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Unformatted text preview: 119 2.1. CONIC SECTIONS Here, we give a simpliﬁed and anachronistic version of the basic ideas in
Book I, bowlderizing [25, pp. 355-9].
Conical Surface: Start with a horizontal circle C ; on the vertical line
through the center of C (the axis6 ) pick a point A distinct from the
center of C . The union of the lines through A intersecting C (the
generators) is a surface K consisting of two cones joined at their
common vertex (Figure 2.1). If we put the origin at A, the axis Figure 2.1: Conical Surface K
coincides with the z -axis, and K is the locus of the equation in
z 2 = m2 (x2 + y 2 ) (2.1) where
m = cot α
is the cotangent of the angle α between the axis and the generators.
Horizontal Sections: A horizontal plane H not containing A intersects
K in a circle centered on the axis. The yz -plane intersects H in a line
which meets this circle at two points, B and C ; clearly the segment
BC is a diameter of the circle. Given a point Q on this circle distinct
from B and C (Figure 2.2), the line through Q parallel to the x-axis
intersects the circle in a second point R, and the segment QR is
bisected at V , the intersection of QR with the yz -plane. A basic
property of circles, implicit in Prop. 13, Book VI of Euclid’s
Elements [27, vol. 2, p. 216] and equivalent to the equation of a
circle in rectangular coordinates (Exercise 3), is
of Eutocius’ edition by Thabit ibn Qurra (826-901); the eighth book is lost. A modern
translation of Books I-IV is . An extensive detailed and scholarly examination of the
Conics has recently been published by Fried and Unguru .
Apollonius allows the axis to be oblique —not necessarily normal to the plane of C . ...
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