Osculating Conic Biarcs

A circular biarc can be defined by using two points K and L with their (oriented) tangents g _K and g _L as input. It is well-known that one can determine a one parametric set of circular arc pairs k,ℓ such that k starts at K with tangent g _K, ℓ ends at L with tangent g _L and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace the circle biarcs by pairs of conic arcs. It turns out that in this case we can prescribe a conic k ₀ with a point K on it, another conic ℓ ₀ with a point L on it, and moreover an intermediate point P to obtain a unique pair k,ℓ of conics such that k osculates k ₀ in K, ℓ osculates ℓ ₀ in L and k and ℓ osculate each other in P. This also confirms a result of H. Pottmann from 1991. We use our method to solve an interpolation task of Hermite type whose input consists of a series of points with their curvature circles and another series of intermediate points. The output is a GC ² spline curve with conic arc segments.