 
Summary: *Hassan Azad, *Abdallah Laradji and **Asghar Qadir
*Department of Mathematical Sciences
King Fahd University of Petroleum & Minerals
Dhahran 31261, Saudi Arabia
**Department of Mathematics
QuaidiAzam University
Islamabad, Pakistan
1
FOCUSING SURFACES
1 Introduction
In standard textbooks (e.g., [7, p. 877]) as well as in Encylopaedia Britannica [6, Vol.
13, p.611], one finds variants of the following statement:
In spherical mirrors, rays parallel and very close to the principal axis are all reflected
through a single point.
This statement is actually false, and can only be understood in the following sense: It
is true only up to a linear approximation, and in fact, as soon as a mirror focuses parallel
rays, it must be a parabolic mirror.
This note makes these ideas precise by providing in Section 2 a novel approach to re
flecting properties of surfaces. There, inward and outward pointing vectors are introduced,
leading to a straightforward derivation of reflecting properties of surfaces of revolution. In
