.7669,1.9381,1.8368,2.1508(b)
Strain energy minimization (viz.k=2),(42)α2,0,α2,1,⋯,α2,5=1.1326,1.6069,1.8997,1.8654,1.8644,1.9352(c)
Curvature variation energy minimization (viz.k=3),(43)α3,0,α3,1,⋯,α3,5=0.8390,1.5787,1.9148,1.8552,1.8832,1.9413
The CTHI curves generated by taking inappropriate free parameters and minimizing the internal energy are shown in Figure12.Example 15.
Let us consider the following data taken from a unit circle,
pi=costi,sinti, mi=−sinti,costi, ti=iπ/3, i=0,1,⋯,6.Figure 11
The CTHI curves generated in Example13. (a) Taking inappropriate free parameters αi=1, i=0,1,⋯,4. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)Figure 12
The CTHI curves generated in Example14. (a) Taking inappropriate free parameters αi=1.2, i=0,1,⋯,5. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)The corresponding CTHI curve is composed of six segments. By computing from Equation (37), the optimal free parameters can be obtained as follows,
(a)
Stretch energy minimization (viz.k=1),(44)α1,0,α1,1,⋯,α1,6=1.0435,1.3575,1.3140,1.3192,1.3244,1.2809,1.5949(b)
Strain energy minimization (viz.k=2),(45)α2,0,α2,1,⋯,α2,6=1.5268,1.6235,1.6179,1.6182,1.6185,1.6130,1.7097(c)
Curvature variation energy minimization (viz.k=3),(46)α3,0,α3,1,⋯,α3,6=1.4231,1.6655,1.6836,1.6851,1.6866,1.7048,1.9471The CTHI curves (solid lines) generated by taking inappropriate free parameters and minimizing the internal energy and the unit circle (dotted lines) are shown in Figure13.Figure 13
The CTHI curves generated in example15. (a) Taking inappropriate free parameters α0,α1,⋯,α6=−2,−1,1,0,2,−1,1. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)Figures11–13 show that planar CTHI curves generated by the internal energy minimization are satisfactory compared with the curves whose free parameters are not appropriate.


### 5.2. Shape Optimization of Spatial CTHI Curve

As we know, planar curves only need to consider bending, while spatial curves should consider not only bending but also twisting. For a spatial curvebta≤t≤b, its bending energy and twisting energy can be described by [20]
(47)Ebend=∫abκ2tb′t2dt,Etwist=∫abτ2tb′t2dt,where κt and τt represents the curvature and the torsion of the curve, respectively.To facilitate the calculation, Equation (47) can be approximately described by [20]
(48)E^bend=∫abb″t2dt,E^twist=∫abb‴t2dt.It should be noted that Equation (48) is regarded as approximate bending energy (or strain energy) and curvature variation energy (or Jerk’s energy) of planar curves. Here, Equation (48) is adopted to describe the approximate bending energy and twisting energy of spatial curves. According to Definition 1, the blending energy and the twisting energy of CTHI curve can be approximately expressed as E2 and E3, respectively. Since spatial CTHI curve should consider bending and twisting at the same time, we can define the internal energy of spatial CTHI curve as follows.Definition 16.
The internal energy of spatial CTHI curve is approximately expressed as(49)E=λE2+1−λE3,where λ0<λ<1 is the weight, and
(50)E2=∑i=0n−1∫0π/2THi″αi,αi+1;t2dt,E3=∑i=0n−1∫0π/2THi‴αi,αi+1;t2dt.From Equation (33), we can find that the energy expressed in Equation (49) is a quadratic function of αii=0,1,⋯,n. Then, we can obtain the following model for generating spatial CTHI curve with minimal internal energy,
(51)minα0,α1,⋯,αn∈REα0,α1,⋯,αn=λE2α0,α1,⋯,αn+1−λE3α0,α1,⋯,αn.Before solving Equation (51), the value of λ needs to be selected. Here, we use the sorting algorithm [29] to select the value of λ, which is described in Algorithm 1.Algorithm 1: Determining the value of weight λ in Equation (51).
Step 1: compute the deviations between the bending energyE2 and the twisting energy E3, that is,δ1=E2α3,0,α3,1,⋯,α3,n−E2α2,0,α2,1,⋯,α2,n,δ2=E3α2,0,α2,1,⋯,α2,n−E3α3,0,α3,1,⋯,α3,n,whereαk,i, k=2,3, i=0,1,2,⋯,n are obtained by Equation (37).Step 2: compute the initial weights of the bending energyE2 and the twisting energy E3, that is, λ1=δ1/δ1+δ2, λ2=δ2/δ1+δ2.Step 3: ifλ1≥λ2, the weight is taken as λ=λ2; else, the weight is taken as λ=λ1.Then, according to Equations (35) and (49), we have
(52)∂E∂α0=2λa2,0+1−λa3,0α0+2λd2,0+1−λd3,0α1+2λe2,0+1−λe3,0,∂E∂αi=2λd2,i−1+1−λd3,i−1αi−1+2λa2,i+b2,i−1+1−λa3,i+b3,i−1αi+2λd2,i+1−λd3,iαi+1+2λe2,i+f2,i−1+1−λe3,i+f3,i−1,i=1,2,⋯,n−1,∂E∂αn=2λd2,n−1+1−λd3,n−1αn−1+2λb2,n−1+1−λb3,n−1αn+2λf2,n−1+1−λf3,n−1.Since the solution of Equation (51) can be obtained by solving ∂E/∂αi=0, i=0,1,⋯,n, from Equation (52), we get the following equation system,
(53)λa2,0+1−λa3,0α0+λd2,0+1−λd3,0α1=−λe2,0+1−λe3,0,λd2,i−1+1−λd3,i−1αi−1+λa2,i+b2,i−1+1−λa3,i+b3,i−1αi+λd2,i+1−λd3,iαi+1=−λe2,i+f2,i−1+1−λe3,i+f3,i−1,i=1,2,⋯,n−1,λd2,n−1+1−λd3,n−1αn−1+λb2,n−1+1−λb3,n−1αn=−λf2,n−1+1−λf3,n−1.LetB be the coefficient matrix of Equation (53). It is clear that B is tridiagonal; then, the solution of Equation (53) can be easily obtained by using the LU method as long as B−1 exists for the given pj and mjj=0,1,⋯,n. Then, we can get the following theorem without proving.Theorem 17.
SupposeB−1 exists for the given pj and mjj=0,1,⋯,n, let
s0≔−λe2,0+1−λe3,0, sj≔−λe2,j+f2,j−1+1−λe3,j+f3,j−1, j=1,2,⋯,n−1,
sn≔−λf2,n−1+1−λf3,n−1, ri,j≔B−1.
The solution of Equation (51) is expressed as
(54)αi=∑j=0nri,jsj,i=0,1,2,⋯,n.Example 18.
Let us consider the following data taken from a cylindrical spiral,pi=costi,sinti,2ti, mi=−sinti,costi,2, t=iπ/2, i=0,1,2,3,4.
The corresponding CTHI curve is composed of four segments. By computing from Equation (54), the optimal free parameters of CTHI curve with minimal internal energy are
(55)α0,α1,⋯,α4=1.4215,1.4992,1.4992,1.4992,1.5768.
The CTHI curves (solid lines) generated by taking inappropriate free parameters and minimizing the internal energy and the cylindrical spiral (dotted lines) are shown in Figure14.Figure 14
The CTHI curves generated in Example18. (a) Taking inappropriate free parameters α0,α1,⋯,α4=1,0,−2,1,−1. (b) Internal energy minimization.
(a)(b)Figure14 shows that spatial CTHI curve generated by minimizing the internal energy is satisfactory compared with the curve with inappropriate free parameters.


## 5.1. Shape Optimization of Planar CTHI Curve

Three types of commonly used internal energy of a planar curve are stretch energy, strain energy (also called bending energy), and curvature variation energy (often replaced by Jerk’s energy). For a planar curvebta≤t≤b, its stretch energy, strain energy, and Jerk’s energy are usually approximately described by ∫abb′t2dt, ∫abb″t2dt and ∫abb‴t2dt, respectively, see [20, 21]. Since CTHI curve expressed in Equation (28) is composed of n segments, we can define the internal energy of planar CTHI curve as follows.Definition 11.
Three types of internal energy of planar CTHI curve are approximately expressed as(30)Ek=∑i=0n−1∫0THikαi,αi+1;t2dt,k=1,2,3,where THikαi,αi+1;t represents the kth derivative of THiαi,αi+1;t about t. Specifically, E1, E2 and E3 is the stretch energy, the strain (or bend) energy, and the curvature variation (or Jerk’s) energy of the curve, respectively.From Equations (29), we can rewrite Equation (28) as
(31)THiαi,αi+1;t=Aitαi+Bitαi+1+Cit,i=0,1,⋯,n−1,where
(32)Ait≔−1+3S2−2S3+C3pi+miBit≔2−3S2+S3−2C3pi+1+mi+1Cit≔1−3S2+2S3pi+−2+3S2+2C3pi+1+S−2S2+S3mi+13−10−3C+18S2−8S3+13C3mi+1,S≔sint,C≔cost,αi,αi+1∈R.By substituting Equation (31) into Equation (30) and by derivation, we obtain
(33)Ek=∑i=0n−1ak,iαi2+bk,iαi+12+ck,i+2dk,iαiαi+1+2ek,iαi+2fk,iαi+1,whereak,i≔∫0π/2Aikt2dt, bk,i≔∫0π/2Bikt2dt, ck,i≔∫0π/2Cikt2dt,dk,i≔∫0π/2Aikt⋅Biktdt, ek,i≔∫0π/2Aikt⋅Ciktdt, fk,i≔∫0π/2Bikt⋅Ciktdt.Becauseak,i, bk,i, ck,i, dk,i, ek,i, fk,i are constants once pi+j, mi+jj=0,1 and k are selected, the internal energy expressed in Equation (33) are quadratic functions of αii=0,1,⋯,n. Then, the following model can be gotten for generating planar CTHI curve with minimal internal energy,
(34)minα0,α1,⋯,αn∈REkα0,α1,⋯,αn,where k=1,2,3.From Equation (34), we have
(35)∂Ek∂α0=2ak,0α0+2dk,0α1+2ek,0,∂Ek∂αi=2dk,i−1αi−1+2ak,i+bk,i−1αi+2dk,iαi+1+2ek,i+fk,i−1,i=1,2,⋯,n−1,∂Ek∂αn=2dk,n−1αn−1+2bk,n−1αn+2fk,n−1.Since the solution of Equation (34) can be obtained by solving ∂Ek/∂αi=0, i=0,1,⋯,n, from Equation (35), we get the following equation system,
(36)ak,0α0+dk,0α1=−ek,0,dk,i−1αi−1+ak,i+bk,i−1αi+dk,iαi+1=−ek,i+fk,i−1,i=1,2,⋯,n−1,dk,n−1αn−1+bk,n−1αn=−fk,n−1,where k=1,2,3.LetAk be the coefficient matrix of Equation (36). It is clear that Ak is tridiagonal; then, the solution of Equation (36) can be easily obtained by using the LU method as long as Ak−1 exists for the given pj and mjj=0,1,⋯,n. Then, we can get the following theorem without proving.Theorem 12.
For selectedkk=1,2,3, suppose Ak−1 exists for the given pj and mjj=0,1,⋯,n, let hk,0≔−ek,0, hk,j≔−ek,j+fk,j−1, j=1,2,⋯,n−1, hk,n≔−fk,n−1, gi,jk≔Ak−1; the solution of Equation (34), denoted by αk,i, i=0,1,2,⋯,n, is expressed as
(37)αk,i=∑j=0ngi,jkhk,j,i=0,1,2,⋯,n.Example 13.
Given
p0=1,2, p1=3,4, p2=5,2, p3=7,4, p4=9,2,
m0=1,2, m1=−4,0, m2=2,0, m3=−4,0, m4=1,2.The corresponding CTHI curve is composed of four segments. By computing from Equation (37), the optimal free parameters can be obtained as follows:
(a)
Stretch energy minimization (viz.k=1),(38)α1,0,α1,1,⋯,α1,4=1.3955,1.3959,2.1150,1.7660,2.3859(b)
Strain energy minimization (viz.k=2),(39)α2,0,α2,1,⋯,α2,4=1.8858,1.6040,1.9430,2.1402,1.8706(c)
Curvature variation energy minimization (viz.k=3),(40)α3,0,α3,1,⋯,α3,4=1.8262,1.5796,1.7821,1.9774,1.5931The CTHI curves generated by taking inappropriate free parameters and minimizing the internal energy are shown in Figure11.Example 14.
Givenp0=0,0, p1=3,1, p2=5,2, p3=6,4, p4=8,5, p5=11,6, m0=0.8,1.5, m1=3.5,0.9, m2=0.9,0.6, m3=1.6,1.0, m4=1.3,1.3, and m5=4.9,1.3.
The corresponding CTHI curve is composed of five segments. By computing from Equation (37), the optimal free parameters can be obtained as follows:
(a)
Stretch energy minimization (viz.k=1),(41)α1,0,α1,1,⋯,α1,5=1.4738,1.6062,1.7669,1.9381,1.8368,2.1508(b)
Strain energy minimization (viz.k=2),(42)α2,0,α2,1,⋯,α2,5=1.1326,1.6069,1.8997,1.8654,1.8644,1.9352(c)
Curvature variation energy minimization (viz.k=3),(43)α3,0,α3,1,⋯,α3,5=0.8390,1.5787,1.9148,1.8552,1.8832,1.9413
The CTHI curves generated by taking inappropriate free parameters and minimizing the internal energy are shown in Figure12.Example 15.
Let us consider the following data taken from a unit circle,
pi=costi,sinti, mi=−sinti,costi, ti=iπ/3, i=0,1,⋯,6.Figure 11
The CTHI curves generated in Example13. (a) Taking inappropriate free parameters αi=1, i=0,1,⋯,4. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)Figure 12
The CTHI curves generated in Example14. (a) Taking inappropriate free parameters αi=1.2, i=0,1,⋯,5. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)The corresponding CTHI curve is composed of six segments. By computing from Equation (37), the optimal free parameters can be obtained as follows,
(a)
Stretch energy minimization (viz.k=1),(44)α1,0,α1,1,⋯,α1,6=1.0435,1.3575,1.3140,1.3192,1.3244,1.2809,1.5949(b)
Strain energy minimization (viz.k=2),(45)α2,0,α2,1,⋯,α2,6=1.5268,1.6235,1.6179,1.6182,1.6185,1.6130,1.7097(c)
Curvature variation energy minimization (viz.k=3),(46)α3,0,α3,1,⋯,α3,6=1.4231,1.6655,1.6836,1.6851,1.6866,1.7048,1.9471The CTHI curves (solid lines) generated by taking inappropriate free parameters and minimizing the internal energy and the unit circle (dotted lines) are shown in Figure13.Figure 13
The CTHI curves generated in example15. (a) Taking inappropriate free parameters α0,α1,⋯,α6=−2,−1,1,0,2,−1,1. (b) Stretch energy minimization. (c) Strain energy minimization. (d) Curvature variation energy minimization.
(a)(b)(c)(d)Figures11–13 show that planar CTHI curves generated by the internal energy minimization are satisfactory compared with the curves whose free parameters are not appropriate.


## 5.2. Shape Optimization of Spatial CTHI Curve

As we know, planar curves only need to consider bending, while spatial curves should consider not only bending but also twisting. For a spatial curvebta≤t≤b, its bending energy and twisting energy can be described by [20]
(47)Ebend=∫abκ2tb′t2dt,Etwist=∫abτ2tb′t2dt,where κt and τt represents the curvature and the torsion of the curve, respectively.To facilitate the calculation, Equation (47) can be approximately described by [20]
(48)E^bend=∫abb″t2dt,E^twist=∫abb‴t2dt.It should be noted that Equation (48) is regarded as approximate bending energy (or strain energy) and curvature variation energy (or Jerk’s energy) of planar curves. Here, Equation (48) is adopted to describe the approximate bending energy and twisting energy of spatial curves. According to Definition 1, the blending energy and the twisting energy of CTHI curve can be approximately expressed as E2 and E3, respectively. Since spatial CTHI curve should consider bending and twisting at the same time, we can define the internal energy of spatial CTHI curve as follows.Definition 16.
The internal energy of spatial CTHI curve is approximately expressed as(49)E=λE2+1−λE3,where λ0<λ<1 is the weight, and
(50)E2=∑i=0n−1∫0π/2THi″αi,αi+1;t2dt,E3=∑i=0n−1∫0π/2THi‴αi,αi+1;t2dt.From Equation (33), we can find that the energy expressed in Equation (49) is a quadratic function of αii=0,1,⋯,n. Then, we can obtain the following model for generating spatial CTHI curve with minimal internal energy,
(51)minα0,α1,⋯,αn∈REα0,α1,⋯,αn=λE2α0,α1,⋯,αn+1−λE3α0,α1,⋯,αn.Before solving Equation (51), the value of λ needs to be selected. Here, we use the sorting algorithm [29] to select the value of λ, which is described in Algorithm 1.Algorithm 1: Determining the value of weight λ in Equation (51).
Step 1: compute the deviations between the bending energyE2 and the twisting energy E3, that is,δ1=E2α3,0,α3,1,⋯,α3,n−E2α2,0,α2,1,⋯,α2,n,δ2=E3α2,0,α2,1,⋯,α2,n−E3α3,0,α3,1,⋯,α3,n,whereαk,i, k=2,3, i=0,1,2,⋯,n are obtained by Equation (37).Step 2: compute the initial weights of the bending energyE2 and the twisting energy E3, that is, λ1=δ1/δ1+δ2, λ2=δ2/δ1+δ2.Step 3: ifλ1≥λ2, the weight is taken as λ=λ2; else, the weight is taken as λ=λ1.Then, according to Equations (35) and (49), we have
(52)∂E∂α0=2λa2,0+1−λa3,0α0+2λd2,0+1−λd3,0α1+2λe2,0+1−λe3,0,∂E∂αi=2λd2,i−1+1−λd3,i−1αi−1+2λa2,i+b2,i−1+1−λa3,i+b3,i−1αi+2λd2,i+1−λd3,iαi+1+2λe2,i+f2,i−1+1−λe3,i+f3,i−1,i=1,2,⋯,n−1,∂E∂αn=2λd2,n−1+1−λd3,n−1αn−1+2λb2,n−1+1−λb3,n−1αn+2λf2,n−1+1−λf3,n−1.Since the solution of Equation (51) can be obtained by solving ∂E/∂αi=0, i=0,1,⋯,n, from Equation (52), we get the following equation system,
(53)λa2,0+1−λa3,0α0+λd2,0+1−λd3,0α1=−λe2,0+1−λe3,0,λd2,i−1+1−λd3,i−1αi−1+λa2,i+b2,i−1+1−λa3,i+b3,i−1αi+λd2,i+1−λd3,iαi+1=−λe2,i+f2,i−1+1−λe3,i+f3,i−1,i=1,2,⋯,n−1,λd2,n−1+1−λd3,n−1αn−1+λb2,n−1+1−λb3,n−1αn=−λf2,n−1+1−λf3,n−1.LetB be the coefficient matrix of Equation (53). It is clear that B is tridiagonal; then, the solution of Equation (53) can be easily obtained by using the LU method as long as B−1 exists for the given pj and mjj=0,1,⋯,n. Then, we can get the following theorem without proving.Theorem 17.
SupposeB−1 exists for the given pj and mjj=0,1,⋯,n, let
s0≔−λe2,0+1−λe3,0, sj≔−λe2,j+f2,j−1+1−λe3,j+f3,j−1, j=1,2,⋯,n−1,
sn≔−λf2,n−1+1−λf3,n−1, ri,j≔B−1.
The solution of Equation (51) is expressed as
(54)αi=∑j=0nri,jsj,i=0,1,2,⋯,n.Example 18.
Let us consider the following data taken from a cylindrical spiral,pi=costi,sinti,2ti, mi=−sinti,costi,2, t=iπ/2, i=0,1,2,3,4.
The corresponding CTHI curve is composed of four segments. By computing from Equation (54), the optimal free parameters of CTHI curve with minimal internal energy are
(55)α0,α1,⋯,α4=1.4215,1.4992,1.4992,1.4992,1.5768.
The CTHI curves (solid lines) generated by taking inappropriate free parameters and minimizing the internal energy and the cylindrical spiral (dotted lines) are shown in Figure14.Figure 14
The CTHI curves generated in Example18. (a) Taking inappropriate free parameters α0,α1,⋯,α4=1,0,−2,1,−1. (b) Internal energy minimization.
(a)(b)Figure14 shows that spatial CTHI curve generated by minimizing the internal energy is satisfactory compared with the curve with inappropriate free parameters.


## 6. Conclusion

We first proposed a cubic trigonometric Hermite interpolation curve in this paper. Different from cubic Hermite interpolation curve, the proposed curve achievesC2 continuity, has local and global adjustability, and can accurately represent some engineering curves. These features are very conducive to geometric modeling. To generate the trigonometric Hermite interpolation curve with specific shapes, the schemes for determining the optimal values of the free parameters based on internal energy minimization are presented. The planar curves and spatial curves with minimal internal energy are satisfactory compared with the curves with inappropriate free parameters. In contrast to cubic Hermite interpolation curve, the cubic trigonometric Hermite interpolation curve is more practical in geometric modeling.



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