1 Analytic Geometry

Analytic geometry studies geometric objects through the equations their points satisfy. The gain is that questions about shape become questions about algebra: the property that defines a curve is written as a relation between coordinates, and the relation can then be manipulated by ordinary calculation.

This section works that idea through the conic sections — the circle, parabola, ellipse and hyperbola — so called because each arises as the intersection of a plane with a cone, the curve obtained depending only on how the plane is tilted. Each is defined by a distance property, and in every case the same three steps turn that property into an equation: write the distances with the distance formula, clear the square roots by squaring, and simplify.

Two later ideas unify what the derivations produce separately. The eccentricity \(e\) is a single number distinguishing the curves, so that the four cases become one family; and in polar coordinates that family collapses into the single equation \(r = \dfrac {ep}{1 + e\cos \theta }\). The section closes with the change of axes — translation and rotation — needed to recognise a conic whose axes are not those of the coordinate system.

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