Takashi Nishimura (Yokohama National University, Japan)
Wednesday, August 14th , 2019. 4:00 p.m. Room 5004.
Title: Anti-orthotomics of frontals and their applications
Abstract:
Let f : Nn → Rn+1 be a frontal with its Gauss mapping ν : N → Sn and
let P ∈ Rn+1 be a point such that (f(x) − P) · ν(x) ̸= 0 for any x ∈ N. In
this talk, for the mapping e f : N → Rn+1 de ned by
e f(x) = f(x) −
||f(x) − P||2
2(f(x) − P) · ν(x)
ν(x),
the following four are shown. (1) e f is a frontal with its Gauss mapping
eν(x) = f(x)−P
||f(x)−P|| at e f(x). (2) e f is the unique anti-orthotomic of f relative to
P. (3) The property ( e f(x) − P) · eν(x) ̸= 0 holds for any x ∈ N. (4) The
equality || e f(x) − P|| = || e f(x) − f(x)|| holds for any x ∈ N.
Moreover, three applications of the main result are given. As the rst
application, a generalization of Cahn-Ho man vector formula is given. The
second application is to clarify an optical meaning of anti-orthotomics. The
third application gives a criterion to be a front for a given frontal.
This is a joint work with Stanis law Janeczko.