Blending canal surfaces along given circles using Dupin cyclides

Abstract

We study blends between canal surfaces using Dupin cyclides via the space of spheres. We have already studied the particular case where it is possible to blend two canal surfaces using one piece of Dupin cyclide bounded by two characteristic circles, but this is not possible in the general case. That is why we solve this problem using two pieces of different cyclides, which is always possible. To get this conclusion and give the algorithms allowing to obtain such a result, we study, at first, the blend between two circles by a piece of cyclide. We impose to the cyclide to be tangent to a given sphere containing one of the circles. We give the existence condition on the previous circles to have a cyclide making the blend. Then, we show how to obtain a G¹-blend between two canal surfaces using two Dupin cyclides by imposing conditions on the blending circles between the two cyclides.

Keywords:

• Dupin cyclides
• space of spheres
• G¹-blend

2010 AMS Subject Classifications:

• 68U05
• 65D18

Notes

1. The circles Aff(O₅, σ, τ)∩Λ⁴, where τ describes Λ³(σ), form a foliation of the region the leaves of which arrive orthogonally on Λ³(σ). The arcs cut on these circles by σ and −σ define a projection of on Λ³(σ).