Parametrization is applied when an object is “really” of lower dimension than the space it lives in. Curves, whether they live in 2D or 3D, are really only one-dimensional. A surface, which might live in 3D (such as a sphere), is really only two-dimensional. The “really” is made rigorous by our ability to represent such surfaces in terms of the lower number of variables. Curves may be parametrized by a vector function r(t), and surfaces by a function r(s,t). These are just vector-valued functions with only one or two inputs. One could think of vector fields as parametrizations, but once you get up to three inputs, what you’re parametrizing is hard to envision.
Throughout this entry I am going to be conflating points and their position vectors, or alternatively vectors and the location of their heads in standard position.
Curves in 2-space that are graphs of some y = f(x) are immediately parametrizable by x = t, y = f(t).
The easiest curve to parametrize in 3-space is a line. If P is a point on the line and v is a nonzero vector parallel to the line, then every point on the line is some multiple of v added to P. We represent the variable multiples using the parameter t: r(t) = P + tv.
If the line is given by two points v1 and v2, r(t) = tv2 + (1-t)v1 = v1 + t(v2 – v1). The first formulation comes from the point of view of parametrizing just the line segment from v1 to v2, hitting the former point as t = 0 and the latter at t = 1. If you remove the bounds on t you get the entire line containing that line segment. The second formulation comes from computing a direction vector by v2 – v1 and using v1 as the point anchoring the line in space.
Circles and ellipses in 2-space are all based on the trigonometric identity sin2t + cos2t = 1. Generally x is some modification of cos t and y of sin t. For a circle of radius r, for instance, we need x2 + y2 = r2. By multiplying the identity by r2 and rewriting a bit, we get (r sin t)2 + (r cos t)2 = r2, so letting x = r cos t and y = r sin t will work. These are not the only possible parametrizations, but they are typically the easiest to get at and work with.
If your circle is moved, with equation, say, (x – 4)2 + (y + 2)2 = 25, you need to parametrize so the left hand side becomes 25(cos2t + sin2t). This is easily taken care of by making the multiplicative adjustment as usual and then “undoing” the additive adjustment: x = 5 cos t + 4; y = 5 sin t – 2.
For ellipses, the multiplicative adjustment is different for each variable. If the axes of the ellipse are not parallel to the coordinate axes, life gets difficult, but if they are, it’s essentially the same process as above: (5x/2 – 1)2 + y2 = 1 (always solve for 1 on the right) may be parametrized as x = 2/5 cos t + 1; y = sin t.
A note on arc length versus the integral of a vector-valued function: you integrate velocity to get displacement, but you integrate speed to get distance. That is why you integrate the magnitude of the derivative to get the length of the curve.
Curvature may be thought of in these terms as well: the curvature κ at time t is |T'(t)|/|r'(t)| = speed of change in tangent direction divided by speed of change in position. You use the unit tangent T to isolate the tangent direction.
Note that every curve has infinitely many possible parametrizations. As t ranges over the real numbers, the same points in space will be traced out. Different parametrizations, however, may hit different points at a given value of t, trace out more or less distance in a unit of t change, and proceed in different directions as t changes positively or negatively.