Spiral Arcs

Spiral Arcs and Curvature Elements

Definition: a convex curve is a curve such that no line crosses the curve more than twice.

Curves with turning angle less than pi (3.14159....) are always convex.

Definition: A spiral arc is a portion of a convex curve that has monotonic curvature.

For example, the curve in the figure at the right shows a single spiral arc.

The curve in this figure shows two spiral arcs, the first green, and the second blue. The place where they join is a local maximum in curvature.

This figure shows a curve broken into spiral arcs in different colors and each joint is at a vertex.

Definition: Examples of a curvature element include a point, a tangent vector, a tangent line, and an osculating circle at a point on the curve.

There is a special relation between two curvature elements on a spiral arc. Try to find out what it is on the discover page.

Relation between Curvature Elements

The curvature elements on a spiral arc are related in that one circle must completely contain the other, as shown in the figure, l1 > = l2. Go to the practice page and see if you can spot which pairs of curvature elements could come from two points on a spiral arc.

Designing Spiral Arcs

One way of designing aesthetic curves is to try to first pick the curvature elements and then have a computer algorithm automatically fill in a curve that fits the curvature elements. If one is very careful, the curvature elements can be chosen so that the computer algorithm can fit a spiral arc between each pair. This provides very direct control over the shape of the curve, but one must understand spiral arcs and curvature to use it.


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