Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines

Abstract

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a global 𝐺² continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.

1. Introduction

Parametric curves/surfaces and implicit curves/surfaces are two important topics in computer-aided geometry design and geometric modelling. With the parametric form, it is easy to generate points on a general curve/surface and plot it. On the other hand, it is convenient to determine whether a point is on, inside, or outside a given solid with the implicit treatments.

For any rational parametric curve/surface, we can convert it into implicit form. However, for a general parametric curve/surface, we usually cannot compute its exact implicit form. Even though its exact implicit form can be computed, the curve/surface implicitization always involves relatively complicated computation and the degree of the implicit curves/surfaces is high. Another difficulty is that implicit curves/surfaces may have unexpected components and self-intersections which lead to computational instability and topological inconsistency in geometric modeling. All these unsatisfied properties limit the applications of the exact implicitization (especially surface implicitization) in practical fields.

Due to these reasons, finding approximate implicitization of parametric curves/surfaces has some practical significance. In recent years, many researches have proposed several approaches to solve this problem [1–10]. The earlier work on approximate implicitization was done by Velho and Gomes [1], who presented an approximate implicitization scheme from parametric surfaces to implicit surfaces based on wavelet analysis. In 1999, Sederberg et al. [2] proposed an approach to solve approximate implicitization problem by using monoid curves and surfaces. The method used by Sederberg was made more available in Dokken’s work [3, 4]. In 2004, Chen and Deng [6] presented the concept of interval implicitization of rational curves and developed the corresponding effective algorithm. In 2006, Li et al. [7] considered the approximate implicitization of planar parametric curves by using the piecewise quadratic Bézier spline curves with 𝐺¹ continuity. In 2007, Wang and wu [8] discussed the approximate implicitization of general parametric curves based on radial basis function networks and multiquadric (MQ) quasi-interpolation. Very recently, Wu et al. [9, 10] discussed the approximate implicitization of parametric surfaces with the introduction of normal constraint points based on multivariate interpolation by using compactly supported radial basis functions, and approximate implicitization of parametric curves by using quadratic algebraic splines.

In this paper, an algorithm is proposed to solve the approximate implicitization of planar parametric curves using cubic Bernstein-Bézier implicit curves. Our piecewise cubic curves are used to give a global 𝐺² continuity approximation, because they keep the same endpoints, the corresponding tangent directions, and curvatures at the separated points with the approximated segments. Approximation error on rational curves is also given.

2. Cubic Bernstein-Bézier Implicit Curve

In this section, some concepts and results on Bernstein-Bézier implicit curve are presented. For more details, the readers may refer to [11–13] and references therein.

By 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] we denote a triangle with vertices 𝑝₁=(𝑥₁,𝑦₁),𝑝₂=(𝑥₂,𝑦₂), and 𝑣₁₂=(𝑥₁₂,𝑦₁₂), and by [𝑝₁𝑝₂] we denote the line passing through the points 𝑝₁ and 𝑝₂. If we denote area ([𝑣₁𝑣₂𝑣₃]) as the area of triangle [𝑣₁𝑣₂𝑣₃], then the barycentric coordinates (𝑢,𝑣,𝑤) of any point 𝑝=(𝑥,𝑦) with respect to 𝑇 are defined by 𝑢=area(𝑝,𝑣₂,𝑝₁₂)area(𝑣₁,𝑣₂,𝑝₁₂),𝑣=area(𝑣₁,𝑝,𝑝₁₂)area(𝑣₁,𝑣₂,𝑝₁₂),𝑤=area(𝑣₁,𝑣₂,𝑝)area(𝑣₁,𝑣₂,𝑝₁₂). (2.1)

Thus, any point 𝑝=(𝑥,𝑦) with respect to 𝑇 can be described as 𝑝=𝑢𝑝₁+𝑣𝑝₂+𝑤𝑣₁₂,𝑢+𝑣+𝑤=1. (2.2)

The Bernstein polynomials are shown as follows: 𝐵³_𝑖𝑗𝑘(𝑢,𝑣,𝑤)=3!𝑢𝑖!𝑗!𝑘!^𝑖𝑣^𝑗𝑤^𝑘,𝑖+𝑗+𝑘=3. (2.3)

When any of the following is true: 𝑖,𝑗,𝑘<0 and 𝑖,𝑗,𝑘>3, the Bernstein polynomial 𝐵³_𝑖𝑗𝑘(𝑢,𝑣,𝑤) is set to zero.

Therefore, the Bézier triangle patch of degree three in Bernstein form is 𝑓(𝑢,𝑣,𝑤)=_{𝑖+𝑗+𝑘=3}𝑏_𝑖𝑗𝑘𝐵³_𝑖𝑗𝑘(𝑢,𝑣,𝑤), (2.4) where all 𝑏_𝑖𝑗𝑘 are called Bézier control points (see Figure 1).

Figure 1 

The control points of a Bézier triangle patch of degree three.

Definition 2.1. Let 𝑓(𝑢,𝑣,𝑤) be defined as (2.4), the cubic Bernstein-Bézier implicit curve 𝒞 on the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] is defined to be the zero contour of 𝑓(𝑢,𝑣,𝑤), that is, 𝒞∶={(𝑢,𝑣,𝑤)∣𝑓(𝑢,𝑣,𝑤)=0}. (2.5)

Theorem 2.2 (see [12]). The directional derivative of Bézier triangle patch at the point 𝑝=(𝑢,𝑣,𝑤) with respect to the direction 𝛼=(𝛼₁,𝛼₂,𝛼₃) is given by 𝐷_𝛼𝑓(𝑢,𝑣,𝑤)=3_{𝑖+𝑗+𝑘=2}[𝛼₁𝑏_{𝑖+1𝑗𝑘}+𝛼₂𝑏_{𝑖𝑗+1𝑘}+𝛼₃𝑏_{𝑖𝑗𝑘+1}]𝐵²_𝑖𝑗𝑘(𝑢,𝑣,𝑤). (2.6)

Lemma 2.3. For the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] and 𝑓(𝑢,𝑣,𝑤) defined as in (2.4), if 𝑏₃₀₀=𝑏₂₀₁=0, then the curve 𝒞 passes through 𝑝₁ and is tangent with the line [𝑝₁𝑣₁₂] at 𝑝₁. Similarly, if 𝑏₀₃₀=𝑏₀₂₁=0, then 𝒞 passes through 𝑝₂ and is tangent with the line [𝑝₂𝑣₁₂] at 𝑝₂.

Proof. Since the barycentric coordinate of 𝑝₁ and direction [𝑝₁𝑣₁₂] with respect to the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] is (1,0,0) and (1,0,−1), respectively. Then the curve 𝒞 passes through 𝑝₁ and is tangent with the line [𝑝₁𝑣₁₂] at 𝑝₁ if and only if 𝑓(1,0,0)=0 and 𝐷_(1,0,−1)𝑓(1,0,0)=0. Thus, we get 𝑏₃₀₀=𝑏₂₀₁=0 with Theorem 2.2. The later of this lemma can be proved similarly.

Lemma 2.4. Let 𝑓(𝑢,𝑣,𝑤) be defined as (2.4). Its curvature of 𝒞 at 𝑝₁ is given by 𝜅₁𝑏=|¹⁰²𝑏₂₁₀|𝜇₁,where𝜇₁=2|(𝑥₂−𝑥₁)(𝑦₁₂−𝑦₁)−(𝑦₂−𝑦₁)(𝑥₁₂−𝑥₁)|((𝑥₁₂−𝑥₁)²+(𝑦₁₂−𝑦₁)²)^3/2. (2.7) Similarly, its curvature of 𝒞 at 𝑝₂ is given by 𝜅₂𝑏=|⁰¹²𝑏₁₂₀|𝜇₂,where𝜇₂=2|(𝑥₁−𝑥₂)(𝑦₁₂−𝑦₂)−(𝑦₁−𝑦₂)(𝑥₁₂−𝑥₂)|((𝑥₁₂−𝑥₂)²+(𝑦₁₂−𝑦₂)²)^3/2. (2.8)

Proof. It can be derived from the curvature formula [14] of implicit curves: 𝜅=−(−𝑓_𝑦,𝑓_𝑥)(^{𝑓𝑥𝑥𝑓𝑥𝑦}_{𝑓𝑦𝑥𝑓𝑦𝑦})(^−𝑓𝑦_𝑓𝑥)(𝑓²_𝑥+𝑓²_𝑦)^3/2, (2.9) where 𝑓_𝑥=(𝜕𝑓/𝜕𝑢)(𝜕𝑢/𝜕𝑥)+(𝜕𝑓/𝜕𝑣)(𝜕𝑣/𝜕𝑥)+(𝜕𝑓/𝜕𝑤)(𝜕𝑤/𝜕𝑥) and the other expressions can be understood similarly.

If the equation of cubic Bézier curve 𝑓(𝑢,𝑣,𝑤) in the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] is expressed in the form of 𝑓(𝑢,𝑣,𝑤)=𝑏₂₁₀3𝑢²𝑣+𝑏₁₂₀3𝑢𝑣²+𝑏₁₁₁𝜅6𝑢𝑣𝑤−¹𝜇₁𝑏₂₁₀3𝑢𝑤²−𝜅₂𝜇₂𝑏₁₂₀3𝑣𝑤²−𝑤³, (2.10) with the restrictions of 𝑏₂₁₀>0 and 𝑏₁₂₀>0, then from Lemmas 2.1 and 2.2, we can easily have the following.

Proposition 2.5. For the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] and 𝑓(𝑢,𝑣,𝑤) is defined as (2.10), then the curve 𝒞∶={(𝑢,𝑣,𝑤)∣𝑓(𝑢,𝑣,𝑤)=0} has the following properties: (i)𝒞 passes through the points 𝑝₁ and 𝑝₂.,(ii)𝒞 is tangent with line [𝑝₁𝑣₁₂] at 𝑝₁, and tangent with line [𝑝₂𝑣₁₂] at 𝑝₂, respectively,(iii)the curvatures of 𝒞 at the points 𝑝₁ and 𝑝₂ are 𝜅₁ and 𝜅₂, respectively.

The following result is due to Xu et al. [13].

Proposition 2.6. For the triangle 𝑇∶=[𝑝₁𝑝₂𝑣₁₂] and let 𝑓(𝑢,𝑣,𝑤) be defined as in (2.10), then the curve 𝒞 is 𝐷₁([𝑝₁𝑝₂𝑣₁₂],𝑣₁₂,𝑝₁𝑝₂)-regular, that is, the straight lines that passes through 𝑣₁₂ and the point on the line [𝑝₁𝑝₂] intersecting the curve 𝒞 exactly once in the interior of the triangle 𝑇. Moreover, 𝒞 has an arc from 𝑝₁ to 𝑝₂, no inflection point or singular point in the interior of 𝑇, and convex inside the triangle.

It is noted out that our constructed cubic algebraic curve (2.10) is coincided with the reduced form in [15, 16], where they use them to construct a family of 𝐺¹ and 𝐺² continuous algebraic splines.

3. Approximate Implicitization of Parametric Curves

Given a planar parametric curve 𝐶(𝑡)=(𝑥(𝑡),𝑦(𝑡)),𝑡∈[0,1], where 𝑥(𝑡) and 𝑦(𝑡) are arbitrary functions, such as trigonometric and exponential functions.

3.1. Curve Segments

In order to solve the approximate implicitization problem using cubic algebraic splines, a basic problem is how to divide the planar parametric curves into several segments. Some concepts and definitions are reviewed. For more details, the readers may refer to [7].

A natural idea is to divide the parametric curve into several curve segments possessing relatively good shape, separated by the following three types of critical points.

(i)A point 𝐶(𝑡₀) is called a cusp point of parametric curve 𝐶(𝑡) if 𝑥^′(𝑡₀)=𝑦^′(𝑡₀)=0. (ii)A point 𝐶(𝑡₀) is called an inflection point of 𝐶(𝑡) if 𝑥^′(𝑡₀)𝑦^′′(𝑡₀)−𝑥^′′(𝑡₀)𝑦^′(𝑡₀)=0 and 𝑥^′(𝑡₀)≠0. (iii)A point 𝐶(𝑡₀) is called a vertical point of 𝐶(𝑡) if 𝑥^′(𝑡₀)=0 and 𝑦^′(𝑡₀)≠0.

A parametric curve 𝐶(𝑡) is called normal if it has a finite number of critical points and at each critical point the tangent direction can be defined as follows.

(1)If 𝐶(𝑡₀) is not a cusp point, then the tangent direction is (𝑥^′(𝑡₀),𝑦^′(𝑡₀)). (2)If 𝐶(𝑡₀) is a cusp point, we assume that 𝑠₋=lim_{𝑡→𝑡0−}𝑦^′(𝑡)/𝑥^′(𝑡) and 𝑠₊=lim_{𝑡→𝑡0+}𝑦^′(𝑡)/𝑥^′(𝑡). If 𝑠₋(𝑠₊) is a finite number, we define 𝑇₋=(1,𝑠₋)(𝑇₊=(1,𝑠₊)). If 𝑠₋(𝑠₊) approaches to infinity, we define 𝑇₋=(0,1)(𝑇₊=(0,1)), where 𝑇₋ and 𝑇₊ are called the left and right tangent directions of point 𝐶(𝑡₀).

Let 𝑝₀ be a cusp point, and let 𝑇₋ and 𝑇₊ be the left and right tangent directions. Then the lines passing through 𝑝₀ and with directions 𝑇₋ and 𝑇₊ are called the left tangent line and right tangent line of 𝐶(𝑡) at the point 𝑝₀, respectively.

A curve segment 𝐶(𝑡)=(𝑥(𝑡),𝑦(𝑡)),𝑡∈[𝑡₁,𝑡₂] is said to be triangle convex if the left tangent line and right tangent line meet at 𝑣₁₂ and the line segment 𝑝₁𝑝₂ and the curve segment 𝐶(𝑡),𝑡∈[𝑡₁,𝑡₂] form a convex region inside the triangle [𝑝₁𝑝₂𝑣₁₂].

For any parametric curve 𝐶(𝑡), its curvature formula at any regular point 𝐶(𝑡) is given by 𝜅(𝑡)=|𝑥^′(𝑡)𝑦^′′(𝑡)−𝑥^′′(𝑡)𝑦^′(𝑡)|(𝑥^′(𝑡)²+𝑦^′(𝑡)²)^3/2. (3.1)

The curvature at each critical point can be defined as follows.

(1)If 𝑝₀=𝐶(𝑡₀) is a vertical point, then its curvature is 𝜅(𝑡₀).(2)If 𝑝₀=𝐶(𝑡₀) is a cusp point, we assume 𝜅₋=lim_{𝑡→𝑡0−0}𝜅(𝑡) and 𝜅₊=lim_{𝑡→𝑡0+0}𝜅(𝑡), where 𝜅₋ and 𝜅₊ are called the left and right curvatures of the curve 𝐶(𝑡) at the point 𝑝₀. (3)If 𝑝₀=𝐶(𝑡₀) is an inflection point, then its curvature is zero.

Throughout this paper, we directly adopt the dividing algorithm in [7] to divide the input normal parametric curve into several triangle convex segments, separated by the above three types of critical points.

3.2. Segments Approximation

Let 𝑡_𝑖,𝑖=0,1,…,𝑛 be the parametric values corresponding to the separated points and two endpoints. For each 𝑖,𝑖=0,1,…,𝑛−1, let 𝑣_𝑖,𝑖+1=(𝑥_𝑖,𝑖+1,𝑦_𝑖,𝑖+1) be the intersection point of the right tangent line at 𝑝_𝑖=(𝑥(𝑡_𝑖),𝑦(𝑡_𝑖)) and the left tangent line at 𝑝_𝑖+1=(𝑥(𝑡_𝑖+1),𝑦(𝑡_𝑖+1)). Here, all the triangles [𝑝_𝑖𝑝_𝑖+1𝑣_𝑖,𝑖+1],𝑖=0,1,…,𝑛−1, are called the control triangles of 𝐶(𝑡).

Next, we show how to approximate each curve segment 𝐶(𝑡),𝑡∈[𝑡_𝑖,𝑡_𝑖+1] in its control triangle [𝑝_𝑖𝑝_𝑖+1𝑣_𝑖,𝑖+1] by using a cubic Bernstein-Bézier implicit curve 𝒞_𝑖={(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)∣𝑓(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)=0} (see Figure 2). Here, 𝑓(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖) is assumed to be 𝑓(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)=𝑏^(𝑖)₂₁₀3𝑢²_𝑖𝑣_𝑖+𝑏^(𝑖)₁₂₀3𝑢_𝑖𝑣²_𝑖+𝑏^(𝑖)₁₁₁6𝑢_𝑖𝑣_𝑖𝑤_𝑖−𝜅₁𝜇₁𝑏^(𝑖)₂₁₀3𝑢_𝑖𝑤²_𝑖−𝜅₂𝜇₂𝑏^(𝑖)₁₂₀3𝑣_𝑖𝑤²_𝑖−𝑤³_𝑖, (3.2) where (𝑢_𝑖,𝑣_𝑖,𝑤_𝑖) are the barycentric coordinates with respect to the triangle [𝑝_𝑖𝑝_𝑖+1𝑣_𝑖,𝑖+1].

Figure 2 

Curve segment 𝐶(𝑡),𝑡∈[𝑡𝑖,𝑡𝑖+1] is approximated by using 𝒞𝑖

The remaining three free parameters 𝑏^(𝑖)₂₁₀,𝑏^(𝑖)₁₂₀, and 𝑏^(𝑖)₁₁₁ in (3.2) can be determined by the following optimization problem: min𝐻(𝑏^(𝑖)₂₁₀,𝑏^(𝑖)₁₂₀,𝑏^(𝑖)₁₁₁),where𝐻(𝑏^(𝑖)₂₁₀,𝑏^(𝑖)₁₂₀,𝑏^(𝑖)₁₁₁)=^𝑡𝑖+1_𝑡𝑖(𝑓(𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡)))²𝑑𝑡, (3.3) under the constraints 𝑏^(𝑖)₂₁₀>0 and 𝑏^(𝑖)₁₂₀>0.

Here, (𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡)) are the barycentric coordinates of the point 𝑝=(𝑥(𝑡),𝑦(𝑡)) with respect to the triangle [𝑝_𝑖𝑝_𝑖+1𝑣_𝑖,𝑖+1], they are univariate functions in variable 𝑡. So, by 𝐺_𝑖(𝑡), we denote 𝐺_𝑖(𝑡)=𝑓(𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡)).

The integral involves complicated computations and can be evaluated by numerical method such as Gaussian quadrature [17].

Proposition 3.1 (see [17]). Let 𝑥_𝑘,𝑘=1,2,…,𝑛 be the zeros of the orthogonal polynomial Legendre 𝑃_𝑛(𝑥)=(1/𝑧^𝑛𝑛!)(𝑑^𝑛/𝑑𝑥^𝑛)(𝑥²−1)^𝑛 and let 𝐴_𝑘=2/((1−𝑥²_𝑘)[𝑃^′_𝑛(𝑥_𝑘)]²),𝑘=1,2,…,𝑛, be the corresponding weights related to 𝐿_𝑛(𝑥). Then the quadrature formula ¹₋₁𝑓(𝑥)𝑑𝑥=^𝑛_𝑘=1𝐴_𝑘𝑓(𝑥_𝑘) (3.4) of this type has algebraic accuracy 2𝑛+1.

Any other interval [𝑡_𝑖,𝑡_𝑖+1] of integration must be transformed into the standard interval [−1,1]. From now on, we let 𝑦=(1/2)[(𝑡_𝑖+1−𝑡_𝑖)𝑥+(𝑡_𝑖+𝑡_𝑖+1)] and 𝑥∈[−1,1] is transformed into 𝑦∈[𝑡_𝑖,𝑡_𝑖+1]. Therefore, the numerical integration that we wish to minimize in (3.3) can be reduced: 𝐻(𝑏^(𝑖)₂₁₀,𝑏^(𝑖)₁₂₀,𝑏^(𝑖)₁₁₁)=^𝑡𝑖+1_𝑡𝑖𝐺_𝑖(𝑡)²𝑡𝑑𝑡=^𝑖+1−𝑡_𝑖2^𝑁_𝑘=1𝐴_𝑘𝐺_𝑖(𝑦_𝑘)², (3.5) where 𝑦_𝑘=(1/2)[(𝑡_𝑖+1−𝑡_𝑖)𝑥_𝑘+(𝑡_𝑖+𝑡_𝑖+1)],𝑘=1,2,…,𝑛.

3.3. Approximation Error

Given the rational parametric curve segment, 𝐶(𝑡)=(𝑥(𝑡),𝑤(𝑡)𝑦(𝑡)𝑤(𝑡)),𝑡∈[𝑡_𝑖,𝑡_𝑖+1], (3.6) where 𝑥(𝑡),𝑦(𝑡),𝑤(𝑡) are polynomials.

Suppose its approximated curve in the interior of its control triangle [𝑝_𝑖𝑝_𝑖+1𝑣₁₂] is 𝒞_𝑖={(𝑢,𝑣,𝑤)∣𝑠_𝑖(𝑢,𝑣,𝑤)=0}. (3.7) In this section, we will discuss the approximation error between 𝐶(𝑡),𝑡∈[𝑡_𝑖,𝑡_𝑖+1] and 𝒞_𝑖.

Let (𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡)) be the barycentric coordinates of the point 𝑝=(𝑥(𝑡)/𝑤(𝑡),𝑦(𝑡)/𝑤(𝑡)) with respect to the triangle [𝑝_𝑖𝑝_𝑖+1𝑣_𝑖,𝑖+1]. The approximation error is defined by 𝐸(𝑠_𝑖)=max_{𝑡𝑖<𝑡<𝑡𝑖+1}|𝐸_𝑖(𝑡)|,𝐸_𝑖(𝑡)=𝑠_𝑖(𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡)). (3.8)

Here |𝑠_𝑖(𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖(𝑡))| denotes the algebraic distance between the point 𝑝=𝐶(𝑡),𝑡∈(𝑡_𝑖,𝑡_𝑖+1) and its approximated curve 𝒞_𝑖.

Theorem 3.2. With the above notations, 𝐸(𝑠_𝑖)≤𝑀ℎ⁴_𝑖,ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖, (3.9) where 𝑀 is a positive number.

Proof. Obviously, we have 𝐸_𝑖(𝑡)=𝑠_𝑖(𝑢_𝑖(𝑡),𝑣_𝑖(𝑡),𝑤_𝑖𝐺(𝑡))=^𝑖(𝑡)𝑤(𝑡)³,𝑡∈[𝑡_𝑖,𝑡_𝑖+1]. (3.10)
Since 𝒞_𝑖 interpolates the two endpoints 𝑝_𝑖=𝐶(𝑡_𝑖) and 𝑝_𝑖+1=𝐶(𝑡_𝑖+1), and keeps tangent directions at them, then it follows easily that
𝐸_𝑖(𝑡_𝑖)=𝐸_𝑖(𝑡_𝑖+1)=0,𝐸^′_𝑖(𝑡_𝑖)=𝐸^′_𝑖(𝑡_𝑖+1)=0. (3.11) This fact is equal to 𝐺_𝑖(𝑡_𝑖)=𝐺_𝑖(𝑡_𝑖+1)=0 and 𝐺^′_𝑖(𝑡_𝑖)=𝐺^′_𝑖(𝑡_𝑖+1)=0. It yields 𝐺_𝑖(𝑡)=(𝑡−𝑡_𝑖)²(𝑡−𝑡_𝑖+1)²𝑟_𝑖(𝑡). (3.12) If we let ℎ_𝑖=𝑡_𝑖+1−𝑡_𝑖, then max_{𝑡𝑖<𝑡<𝑡𝑖+1}(𝑡−𝑡_𝑖)²(𝑡−𝑡_𝑖+1)²=ℎ⁴_𝑖/4 from simple computation.
Thus, if we set
max_{𝑡𝑖≤𝑡≤𝑡𝑖+1}𝑟_𝑖(𝑡)𝑤(𝑡)³=4𝑀, (3.13) then 𝐸(𝑠_𝑖)=max_{𝑡𝑖<𝑡<𝑡𝑖+1}|𝐸_𝑖(𝑡)|≤𝑀ℎ⁴_𝑖. This completes the proof.

Theorem 3.3. With the above proposed method, one obtains a piecewise 𝐺² continuous cubic algebraic curve which keeps the convexity of the original normal curve.

Proof. The 𝐺² continuity of the piecewise cubic approximate splines is a direct consequence of the fact that the cubic algebraic curves have the same tangent directions and curvature with the original curve. Furthermore, the curve is divided into triangle convex segments and the cubic curve segments are convex with no inflection points, which also keep the convexity of the curve.

3.4. Main algorithm

The algorithm of approximate implicitization for planar parametric curves using a cubic algebraic spline is outlined in what follows.

Algorithm 3.4. Approximate implicitization using cubic algrbraic spline.Input:A normal parametric curve 𝐶(𝑡)=(𝑥(𝑡),𝑦(𝑡)),𝑡∈[0,1], and a sufficiently small positive number 𝜀.Output:A cubic algebraic spline 𝒞={(𝑢,𝑣,𝑤)∣𝑠(𝑢,𝑣,𝑤)=0} satisfying each 𝐸(𝑠_𝑖)<𝜀.Step 1:Divide the normal parametric curve into several triangle convex segments using the dividing algorithm [7]. Let 𝑡_𝑖,𝑖=0,1,…,𝑛 be the parametric values corresponding to the critical points and two endpoints. For each 𝑖=0,1,…,𝑛, compute the left and right directions 𝑇_𝑖− and 𝑇_𝑖+, left and right curvatures 𝜅_𝑖− and 𝜅_𝑖+ at 𝐶(𝑡_𝑖).Step 2:On each interval [𝑡_𝑖,𝑡_𝑖+1],𝑖=0,1,…,𝑛−1, perform the optimization problem (3.3) to compute the cubic curve segment 𝒞_𝑖={(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)∣𝑠_𝑖(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)=0}. Step 3:If 𝐸(𝑠_𝑖)>𝜀, then we subdivide the interval [𝑡_𝑖,𝑡_𝑖+1] and repeat Step 2 on each subinterval.

4. Numerical Examples

In this section, some numerical examples are provided to illustrate that the proposed approximate implicitization method is flexible and effective.

Example 4.1. Consider the following curves from [6, 7]: 𝐶₁(𝑡)=(5𝑡³+2𝑡²,𝑡⁴−3𝑡³+2𝑡²𝐶),₂(𝑡)=(−5𝑡−100𝑡²+250𝑡³−240𝑡⁴+87𝑡⁵−1−30𝑡²+80𝑡³−75𝑡⁴+25𝑡⁵,−5𝑡−60𝑡²+150𝑡³−120𝑡⁴+35𝑡⁵−1−30𝑡²+80𝑡³−75𝑡⁴+25𝑡⁵𝐶),₃(𝑡)=(sin(2𝑡)+ln(5𝑡⁴+2)+3𝑡²,3𝑒^𝑡2−1𝑡+cos(5)+2𝑡⁷). (4.1) The parameters for curves of 𝐶₁(𝑡),𝐶₂(𝑡), and 𝐶₃(𝑡) take values in [−1,1],[0,1], and [−1,1]. Their approximate cubic algebraic splines are shown in Figures 3, 4, and 5 and their approximation errors are listed in Tables 1, 2, and 3.
In the following figures, we simultaneously give the original parametric curves, the cubic algebraic splines, and the separated points, denoted by black line, red line, and black dots, respectively. By 𝑠₁ and 𝑠₂, we denote the two segments in the left picture of Figure 3. By 𝑠₁₁ and 𝑠₁₂, we denote the two segments of which 𝑠₁ is subdivided in Figure 3(b). Other notations can be understood similarly.
We list the exact implicit form of the first two curves with Gröbner bases method as 𝑔_𝑖(𝑥,𝑦)=0,𝑖=1,2. Whereas, curve 𝐶₃(𝑡) does not have an exact implicit form. 𝑔₁(𝑥,𝑦)=336𝑥²−55𝑥³+𝑥⁴−672𝑥𝑦−683𝑥²𝑦+336𝑦²−1325𝑥𝑦²−625𝑦³,𝑔₂(𝑥,𝑦)=608755200000𝑥−3333251200000𝑥²[phone]𝑥³−249967600000𝑥⁴[phone]𝑥⁵−608755200000𝑦+9279481600000𝑥𝑦−2693703200000𝑥²𝑦+373486700000𝑥³𝑦−18653234375𝑥⁴𝑦−6920238720000𝑦²−1644719360000𝑥𝑦²[phone]𝑥²𝑦²[phone]𝑥³𝑦²[phone]𝑦³[phone]𝑥𝑦³−55873524750𝑥²𝑦³−1083739330400𝑦⁴[phone]𝑥𝑦⁴[phone]𝑦⁵. (4.2)
With comparison to the expressions of exact implicitization, we also list the implicit forms of the two approximate segments for 𝐶₁(𝑡) in Figure 3(a) as follows:
𝑠₁(𝑥,𝑦)=−1.001𝑥+0.321𝑥²−0.211𝑥³+1.001𝑦−2.676𝑥𝑦+0.284𝑥²𝑦−1.891𝑦²+0.939𝑥𝑦²+0.409𝑦³,𝑠₂(𝑥,𝑦)=0.464𝑥−0.101𝑥²+0.005𝑥³−0.464𝑦+0.252𝑥𝑦−0.111𝑥²𝑦−2.118𝑦²−4.625𝑥𝑦²−12.972𝑦³. (4.3)

Table 1 

Approximation error of curve 𝐶₁(𝑡).

Table 2 

Approximation error of curve 𝐶₂(𝑡).

Table 3 

Approximation error of curve 𝐶₃(𝑡).

(a)

(b)

Figure 3 

𝐶1(𝑡) and its approximate cubic algebraic splines.

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(b)

Figure 4 

𝐶2(𝑡) and its approximate cubic algebraic splines.

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(b)

Figure 5 

𝐶3(𝑡) and its approximate cubic algebraic splines.

5. Conclusion

We have described an algorithm to solve approximate implicitization of planar parametric curves using piecewise cubic algebraic splines. With the proposed algorithm, we obtain a global 𝐺² continuous cubic algebraic spline which keeps the direction, the curvature, and the convexity of the original normal parametric curve with simple computation. The proposed method is flexible and effective from the numerical examples.

However, the proposed algorithm is hard to be generalized to solve approximate implicitization of parametric surfaces directly. Therefore, the problem on approximate implicitization of parametric surfaces by algebraic spline surfaces remains to be our future work.

Acknowledgments

(1)This work was supported by the Natural Science Foundation of Zhejiang Province (nos.Y7080068 and Y6090211), and the Foundation of Department of Education of Zhejiang Province (nos. Y200802999 and 20070628).