The Mathematics Of Conic Sections

Preface
A circle can look like an ellipse when we move our head. A thrown stone draws something close to a parabola. A spacecraft arriving from beyond the Solar System may follow a hyperbolic path around the Sun. These observations do not yet explain why such different situations should share a small family of equations. This book follows that question from a cut through a cone to some of the most useful ideas in modern mathematics.
The subject is unusually hospitable. Its first pictures can be made with string, paper and a torch. Its later arguments use matrices, derivatives, homogeneous coordinates and optimisation. There is no need to pretend that these later tools are elementary. They become intelligible when we see which obstacle each tool removes. Coordinates let us calculate what a picture suggests. Matrices organise transformations. Projective geometry explains what perspective preserves. Calculus turns a curve into a local direction, an area or a changing motion.
I have written for readers who enjoy the moment when an apparently isolated fact becomes part of a larger structure. The equations therefore belong in the narrative. A formula is not a badge of difficulty. It is a compact claim that can be opened, tested and understood. Worked examples develop the claims rather than merely supplying answers. Diagrams show configurations that words alone would make cumbersome. Tables provide comparisons at the points where comparison matters.
Several boundaries deserve attention. A photograph of an oval does not prove that the photographed object is an ellipse. An exact conic orbit belongs to an ideal two body model. A curved roof is not automatically a parabola, and a catenary is a different curve. Mathematical beauty gains precision when these distinctions are kept.
We move from distance and cutting to transformations and projective geometry, then to mechanics, light, measurement, arithmetic and design. Later chapters examine reliable calculation, wave and quantum applications, projective flows and the limits of conic models.