Interpolation on quadric surfaces with rational quadratic spline curves

X n Given a sequence of points { i}i=l on a regular quadric S: X T A X = 0 C E d, d >1 3, we study the problem of constructing a G I rational quadratic spline curve lying on 5' that interpolates { X ~ } ~ j . It is shown that a necessary condition for the existence of a nontrivial interpolant is ( X ~ A X 2 ) ( X T i A X i + I ) > 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 G ~ 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.