Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse

Abstract

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.

Area-Invariant Pedal-Like Curves Derived from the Ellipse

“…One open question is whether a common thread exists which links the Steiner Hat [4], the Hybrid, and Pseudo-Talbot curves, since all of them are area-invariant over M on the ellipse. Furthermore, if a continuous family of curves exists with this area-invariance property.…”

“…Three cases are shown, for M (i) interior (top left), (ii) on the boundary (top right), and (iii) at the center (bottom) of the ellipse. For case (ii) the area of the curve is invariant for all M[4]. Case (iii) yields Talbot's Curve[11] (in general it does not pass through the foci, but for the case shown, a/b = 2, it does).…”

“…Let E n denote the negative-pedal curve with respect to M [6], i.e., the envelope of lines L(t) through a point P (t) on E and perpendicular to P (t) − M ; see Figure 1. This article was motivated by a recent result [4]: E n is a three-cusp area-invariant deltoid for all M on E; see Figure 1 (top right).…”

Intriguing Invariants of Centers of Ellipse-Inscribed Triangles

“…Also shown are envelopes (pink) to the ellipse families. For X3 (top right), the envelope is a half-sized Steiner's hat [9]; for X2 (bottom left) the envelope is an ellipse of fixed axes, internally tangent to E at one point. Interestingl, all envelopes are area-invariant wrt V1.…”

“…More recently, with the help of a computer algebra system (CAS), we showed that 29 out of the first 100 entries in [11] are ellipses over billiard 3-periodics, though what determines ellipticity is still not understood [8]. We've also studied properties (e.g., area invariance) of the negative pedal curve (NPC) of the ellipse with respect to a point on its boundary [9] as well as its pedal-like [11] which are both on the Euler line and which are fixed linear combinations of X2 and X4.…”