Fold Curve:

The fold curve of a smooth surface for a given direction z is the collection of points on the surface such that the tangent plane to the surface at that point contains the direction z.

For a polyhedral surface, a fold edge for a given direction z is one such that the two faces containing the edge both lie on the same side of the plane determined by the edge and the direction z. The fold curve is the collection of fold edges for the given direction.

Alternatively, the fold curve in a given direction is the set of singular points for the projection of the surface into the plane perpendicular to that direction. In other words, the fold curve is the "edge" of the surface as you look at it from a given direction.

For a convex surface, the fold curves are simply the outer edges of the "shadow" of the surface in the projection of the surface in the given direction. For non-convex surfaces, there may be additional components to the fold curve that come from the holes or bumps in the surface.

In general, the fold curve is a collection of closed curves. The term fold set is sometimes used to indicate the fold curves on the surface itself, and fold curve may refer to the projections of these curves into the plane.

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8/12/94 -- The Geometry Center