Chapter 2: Curves and Coordinate Systems

A moving point traces a curve, and the honest way to describe the motion is one coordinate at a time: three ordinary functions $f(t)$, $g(t)$, $h(t)$ woven into a single vector-valued function $𝐫(t)=(f(t),g(t),h(t))$. This chapter is about learning to read that weave. We begin with the helix, whose three coordinate-plane shadows untangle it into the familiar graphs hiding inside, and then spend time in a parametric playground where a rolling wheel and a polar rose show how much personality two small formulas can carry — and how a single slider changes it.

Curves also force the question of which coordinates to use. Rectangular coordinates treat every direction alike, but many objects — spirals, spheres, cones — have a center or an axis, and coordinates adapted to that symmetry make their equations nearly trivial. We meet cylindrical and spherical coordinates through their coordinate surfaces, and discover that a "box" in spherical coordinates has almost no flat sides. The distortions you see there are not a nuisance; they are the geometric raw material for the volume elements of the integration chapter to come.