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