Applications of envelopes

Abstract

Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore relationships between numerical ranges of 2 × 2 matrices and families of circles to study the elliptical range theorem. Then we deduce a relationship between envelopes and the boundaries of families of intersecting circles and use it to find the boundaries of various families of pseudohyperbolic disks.

Numerical range and compressions of the shift

“…However in this case, the answer is yes. For the details about that and the fact that W (T ) is the closed elliptical disk with boundary (5), see [13]. Then the elliptical range theorem follows immediately from this result about W (T ).…”

“…One can also use envelopes of families of circles to prove the elliptical range theorem. The proof idea described here is due to Donoghue [27], but many of the details appear in [13]. First observe that each A is unitarily equivalent to…”

A shadow of the repulsive Rutherford scattering and Hamilton vector

“…where φ 0 is the polar angle of the symmetry axis of the hyperbolic trajectory. Then it is known from mathematics (for inquisitive readers: there are some subtleties in the mathematical definition of an envelope, see [12][13][14]) that the envelope equation is obtained by excluding the parameter φ 0 from the system…”