This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach.
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. So 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 of a curve is primarily defined by its curvature distribution, monotonicity of curvature is not easily achieved and controlled. To overcome this problem,
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