Abstract
We study functions defined in the plane E 2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections.
Highlights
The most well-known plane curves are straight lines and circles, which are characterized as the plane curves with constant Frenet curvature
The most familiar plane curves might be the conic sections: ellipses, hyperbolas and parabolas. They are characterized as plane curves with constant affine curvature ([1], p. 4)
The region D bounded by the ellipse Xl and the chord AB outside Xk has constant area independent of the point p ∈ Xk
Summary
The most well-known plane curves are straight lines and circles, which are characterized as the plane curves with constant Frenet curvature. The most familiar plane curves might be the conic sections: ellipses, hyperbolas and parabolas. They are characterized as plane curves with constant affine curvature We investigate the family of strictly convex level curves Xk , k ∈ Sg of a function g : R2 → R which satisfies the following condition.
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