Representation of Dupin cyclides using quaternions

Dupin cyclides are surfaces characterized by the property that all their curvature lines are circles or lines. Spheres, circular cylinders, cones and tori are particular examples. We introduce a bilinear rational Bézier-like formula with quaternion weights for parametrizing principal patches of Dupin cyclides. The proposed construction is not affine invariant but it is Möbius invariant, has lower degrees compared with the standard representation, and it is convenient for offsetting. Several important properties of Dupin cyclides can be recovered in terms of closed quaternion formulas: implicit equation, principal curvatures, representation as canal surfaces. Advantages of this approach are demonstrated by deriving a new formula for the Willmore energy of a principal patch.