Geometric shapes of C-Bézier curves

doi:10.1016/j.cad.2014.08.007

Computer-Aided Design

Volume 58, January 2015, Pages 242-247

https://doi.org/10.1016/j.cad.2014.08.007 Get rights and content

Highlights

•
Transformation matrixes between any C-Bézier curve and its separated form.
•
Geometric shape and the explicit expressions of the geometric characters of any C-Bézier curve.
•
Sufficient and necessary conditions for the C-Bézier basis constructing some classical curves.
•
Some classical curves can be constructed intuitively.

Abstract

In this paper, we focus on the geometric shapes of the C-Bézier curves for the space span ${1, t, \dots, t^{n}, sin t, cos t}$ . First, any C-Bézier curve is divided into a Bézier curve and a trigonometric part. So any C-Bézier curve describes the trajectory of a point orbiting around a center in an elliptical orbit while the orbital plane is moving as the ellipse center translating along a Bézier curve. Second, the geometric characters of the C-Bézier curve (the control points of the center Bézier curve, the trajectories of the vertices and the foci of the ellipse, etc.), can all be explicitly presented by the control points of the C-Bézier curve. Third, considering some special cases, we give the sufficient and necessary conditions of C-Bézier basis forming Bézier curve, ellipse, circle, common helix, and so on. Lastly, we show how to build some geometrically intuitive curves through the C-Bézier basis without rational forms.

Introduction

The Bézier-like models, defined over some non-algebraic polynomial spaces, may be used as alternatives of the rational Bézier model [1]. They include the $p$ -Bézier [2], the quasi Bézier [3], the C-Bézier [4], [5], the AH-Bézier [6], the AHT-Bézier [7], the Bézier-like models using complex numbers [8], [9], [10], and so on.

These Bézier-like models can build some classical curves without rational forms; for example, C-Bézier for ellipse, and AH-Bézier for hyperbola. Then what about the general cases? This paper considers an arbitrary C-Bézier curve over $Γ_{n} = span {1, t, \dots, t^{n}, sin t, cos t}$ [5]. $P (t) = \sum_{i = 0}^{n + 2} C_{i, n + 2} (t) P_{i}, t \in [0, α],$ where $α \in (0, π)$ is the given shape parameter, ${P_{i}}_{i = 0}^{n + 2}$ are the control points, and ${C_{i, n} (t)}_{i = 0}^{n + 2}$ are the C-Bézier basis functions.

We focus on the geometric shape of the C-Bézier curve (1). Previous studies indicate that any rational quadratic Bézier curve is a conic [11]. Its geometric characters (vertices, eccentricity, etc.) can be explicitly expressed by the control points and weighs [12], [13]. Similarly, for any given C-Bézier curve (1) over $Γ_{n}$ , we ask two questions.

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What is the geometric shape of (1)?
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Can the geometric characters of (1) be expressed by its control points ${P_{i}}_{i = 0}^{n + 2}$ explicitly?

For the first question, [14] gives the answer to planar curves under the condition $n = 1$ . In this paper, we consider more general cases, both planar and space curves, for any positive integer $n$ . We explain what the C-Bézier curve is from a geometric view. This helps the intuitive construction of some classical curves.

Section snippets

Transformation between C-Bézier and its separated form

The C-Bézier curve (1) for $Γ_{n}$ can be represented as $P (t) = Q (t) + R (t), t \in [0, α],$ called the separated form in this paper. Here, $Q (t)$ is the Bézier part $Q (t) = \sum_{i = 0}^{n} B_{i, n} (t) Q_{i}, t \in [0, α],$ where ${Q_{i}}_{i = 0}^{n}$ are the control points and ${B_{i, n} (t)}_{i = 0}^{n}$ are the Bernstein basis functions of degree $n$ on $[0, α]$ . And $R (t)$ is the trigonometric part $R (t) = R_{0} sin t + R_{1} cos t, t \in [0, α] .$

In this paper, both ${R_{i}}_{i = 0}^{1}$ and ${Q_{i}}_{i = 0}^{n}$ are called geometric points of (1) to distinguish them from the C-Bézier’s control points ${P_{i}}_{i = 0}^{n + 2}$ .

In this

Geometric shapes of C-Bézier curves

This section focuses on the geometric shapes of C-Bézier curves. We first consider the general cases, and then some special cases.

Applications

The Shape analysis for the C-Bézier curve may be used to construct some geometrically intuitive curves.

For example, cycloid is traced by a point on the rim of a circle while the circle is rolling along a line at a uniform speed [16]. First, design $L_{0}$ , $L_{1}$ satisfying $‖ L_{0} ‖ = ‖ L_{1} ‖ = r$ , where $r$ is the radius of the circle. Second, choose $λ$ , the angle between the ellipse radius vector of the start point and the circle’s forward direction parallel to the line. Third, design $Q_{0}$ , $Q_{n}$ , which are the two

Conclusions

In this paper, we consider the geometric shapes for C-Bézier curves over $Γ_{n}$ . Theorem 3.1 answers the two questions in Section 1.

Our main results include: the transformation matrixes between any C-Bézier curve and its separated-form; the geometric shape and the explicit expressions of the geometric characters of any C-Bézier curve; the sufficient and necessary conditions for the C-Bézier basis constructing some classical curves (ellipse, helix, etc.). The results show the geometric means of

Acknowledgments

We are very grateful to the referees for their helpful suggestions and comments. The work was partially supported by the National Natural Science Foundation of China (No. 11326243, 61272300, 61402201 and 11371174) and the Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK20130117).

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Geometric shapes of C-Bézier curves☆

Highlights

Abstract

Introduction

Section snippets

Transformation between C-Bézier and its separated form

Geometric shapes of C-Bézier curves

Applications

Conclusions

Acknowledgments

Comput Aided Geom Design

Comput Aided Geom Design

Comput Aided Geom Design

Comput Aided Geom Design

Comput Math Appl

Comput Aided Geom Design

Comput Aided Geom Design

Comput Aided Geom Design