Abstract While considering a mirror and light rays coming either from a point source or from infinity, the reflected light rays may have an envelope, called a caustic curve. In this paper, we study developable surfaces as mirrors. These caustic surfaces, described in a closed form, are also developable surfaces of the same type as the original mirror surface. We provide efficient, algorithmic computation to find the caustic surface of each of the three types of developable surfaces (cone, cylinder, and tangent surface of a spatial curve). We also provide a potential application of the results in contemporary free-form architecture design.

Modifying the factorization of Bernstein polynomials, we propose a system of functions with two free parameters, which can be used for control point based curve modeling. Essential properties of functions such as nonnegativity, partition of unity, Hermite-like end properties, asymptotic behavior, symmetry, linear independence are shown. Consequently, the control point based curve obtained by these functions is closed for the affine transformation of its control points and has the convex hull, endpoint interpolation and endpoint tangency properties. The influence of the free parameters on the shape of the curve is studied as well.

Presents shape control methods for cubic B-spline and NURBS curves by the modification of their knot values and by the simultaneous modification of weights and knots. Theoretical aspects of knot modification are also discussed, concerning the paths of points on a curve and the existence of an envelope for the family of curves resulting from a knot modification for curves of degree k.