So, observing once more that consecutive conic sections
share a common point, the third shaping equation is obtained from (5) at [[theta].sub.F] = [[theta].sub.Fn-1]:
Although we can calculate the next term [[phi].sub.m+2] directly, the conic sections
allow us to easily "track" the movements of the points from [mathematical expression not reproducible].
The second part of Problem 1 considers studying Conic sections
It should be noted that the algebraic expression that refers to all possible parabolas is derived from the general, second-degree equation of the conic sections
, a[x.sup.2] + 2hxy + [by.sup.2] + 2gx + 2fy + c = 0.
On page 81 (Hypatia's work) students learn that conic sections
yield four kinds of curves, but only three are presented (the circle, ellipse, and parabola).
At twelve years of age Pascal discovered Euclid's axioms unaided; at sixteen he wrote a treatise on conic sections
, and at eighteen he invented a calculating machine.
Khayyam states that the solution of this cubic needs the use of conic sections
and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years.
Moreira, "Main-reflector shaping of omnidirectional dual reflectors using local conic sections
," IEEE Transactions on Antennas and Propagation, vol.
The authors provide a precalculus review before covering limits, differentiations, the applications of the derivative, the integral, applications of the integral, techniques of integration, advanced applications of the integral and Taylor polynomials, differential equations, infinite series, parametric equations, polar coordinates, and conic sections
over the bookAEs eleven chapters.
After reviewing prerequisites, the chapters consider such topics as equations and graphs, exponential and logarithmic functions, matrices and determinants, conic sections
, and probability and statistics.
7 Conic sections
cannot be plotted directly with many dynamic geometry software packages, and a conic section
only appears as the locus of some point.
Eight pages of "quickies" are followed by 26 pages of more complex problems in the areas of combinatorics and number theory, functions and polynomials, expression and identities, numerical approximation, algebraic inequalities, trigonometric inequalities, geometric inequalities, the triangle, Cevian lines, central symmetry, conic sections
, solid geometry, higher dimensions, vectors and matrices, and calculus.