Interpolation on quadric surfaces with rational quadratic spline curves

Abstract

Given a sequence of points {Xi}n i=1 on a regular quadric S: XT AX = 0 ⊂ double-struck E signd, d ≥ 3, we study the problem of constructing a G1 rational quadratic spline curve lying on S that interpolates {Xi}n i=1. It is shown that a necessary condition for the existence of a nontrivial interpolant is (XT 1 AX2)(XT i AXi+1) > 0, i = 1, 2, . . . , n - 1. Also considered is a Hermite interpolation problem on the quadric S: a biarc consisting of two conic arcs on S joined with G1 continuity is used to interpolate two points on S and two associated tangent directions, a method similar to the biarc scheme in the plane (Bolton, 1975) or space (Sharrock, 1987). A necessary and sufficient condition is obtained on the existence of a biarc whose two arcs are not major elliptic arcs. In addition, it is shown that this condition is always fulfilled on a sphere for generic interpolation data.