Bitangent

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The Trott curve (black) have 28 real bitangents (red). This image shows 7 of them; the others are symmetric with respect to the origin.

In mathematics, a bitangent to a curve C is a line L that touches C in two distinct points P and Q and that has the same direction to C at these points. That is, L is an tangent line at P and at Q. It differs from a secant line in that a secant line may cross the curve at the two points it intersects it. In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents.

Bézout's theorem implies that a plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents to a general plane quartic curve was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Such bitangents are in general defined over the complex numbers, and are not real (see Salmon's Higher Plane Curves). For an example where all bitangents are real, see Trott curve.