45-135-isoptic of an ellipse (45-bisoptic)
An "apha"-isoptic of a plane curve C is the geometric locus of points through which passes a pair of tangents making an angle of "aplha".
The "alpha"-isoptic and the "180-alpha"-isoptic of the given ellipse are the two components of a quartic, called a spiric of Perseus (an oval). The internal component corresponds to obtuse angle, and the external to accute angle. The director circle of the ellipse (non represented here) lies between them.
Reference:
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.