Abstract

The way we perceive the shape of a smooth object A E IK z with our eye is through its apparent contour, the curve on the object where the lines passing through our eye and tangent to the boundary surface S = OA of A touches S. One may say that this fact has been at the origin of a certain quantity of experimental Mathematics since the invention of perspective. One may also speculate that the origin of the mathematical study of apparent contours is, stated in modern terms, a problem of measure of complexity; first remark that the apparent contour depends only upon the bounding surface S; it is also the apparent contour of S. Remark also that we may measure the complexity of an algebraic surface by the degree of its equation. Now the following is a natural problem: if S is an algebraic surface of degree m, how complicated is its apparent contour; what is its degree (as a space curve)? Intersecting everything with a plane containing the origin 0 (the eye) and not tangent to S reduces the problem to the following: given a nonsingular algebraic curve C of degree m in lR 2, how many of the lines tangent to C pass through a given point 0?

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