Unformatted text preview: 122 CHAPTER 2. CURVES
The coeﬃcient p above is called the parameter of ordinates
for γ .
• Ellipses: If P V is not parallel to AC , then the line P V
(extended) meets the line AB (extended) at a second vertex P ′ .
If φ denotes the (acute) angle between P and a horizontal plane
H, then V lies between P and P ′ if 0 ≤ φ < π − α and P lies
between V and P ′ if π − α < φ ≤ π .
In the ﬁrst case, in contrast to the case of the parabola, the
ratio of |QV |2 to |P V | depends on the point Q on the curve γ .
To understand it, we again form the “orthia” of γ as a line
segment P L perpendicular to P V with length p.
Now let S be the intersection of LP ′ with the line through V
parallel to P L (Figure 2.4).
V C J B
S H P′ Figure 2.4: Deﬁnition of S
One derives the equation (see Appendix A)
|QV |2 = |V S | · |P V | . (2.5) This is like Equation (2.3), but |P L| is replaced by the shorter
length |V S |; in the Pythagorean terminology, the square on the
ordinate is equal to the rectangle with width equal to the
abcissa applied to the segment V S , falling short of P L. The
Greek for “falling short” is ἔλλειψιζ, and Apollonius calls γ an
ellipse in this case.
To obtain the rectangular equation of the ellipse, we set
d = PP′ ...
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- Fall '08
- Calculus, Horizontal plane, rectangular equation, First Case, Pythagorean terminology