## Convex Cubic Spirals

I chose to study mathematics in college and then in graduate school, and I ended up doing research in Computer-Aided Geometric Design. My thesis topic involved using the degrees of freedom present in a planar parametric cubic to ensure monotonic curvature of the cubic. My advisor and I co-authored two papers on the subject, where the second paper expanded the discussion to include rational cubics.

This diagram is of three nested circles of curvature on a (non cubic) spiral. Note that the spiral is always crossing into its circles of curvature as it is followed inward.

The black curves, found numerically, eliminate possible f0-f1 pairs which can yield cubic spirals. Remining white area indicates cubic spirals. The free variables f0-f1 are explained in the papers.

The gray curves, found numerically, eliminate possible f0-f1 pairs which can yield cubic spirals. Remining white area indicates cubic spirals. The black curves are for analytical purposes and are found algebraically. Tangents at ends are fixed.

The gray and black curves in this diagram play exactly the same role as in the diagram to its left, except the two free variables are the curvatues at the endpoints of the cubic. The tangent angles at the endpoints are fixed (0.2 and 0.7).

In this diagram, rational cubics are numerically analysed for
fixed tangential conditions. Squares represent curvature constraints
which may be satisfied by cubics alone. Diamonds indicate additional
curvature combinations gained by allowing rational cubics rather than
just cubics. The curve which begins roughly at (0,10) in the diagram
represents the the boundary beyond which no spiral may occur due to
simple geometric constraints.

Paper 1
Paper 2