Definitions for curves

We start with a reminder of some basic definitions. Most of these definitions will have equivalent concepts with respect to surfaces. A parameterized curve is a map from some interval of the real line to a subset of $\mathbb{R}^3$ . The trace of a curve is its image in $\mathbb{R}^3$ , and is simply a set of points. Note that many curves can have the same trace, for example, there are many ways to parameterize the unit circle. The map does not have to be one-to-one, that is, the curve can intersect itself. The term differentiable or smooth means infinitely differentiable, that is, of class $C^\infty$ .

An example of a curve is the map $\alpha: (a,b) \rightarrow \ensuremath{\mathbb{R}^3}$ where $\alpha(t) = (x(t), y(t), z(t))$ . Since the map is smooth, we can define the velocity of the curve at a point $t$ as $\alpha'(t) = (x'(t), y'(t), z'(t))$ , where the prime indicates differentiation. The velocity is a vector quantity at every point and is also known as the tangent vector. The line containing the point on the curve and the tangent vector is the tangent line. A point on a curve with zero velocity has no tangent, and these points are called singular points. We generally want a tangent at every point, so we restrict ourselves to regular curves, that is, curves without singular points.

The distance measured along a curve is called arc length and it is defined by

$$s(t) = \int_{t_0}^t \vert\alpha'(t)\vert dt,$$

where $\vert\cdot\vert$ denotes Euclidean distance in $\mathbb{R}^3$ . It is possible to find a parameterization of a regular curve such that the parameter $t$ is the arc length measured from some point. The velocity of such curves is one, which is quite useful in practice. We will assume that our curves are parameterized by arc length in the material that follows.