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Section 1.4 Parameterization

In this section, we study how to describe geometric objects like curves and surfaces using parameters. A curve is static like a road on a map, while a parameterization describes motion along that road.
We will parametrize geometric figures using "nice" vector-valued functions. Here "nice functions" means \(C^1\) (continuously differentiable) functions but often smooth functions.

Subsection 1.4.1 Curves in \(\R^n\)

A parametrization of a curve in \(\R^n\) is a vector-valued function \(\vr \colon I \to \R^n\) where \(I \subseteq \R\) is an interval. The variable of \(\vr\) is called the parameter. The range of \(\vr\) is the curve parameterized by \(\vr\text{.}\)
A line passing through a point \(\vp \in \R^n\) with a constant velocity vector \(\vv \in \R^n\) is parameterized by
\begin{equation*} \vr(t) = \vp + t\vv, \quad t \in \R\text{.} \end{equation*}
If we interpret the parameter \(t\) as time, \(\vr(t)\) represents the position of a particle at time \(t\text{.}\) The velocity is given by the derivative \(\vr'(t) = \vv\text{,}\) and the acceleration is \(\vr''(t) = \vz\text{,}\) which characterizes linear motion.

Example 1.4.1.

Find a parameterization of the line passing through \(A=\gv{1, 2}\) and \(B=\gv{3, 5}\text{.}\)
Solution.
The direction vector is \(\vv = B - A = \langle 2, 3 \rangle\text{.}\) Using \(A\) as the base point:
\begin{equation*} \vr(t) = \langle 1, 2 \rangle + t\langle 2, 3 \rangle = \langle 1+2t, 2+3t \rangle \quad t \in \R. \end{equation*}
A line passing through two points A and B.
Figure 1.4.2. Line through \(A\) and \(B\text{.}\)
A circle of radius \(R\) centered at the origin in \(\R^2\) can be parameterized by
\begin{equation*} \vr(t) = \langle R\cos t, R\sin t \rangle, \quad t \in [0, 2\pi). \end{equation*}

Example 1.4.3.

Parameterize the circle of radius 2 centered at the origin, traversed counterclockwise.
Solution.
\begin{equation*} \vr(t) = \langle 2\cos t, 2\sin t \rangle\text{.} \end{equation*}
A circle with an arrow indicating counterclockwise direction.
Figure 1.4.4. Parametrized Circle.
A helix in \(\R^3\) combines circular motion in a plane with linear motion along a normal direction (a direction orthogonal to the plane). For example, if the circular motion of the helix in the \(xy\)-plane and the line motion is in the positive \(z\)-direction, then it is parametrize by
\begin{equation*} \vr(t) = \langle R\cos t, R\sin t, ct \rangle \ c \gt 0\text{.} \end{equation*}

Example 1.4.5.

Parameterize a helix of radius 1 that rises 1 unit for every full turn.
Solution.
We need \(z(2\pi) - z(0) = 1\text{,}\) so \(c(2\pi) = 1 \implies c = 1/(2\pi)\text{.}\)
\begin{equation*} \vr(t) = \left\langle \cos t, \sin t, \frac{t}{2\pi} \right\rangle\text{.} \end{equation*}
A 3D spiral curve.
Figure 1.4.6. A Helix.

Subsection 1.4.2 Planes in \(\R^3\)

A parametrization of a surface in \(\R^3\) uses two parameters:
\begin{equation*} \vr(u,v) \colon D \to \R^3, \qquad D \subseteq \R^2. \end{equation*}
The set \(D\) is the parameter domain, and the range of \(\vr\) is the surface.
A plane through a point \(\vp\) with two nonparallel direction vectors \(\vu\) and \(\vv\) can be parameterized by
\begin{equation*} \vr(s,t) = \vp + s\vu + t\vv, \qquad (s,t)\in \R^2. \end{equation*}
This is the parametric equation of a plane. The vectors \(\vu\) and \(\vv\) span the plane, while \(\vp\) fixes its position in space.

Example 1.4.7.

Parameterize the plane through \(\gv{1,1,0}\) parallel to the vectors \(\gv{1,0,1}\) and \(\gv{0,2,1}\text{.}\)
Solution.
Using the point as the base point and the two given vectors as direction vectors,
\begin{equation*} \vr(s,t)=\gv{1,1,0}+s\gv{1,0,1}+t\gv{0,2,1}. \end{equation*}
Therefore
\begin{equation*} \vr(s,t)=\gv{1+s,\;1+2t,\;s+t}, \qquad (s,t)\in\R^2. \end{equation*}
A plane in three-dimensional space with a marked base point and two spanning vectors.
Figure 1.4.8. A parameterized plane patch.

Subsection 1.4.3 Common Surfaces in \(\R^3\)

Many familiar surfaces in three dimensions are described by choosing two parameters that move independently along the surface. We now record standard parameterizations for several common examples.
A right circular cone with vertex at the origin can be parameterized by using the radius \(r\) and the angle \(\theta\text{.}\) For the portion with \(0 \le z \le h\text{,}\)
\begin{equation*} \vr(r,\theta)=\gv{r\cos\theta,\; r\sin\theta,\; r}, \qquad 0 \le r \le h,\ 0 \le \theta \lt 2\pi. \end{equation*}

Example 1.4.9.

Parameterize the cone \(z=\sqrt{x^2+y^2}\) for \(0 \le z \le 2\text{.}\)
Solution.
Since the distance from the \(z\)-axis is \(r=z\text{,}\) a convenient parameterization is
\begin{equation*} \vr(r,\theta)=\gv{r\cos\theta,\; r\sin\theta,\; r}, \qquad 0 \le r \le 2,\ 0 \le \theta \lt 2\pi. \end{equation*}
A right circular cone in three-dimensional space.
Figure 1.4.10. A parameterized cone.
A sphere of radius \(R\) centered at the origin is parameterized by the longitude angle \(\theta\) and the polar angle \(\phi\text{:}\)
\begin{equation*} \vr(\theta,\phi) = \gv{R\sin\phi\cos\theta,\; R\sin\phi\sin\theta,\; R\cos\phi}, \end{equation*}
where \(0 \le \theta \lt 2\pi\) and \(0 \le \phi \le \pi\text{.}\)

Example 1.4.11.

Parameterize the sphere \(x^2+y^2+z^2=4\text{.}\)
Solution.
Here \(R=2\text{,}\) so
\begin{equation*} \vr(\theta,\phi) = \gv{2\sin\phi\cos\theta,\; 2\sin\phi\sin\theta,\; 2\cos\phi}, \end{equation*}
with \(0 \le \theta \lt 2\pi\) and \(0 \le \phi \le \pi\text{.}\)
A sphere centered at the origin in three-dimensional space.
Figure 1.4.12. A parameterized sphere.
A cylinder of radius \(R\) around the \(z\)-axis is obtained by combining circular motion in the horizontal direction with free motion in the vertical direction:
\begin{equation*} \vr(\theta,z)=\gv{R\cos\theta,\; R\sin\theta,\; z}. \end{equation*}

Example 1.4.13.

Parameterize the cylinder \(x^2+y^2=4\) for \(0 \le z \le 3\text{.}\)
Solution.
Since the radius is \(2\text{,}\)
\begin{equation*} \vr(\theta,z)=\gv{2\cos\theta,\; 2\sin\theta,\; z}, \end{equation*}
where \(0 \le \theta \lt 2\pi\) and \(0 \le z \le 3\text{.}\)
A vertical cylinder centered on the z-axis.
Figure 1.4.14. A parameterized cylinder.
A torus is formed by revolving a circle around an axis outside the circle. If the major radius is \(R\) and the minor radius is \(r\text{,}\) then a standard parameterization is
\begin{equation*} \vr(\theta,\phi) = \gv{(R+r\cos\phi)\cos\theta,\; (R+r\cos\phi)\sin\theta,\; r\sin\phi}, \end{equation*}
with \(0 \le \theta \lt 2\pi\) and \(0 \le \phi \lt 2\pi\text{.}\)

Example 1.4.15.

Parameterize the torus with major radius \(3\) and minor radius \(1\text{.}\)
Solution.
Substituting \(R=3\) and \(r=1\) gives
\begin{equation*} \vr(\theta,\phi) = \gv{(3+\cos\phi)\cos\theta,\; (3+\cos\phi)\sin\theta,\; \sin\phi}, \end{equation*}
where \(0 \le \theta \lt 2\pi\) and \(0 \le \phi \lt 2\pi\text{.}\)
A torus obtained by revolving a circle around the z-axis.
Figure 1.4.16. A parameterized torus.
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