Interpolation on quadric surfaces with rational quadratic spline curves Original Research Article

Given a sequence of points {Xi}i=1n on a regular quadric S: XTAX = 0 ⊂ Ed, d ⩾ 3, we study the problem of constructing a G1 rational quadratic spline curve lying on S that interpolates {Xi}i=1n. It is shown that a necessary condition for the existence of a nontrivial interpolant is (X1TAX2)(XiTAXi+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.