We first saw vector-valued functions and parametrized curves when we were studying curves in the plane. The exact same ideas work in three dimensions. The input of our function is a scalar $t$, and the output is a vector ${\bf f}(t)$, which can be

or a host of other quantities that are described by vectors. The variable $t$, which often means time, is called a parameter.

Let's explore the first interpretation. Let $${\bf r}(t)\ = \ U \subseteq {\mathbb R} \to {\mathbb R}^3 \ = \ x(t)\,{\bf i}+ y(t)\, {\bf j} + z(t)\, {\bf k}$$ be a vector-valued function with real valued components $x(t), \, y(t),$ and $z(t)$ which we assume have continuous derivatives. Think of ${\bf r}(t)$ as a position vector from the origin to the point $(x(t),\, y(t),\, z(t))$ that changes as $t$ varies over the domain $U$ of ${\bf r}$. The terminal point of ${\bf r}(t)$ traces a curve in ${\mathbb R}^3$ as $t$ varies; think of it as the trajectory of a moving particle. Formally,

A Path in ${\mathbb R}^3$ is a map ${\bf r} : U \subseteq {\mathbb R} \to {\mathbb R}^3$. The set $C$ of terminal points ${\bf r}(t)$ as $t$ varies over $U$ is called a space curve, and the path ${\bf r}$ is said to parametrize $C$ or trace out $C$. When ${\bf r} : U \subseteq {\mathbb R} \to {\mathbb R}^2$, the set of terminal points ${\bf r}(t)$ will be called a plane curve.


The vector form ${\bf r}(t) = {\bf a} + t{\bf v}$ for a line is an example we've met already. Also, if $y = f(x)$ is a real-valued function of one variable, then the terminal points of the path ${\bf r}(x) = x\,{\bf i} + f(x)\, {\bf j}$ is just the graph of $y = f(x)$. If a curve $C$ in the plane, such as a circle, is given parametrically by $(x(t),\,y(t))$, then ${\bf r}(t) = x(t)\, {\bf i}+y(t)\, {\bf j}$ parametrizes $C$. We are combining both of these familiar concepts and generalizing them to ${\mathbb R}^3$ (or to any dimensional space):

A vector valued function ${\bf r}(t)$ is essentially the same thing as three scalar functions. Some texts make a distinction between the vector equation ${\bf r}(t) = \cos(t) {\bf i} + \sin(t) {\bf j} + t {\bf k}$ and the corresponding parametric curve or parametrized curve\begin{eqnarray*}x(t)&=& \cos(t) \cr y(t) &=& \sin(t) \cr z(t) &=& t.\end{eqnarray*}But since they mean the exact same thing, we will use these terms interchangably.

As these two examples show, however, it is often easier to understand a space curve by identifying it with a curve on a surface, especially when it's an 'interesting' quadric surface! Usually this means eliminating the parameter $t$ from the components $x(t),\, y(t),$ and $z(t)$ (the trig identity $\cos^2 (.) + \sin^2 (.) = 1$ and double angle formulas are often useful here).

  Example 1: the curve parametrized by $${\bf c}(t) \ = \ \langle\, \cos t,\ \sin t\,, \ t\, \rangle$$ lies on the cylinder $\ x^2 + y^2 \ = \ 1$ because $$x(t)^2 + y(t)^2 \ = \ 1, \quad \hbox{for all} \ t\,.$$   Example 2: the curve parametrized by $${\bf c}(t) \ = \ \langle\, t \cos t,\ t \sin t\,, \ t\, \rangle$$ lies on the cone $z^2 \ = \ x^2 + y^2$ because $$x(t)^2 + y(t)^2 \ = \ z(t)^2, \quad \hbox{for all} \ t\,.$$


Notice that both of these curves spiral counter-clockwise when viewed from overhead. How would the vector functions have to change for the curves to spiral clockwise?

Question: Example $1$ is a spiral staircase. What's its relation to a Double Helix$?$ Which vector function ${\bf r}(t)$ will parametrize a Double Helix?

Many interesting space curves such as ${\bf r}(t) = \langle \cos t,\, \sin t,\,\cos 2t\rangle$ arise as the intersection of two surfaces:

Since $$\cos ^2 t + \sin^2 t \ = \ 1\,, \qquad \qquad \cos 2 t \ = \ \cos^2 - \sin^2t \ = \ 2 \cos^2 t - 1\,,$$ the orange curve, call it a 'Pringle curve', lies on each of the quadric surfaces $$x^2 + y^2 \ = \ 1\,, \qquad\qquad z \ = \ x^2 - y^2\,, \qquad\qquad z \ = \ 2 x^2 - 1\,,$$ as well as in the intersection of these surfaces. Does the Pringle curve lie on any other quadric surfaces? Did we use up all the double angle formulas in the list above?