Parametric Curves and Polar Coordinates: Overview

A Cartesian equation

y = f(x)

describes a curve but says nothing about how a particle traverses it. Parametric equations

x = f(t)

y = g(t)

restore that information — the parameter

t

is typically time, and the same curve

(x,y)

can be swept in any direction and at any speed. This is the natural language for trajectories, and it gives tangent lines, arc length, and speed from the derivatives

\dot{x}, \dot{y}

Polar coordinates replace the Cartesian grid with a radial one centred at the origin: every point is described by its distance $r$ from the origin and its angle $\theta$ from the positive $x$ -axis. Spirals, rose curves, and especially conic sections (ellipses, parabolas, hyperbolas) have vastly simpler polar equations than Cartesian ones, and the unified focus-directrix definition of conics in polar form $r = \frac{ed}{1 - e\cos\theta}$ appears directly in orbital mechanics and in the geometry of zero-coupon bond price paths under yield changes.

By the end of this topic you should be able to:

parametrise standard curves; compute $dy/dx$ , $d^2y/dx^2$ , and arc length from parametric form;
convert between Cartesian and polar coordinates; sketch polar curves; compute polar area and arc length;
identify and sketch the three conics from their focus-directrix definitions; write their polar equations.

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