Interpolation on quadric surfaces with rational quadratic spline curves

Given a sequence of points {X(i)}(i=1)(n) on a regular quadric S: X(T) AX = 0 subset of E(d), d greater than or equal to 3, we study the problem of constructing a G(1) rational quadratic spline curve lying on S that interpolates {X(i)}(i=1)(n). It is shown that a necessary condition for the existence of a nontrivial interpolant is (X(1)(T)AX(2))(X(i)(T)AX(i+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 G(1) 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.