Parametric Curves and Surfaces

Definition

A parametric curve in cartesian coordinates is defined by three coordinate functions: \[ \begin{aligned} x &= x(t), \\ y &= y(t), \\ z &= z(t), \end{aligned} \] where \(t\) is a real parameter that varies over some interval \([t_0, t_1]\). As \(t\) changes, the point \((x(t), y(t), z(t))\) traces out a curve in \(\mathbb{R}^3\).

It is often convenient to collect these into a single vector-valued function \(\bm{r}: [t_0, t_1] \to \mathbb{R}^3\): \[ \bm{r}(t) = \begin{bmatrix} x(t) \\ y(t) \\ z(t) \end{bmatrix}. \] Such vector-valued functions are ubiquitous in engineering, describing such quantities as position, velocity, acceleration, and many other time-varying vectors, sometimes called state vectors (although they are in fact vector-valued functions).

When the curve lies entirely in a plane, one or more of the components \(\bm{r}(t)\) may be constant, reducing to a 2D case.

Example: Helical Motion

A point moving on a helix can be described by \[ x(t) = R \cos(\omega t), \qquad y(t) = R \sin(\omega t), \qquad z(t) = c t. \] Here \(R\) is the radius, \(\omega\) the angular velocity, and \(c\) the rate of ascent. Note that this is not the only possible paramterization of the helix. For instance, we could replace \(t\) with \(t^2\) in \(\bm{r}(t)\) to draw exactly the same helix, but with a different parameterization.

If we treat the parameter \(t\) as time, then \(\bm{r}(t) = \begin{bmatrix} x(t) & y(t) & z(t) \end{bmatrix}^\top\) is a position vector-valued function of time. Then the velocity vector-valued function \(\bm{v}(t)\) is given by \[ \bm{v}(t) = \frac{d\bm{r}}{dt} = \begin{bmatrix} -R \omega \sin(\omega t) \\ R \omega \cos(\omega t) \\ c \end{bmatrix}. \]

Definition

A parametric surface in three-dimensional space is given by three coordinate functions: $$\begin{align} x &= x(u, v), \\ y &= y(u, v), \\ z &= z(u, v). \end{align}$$ Each pair \((u, v)\) defines a point \((x, y, z)\) in \(\mathbb{R}^3\), and the collection of all such points forms a surface. It is often convenient to write this as a vector-valued function \(\bm{r}: [u_0, u_1] \times [v_0, v_1] \to \mathbb{R}^3\): \[ \bm{r}(u, v) = \begin{bmatrix} x(u, v) \\ y(u, v) \\ z(u, v) \end{bmatrix}. \]

As \(u\) varies while \(v\) is held fixed, we trace a curve of constant \(v\) on the surface, and vice versa for curves of constant \(u\).

The Tangent Plane and Normal Vector

The tangent plane to a parametric surface at a point \((u_0, v_0)\) is the plane that touches the surface at that point and is spanned by the tangent vectors \[\partial_u \bm{r}\quad\text{and}\quad\partial_v \bm{r}\] evaluated at \((u_0, v_0)\).

The normal vector to the surface is given by the cross product of the tangent vectors, assuming they are linearly independent: \[ \bm{n} = \partial_u \bm{r} \times \partial_v \bm{r} \] A level surface is a surface defined by a constant value of a function \(f(x, y, z)\), i.e., \(f(x, y, z) = c\). For level surfaces, the normal vector is given by the gradient of the function: \[ \bm{n} = \nabla f \] Here \(\nabla\) is the gradient operator, which will be defined in section 5.5.

The unit normal vector \(\hat{\bm{n}}\) is obtained by normalizing the normal vector: \[ \hat{\bm{n}} = \frac{\bm{n}}{\|\bm{n}\|} \]

Example: Cylinder

A simple and important surface in engineering is the cylinder, parameterized by its radius \(R\) and height \(h\): \[ x(u, v) = R \cos u, \qquad y(u, v) = R \sin u, \qquad z(u, v) = v, \] where \(u \in [0, 2\pi]\) and \(v \in [0, h]\). Each constant-\(v\) curve is a circle, while varying \(v\) moves the circle along the \(z\)-axis.

Computing the tangent vectors: \[ \partial_u \bm{r} = (-R \sin u, R \cos u, 0), \qquad \partial_v \bm{r} = (0, 0, 1). \] Computing the normal vector: \[ \bm{n} = \partial_u \bm{r} \times \partial_v \bm{r} = (R \cos u, R \sin u, 0). \] Computing the unit normal vector: \[ \hat{\bm{n}} = \frac{\bm{n}}{\|\bm{n}\|} = \frac{(R \cos u, R \sin u, 0)}{R} = (\cos u, \sin u, 0). \]