Parametric Curves

Section 2.1 Parametric Curves

Subsection 2.1.1 Definition of Parametric Cuves

The major goal of this course is the extension of calculus to multivariable functions. A parametric curve is a function with multiple outputs; these outputs are coordinates of position in some euclidean space depending on time; as such, parametric curves are used to talk about motion through space. The calculus of parametric curves is a way to understand the physics of such motion, covering both linear and angular velocity and acceleration in a nice, holistic approach. When considering parametric curves, I like to imagine the movement of point-like objects through space under the influence of various forces. Projectiles with gravity and air friction is one imporant example; flying objects such as insects, birds or helicopters is a second; the motions of planets, moons and satellite under gravity is a third.

Subsection 2.1.2 Parametric Curves

Definition 2.1.1.

A parametric curve in \(\RR^n\) is a continuous function \(\gamma :[a,b] \rightarrow \RR^n\text{.}\) As is convention, I will typically use the symbol \(\gamma\) for an arbitrary parametric curve. The output is often writen in component as follows (in \(\RR^3\) for demonstration).

\begin{equation*} \gamma(t) = (x(t), y(t), z(y)) \end{equation*}

The functions \(x(t)\text{,}\) \(y(t)\) and \(z(t)\) are called the component functionsi.

I can identify a parametric curve with its image: that is, I think of a curve as a decription of the set of points in \(\RR^n\) which are the output of the curve. In this interpretation, the curve describes motion along this set of points: it starts at the point \(\gamma(a) \in \RR^n\) and moves along the curve, ending at \(\gamma(b) \in \RR^n\) when it reaches the end of its domain.

The continuity condition is important, since a parametric curve is a connected path. I could define a function which jumps around, but it wouldn’t really fit the notion of a curve — it would not describe reasonable motion through space.

For visualizing parametric curves, it is conventional to graph only the output or image of the curve. There is never a \(t\) axis in any of these graphs; instead, the variable \(t\) is the parameter of movement along the curve. Let me start with some basic examples.

Example 2.1.2.

The curve \(\gamma(t) = (7t, 4t)\) is a straight line in \(\RR^2\text{.}\) At \(t=0\text{,}\) it is at the origin. It then moves upward in the first quadrant: \((7,4)\text{,}\) \((14,8)\text{,}\) \((21,12)\) and so on. It has slope \(\frac{4}[7}\text{.}\) It is a special case of a general construction. In \(\RR^2\text{,}\) any curve \(\gamma(t) = (at, bt)\) is a straight line through the orign with slope \(\frac{b}{a}\text{.}\) In \(\RR^3\text{,}\) any curve \(\gamma(t) = (at, bt, ct)\) is a straight line through the origin as well.

Example 2.1.3.

Figure 2.1.4. The curve \(\gamma(t) = (\cos t, \sin t)\)

The curve \(\gamma(t) = (\cos t, \sin t)\text{,}\) for \(t \in [0, 2\pi]\) traces out a circle, as in Figure 2.1.4. This is an important example: many of the other curves I will use in this course are adaptations of the circle.

Notice that I defined this curve on the domain \([0, 2\pi]\text{.}\) If I extend this domain, the curve just starts to retrace over the circle. It’s good to observe that parametric curves can self-intersect and trace over themselves many times. This is a very different situation from, say, graphs of functions which cannot self-intersect.

Example 2.1.5.

Figure 2.1.6. The curve \(\gamma(t) = \left( \frac{1}{t} , t \right)\)

The curve \(\gamma(t) = \left(\frac{1}{t}, t \right)\) on the domain \(t \in \left[\frac{1}{5},5 \right]\) traces part of the graph of \(f(x) = \frac{1}{x}\text{,}\) as in Figure 2.1.6.

Example 2.1.7.

Figure 2.1.8. The curve \(\gamma(t) = (\cos 2t, \sin t)\)

The form of this curve is similar to the circle, but the first component has a \(2\text{,}\) which doubles the period of the cosine function. The curve \(\gamma(t) = (\cos 2t, \sin t)\) on the domain \(t \in [0, 2\pi]\) oscillates faster in the \(x\) direction than in the \(y\) direction, as in Figure 2.1.8. So, instead of a circle, I get a different kind of rotational movement, where the \(x\) coordinate oscillates faster than the \(y\) coordinate.

Example 2.1.9.

Figure 2.1.10. The logarithmic spiral

A spiral in \(\RR^2\) is a parametric curve of the form \(\gamma(t) = (f(t) \cos t, f(t) \sin t)\) where \(f(t)\) is a monotonic continuous function. It is based on the circle, but instead of having a constant radius, the radius is either increasing or decreasing as the curve traces around the circle. The curve \(\gamma(t) = (2e^{\frac{t}{4}} \cos t, 2e^{\frac{t}{4}} \sin t)\) is a logarithmic spiral, as in Figure 2.1.10. For the logarithmic spiral, the parameter \(t\) be any real number: the spiral will spin outwards without end as \(t \rightarrow \infty\text{.}\)

Example 2.1.11.

Figure 2.1.12. The archimedian spiral

The curve \(\gamma(t) = (t \cos t, t \sin t)\) is the archimedian spiral, as in Figure 2.1.12. For this spiral, I assume the domain \(t \in [0,\infty)\text{;}\) the shape starts at the origin and spins outward. Unlike the logarithmic spiral, the arms of the archimedian spiral are equally spaced.

Example 2.1.13.

Figure 2.1.14. The curve \(\gamma(t) = (t\cos t, t\sin t,t)\) on \([0,20]\) is a spiral in \(\RR^3\text{.}\)

In three dimensions, in addition to spiraling outward or inward, a curve can spiral upward or downward. The curve \(\gamma(t) = (t \cos t, t \sin t, t)\text{,}\) for \(t \in [0,\infty)\text{,}\) is a conical spiral extending infinitely upward, as seen in Figure 2.1.14.

Subsection 2.1.3 Varied Parametrizations

Figure 2.1.15. The graph of a parabola as a parametric curve

A parametric curve is not just its shape: it is also the rate of movement along that shape. Therefore, given a fixed shape of a curve, there are many (infinitely many!) parametric curves that trace the same shape. I can say that the same curve has many parametrizations.

Example 2.1.16.

Consider the following curves.

\begin{align*} \gamma_1(t) \amp = \left( t, t^2 \right) \amp \amp t \in [0, 4]\\ \gamma_2(t) \amp = \left( t^2, t^4 \right) \amp \amp t \in [0, 2]\\ \gamma_3(t) \amp = \left( \sqrt{t}, t \right) \amp \amp t \in [0, 16]\\ \gamma_4(t) \amp = \left( 5t, 25t^2 \right) \amp \amp t \in \left[0, \frac{4}{5} \right] \end{align*}

All four of these have exactly the same parabolic image. They all describe the same curve, shown in Figure 2.1.15.

Subsection 2.1.4 Reparametrization

Since the same shape can have many different parametrizations, I want a process to switch between them. This process is called reparametrization. I’ll state the definition in \(\RR^3\text{,}\) but dropping the third coordinate will give the same idea in \(\RR^2\) and, of course, there is a generalization to \(\RR^n\text{.}\)

Definition 2.1.17.

Let \(\gamma(t): [a,b] \rightarrow \RR^3\) be a parametric curve with coordinates \((\gamma_1(t), \gamma_2(t), \gamma_3(t))\text{.}\) A reparametrization of \(\gamma\) is the replacement of the variable \(t\) by a new variable \(u\text{.}\) The two variables are related by a monotonic increasing function \(t = t(u)\) expressing the parameter \(t\) in terms of a new parameter \(u\text{.}\) I replace \(t\) by the function \(t(u)\) to give a parametric curve in terms of \(u\text{:}\) \(\gamma(u) = (\gamma_1(t(u)), \gamma_2(t(u)), \gamma_3(t(u)))\text{.}\)

Example 2.1.18.

The unit circle in \(\RR^2\) is parametrized by \(\gamma(t) = (\cos t, \sin t)\text{.}\) If \(t = 3u\) then \(\gamma(u) = (\cos 3u, \sin 3u)\) is a reparametrization of the same circle. The first parametrization finishes a revolution in \(t \in [0, 2\pi]\text{,}\) but multiplication by \(3\) in the second parametrization means that a full revolution is completed in \(u \in [0, 2\pi/3]\) — that is, the second parametrization moves along the circle three times as fast.

Many reparametrizations of the circle are possible.

\begin{align*} \text{ If } t = \frac{u}{3} \amp \text{ then } \gamma(u) = \left( \cos \frac{u}{3}, \sin \frac{u}{3} \right)\\ \text{ If } t = u^2 \amp \text{ then } \gamma(u) = \left( \cos u^2, \sin u^2 \right)\\ \text{ If } t = \sqrt{u} \amp \text{ then } \gamma(u) = \left( \cos \sqrt{u}, \sin \sqrt{u} \right) \end{align*}

Even though the shape of the curve is the same, the parametrization affects the rate movement along the curve.

Coming to grips with the fact that the same shape can have multiple parametrizations is a major part of the conceptual challenge of parametric curves. When I calculate, I need to be conscious of the varied parametrizations. If I calculate the length of the shape, I want that length to be the same, regardless of how fast the movement is along the curve. Therefore, the length calculation should be independent of the parametrization. However, if I want to know details about the local movement (speed, acceleration, change in directions), then these calculations should depened on the parametrization. Different parametrizations will move along the curve at different rates, and therefore have different speeds, accelerations, etc. The next two sections work out all this calculus of parametric curves, and everything in those two sections will be careful with the variability of the parametrization.

Prev Top Next