The isoptic of an ellipse as toric intersection
From conic intersection to toric intersection
The isoptic curve of an ellipse is a quartic called a spiric of Perseus (a kind of oval). It is the intersection of a torus with a plane parallel to the torus axis. Note that in the case of a isoptic of an ellipse, the torus is self-intersecting (look at the internal part of the surface, in the upper right window).
The k-slider bar enable to change the eccentricity of the ellipse, the theta-silder bar changes the viewing angle on the ellipse.
The lower right window shows the original plane situation, the upper right window shows the 3D reconstruction of the toric intersection.
Note that for a given ellipse, changing the angle changes both, the torus and the plane.
Spirics of Perseus appear as bisoptics of hyperbolas also, with some restrictions: the points of intersection of the spiric with the asymptotes of the hyperbola do not belong to the isoptic.
Another difference: each component of the bisoptic of an ellipse corresponds to a different angle (theta and 180-theta). For a bisoptic of a hyperbola, on each component 2 arcs correspond to one angle and 2 arcs correspond to the supplementary angle.
This applet has been built with George Ghantous.
References:
Th. Dana-Picard, G. Mann and N. Zehavi (2011): From conic intersections to toric intersections: the case of the isoptic curves of an ellipse, The Montana Mathematical Enthusiast 9 (1), 59-76.
Available: http://www.math.umt.edu/TMME/vol9no1and2/index.html.
Th. Dana-Picard, N. Zehavi and G. Mann (2014): Bisoptic curves of hyperbolas, International
Journal of Mathematical Education in Science and Technology 45 (5), 762-781.