William H. Frey

Designing Bézier conic segments with monotone curvature

Abstract For some applications of computer-aided geometric design it is important to maintain strictly monotone curvature along a curve segment. Here we analyze the curvature distributions of segments of conic sections represented as rational quadratic Bezier curves in standard form. We show that if the end points and the weight are fixed, then the curvature of the conic segment will be strictly monotone if and only if the other control point lies inside well-defined regions bounded by circular arcs. We also show that if the turning angle of the curve is less than or equal to 90°, then there are always values of the weight that ensure strict monotonicity of the curvature distribution. Furthermore, bounds on such values of the weight are easily computed.

Modeling buckled developable surfaces by triangulation

Abstract In the first stage of sheet metal stamping, a binder ring, an annular surface surrounding the die cavity, clamps down on the flat blank, bending it to a developable binder wrap surface which may be smooth or buckled. Buckles generally appear in the binder wrap when the binder ring does not lie on a smooth developable surface that spans the die cavity. However, sometimes buckles can improve the formability of the stamped part, so the ability to design buckled developable surfaces becomes desirable. Designing buckled developable surfaces requires geometric modeling of creases and other singularities in the interior a flat sheet. In this paper we review the properties of such surfaces, show how to approximate buckled binder wrap surfaces by developable three-dimensional triangulations and discuss the insights gained from specific examples.

Controlling the curvature of a quadratic Be´zier curve

Abstract This paper describes a simple geometric condition that indicates when a quadratic Bezier curve segment has monotone curvature. Using this condition, it treats the problem of ‘correcting’ the curvature plot of such a curve by moving the middle control point to a new location. The method guarantees minimization of error between the new and old Bezier curve.