Designing planar cubic B-spline curves with monotonic curvature for curve interpolation
- PDF / 754,180 Bytes
- 7 Pages / 612 x 808 pts Page_size
- 93 Downloads / 195 Views
Designing planar cubic B-spline curves with monotonic curvature for curve interpolation Aizeng Wang1,2 , Chuan He1,2 , Fei Hou3,4 , Zhanchuan Cai5 , and Gang Zhao1,2 (
)
c The Author(s) 2020.
1
of a curve is primarily defined by its curvature distribution, monotonicity of curvature is not easily achieved and controlled. To overcome this problem, we present a construction approach for B-spline curves, which guarantees monotonic curvature by enforcing simple geometric constraints on the control vectors. Euler curves have the useful property that their curvature changes linearly with arc length, so are widely used in modeling and shape interpolation [4–6]. However, Euler curves also have disadvantages for curve interpolation. Firstly, they are defined by transcendental functions, requiring complex mathematical expressions which are difficult to compute. Secondly, given two endpoints with associated tangents, there is no exact solution for an Euler curve with these boundary conditions. Thirdly, Euler curves are not compatible with current CAD software systems, which are based on NURBS. In order to overcome these drawbacks of Euler curves, we have developed a new interpolation algorithm for cubic B-spline curves, with advantages of simple computation, exact interpolation and compatibility with existing CAD systems.
Introduction
Monotonic curvature plays an important role in industrial design and styling of curves with aesthetic shapes, e.g., in automobile and aircraft design [1]. Used in conventional parametric CAD/CAM systems, general B-splines are not adequate for aesthetic requirements. Except for the straight line and circle, monotonic curvature distribution, associated with pleasing shape, is very difficult to achieve. Curvature plays an important role as a shape descriptor. Farin suggested that a fair curve has a curvature plot with relatively few regions of monotonically varying curvature. Starting from this basis, work on B-spline fairing was developed mainly in three direction: knot-removal-reinsertion methods, optimization methods based on minimizing an energy function, and filtering approaches based on B-spline wavelets. Visual curve completion (interpolating a curve segment, with continuity, to fill a gap) is a fundamental problem for human visual understanding [2]. Aesthetically pleasingly shaped curves usually have monotonically varying curvature [3]. While the shape 1 School of Mechanical Engineering & Automation, Beihang University, Beijing 100191, China. E-mail: A. Wang, [email protected]; C. He, [email protected]; G. Zhao, [email protected] ( ). 2 State Key Laboratory of Virtual Reality Technology & Systems, Beihang University, Beijing 100191, China. 3 State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China. E-mail: [email protected]. 4 University of Chinese Academy of Sciences, Beijing 100049, China. 5 Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China. E-mail: [email protected]. Manuscript r
Data Loading...
