Parametric Curves

Section 6.1 Parametric Curves

Subsection 6.1.1 Definition

So far in the study of calculus, a curve was the locus of an equation. In Section 2.1, I defined a special set of loci called algebraic plane curves and worked with implicit derivatives to understand their shapes, slopes and singularities. A curve, as a locus, is a static object presented by an equation. Sometimes, however, this presentation isn’t sufficient. A curve can represent the motion of an object (think of an ellipse representing the path of an orbit or a parabola representing the path of a projectile). In the context of motion, the curve isn’t presented all at once. In addition to the shape of the curve, information is needed about how fast an object is moving and where it is at some point in time. This calls for a new treatment of curves. This section introduces the idea of a parametric curve, which will capture motion along a curve as well as describing the shape of the curve.

Motion along a curve is descried by a parameter, hence the name: parametric curves. This parameter is usually interpreted as time, so the variable \(t\) is conventional. Here is the formal definition. I’ll state the definition in \(\RR^2\text{,}\) since I will only be working with curves in the plane in this course, but the definition could be stated in any \(\RR^n\text{.}\)

Definition 6.1.1.

Let \(t\) be an indepedent variable (the parameter, usually interpreted as time) and let \([a,b]\) be an interval in \(\RR\text{.}\) A parametric curve in \(\RR^2\) is a set of two continuous functions \(x(t)\) and \(y(t)\) on the domain \([a,b]\text{.}\) These functions describe the \(x\) and \(y\) coordinates of the curve at any point \(t\) in time. Though there are two functions, it is conventional to refer to the curve with one symbol, typically the greek letter \(\gamma\text{.}\)

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

To visualize a curve, I can draw all the points \(\gamma(t) \in \RR^2\) as time runs through its domain \(t \in [a,b]\text{.}\) This visualization is notably different from drawing the graph of a function \(y = f(x)\text{.}\) In the graph of a function, both in input and the output are shown; the \(x\) axis is the input and the \(y\) axis is the output. For a parametric curve, only the output it drawn. There is no \(t\) axis; time is an implicit idea, thinking of movement along the shape. Since the time axis is not show, a parametric curve doesn’t need to satisfy any kind of vertical line test. It can meander all over the plane, double back on itself, or cross over previous section of itself. Any direction of movement is possible at any point.

I did insist, in the definition, that the functions must be continuous. This means that the movement along a parametric curve is connected — no teleportation from once point to somewhere else. For paths and movement, this is a reasonable restriction.

Subsection 6.1.2 Examples

Like many mathematical objects, the best way to understand curves is through examples. I’m going to give many examples in this section.

Example 6.1.2.

Figure 6.1.3. Parametric Curve \(\gamma_1(t) = (t,t)\)

Consider the curve \(\gamma_1(t) = (t,t)\) for \(t \in [0,5]\text{.}\) This curves gives the line \(y=x\text{,}\) starting at \((0,0)\) and ending at \((5,5)\text{.}\) At \(t=0\text{,}\) the curve is at \((0,0)\text{;}\) at \(t=1\text{,}\) it is at \((1,1)\text{;}\) and so on. All parametric curves are about movement: this curve is not only the line \(y=x\text{,}\) but also the starting point \((0,0)\text{,}\) the ending point \((5,5)\) and the rate of movement along the line.

Example 6.1.4.

Consider the curve \(\gamma_2(t) = (-t,-t)\) for \(t \in [-5,0]\text{.}\) This is exactly the same line as the previouus example, since it still satisfies \(y=x\text{.}\) However, now the curve starts at \((5,5)\) when \(t=-5\) and moves towards \((0,0)\) when \(t=0\text{.}\) Different parametric curve can describe the same shape with different notions of movement along that shape.

Example 6.1.5.

Now consider the curve \(\gamma_3(t) = (t^2,t^2)\) for \(t \in [0,\sqrt{5}]\text{.}\) Again, this is the same line as the previous two examples. Like \(\gamma_1\text{,}\) it starts at \((0,0)\) and ends at \((5,5)\text{.}\) However, the previous two examples described movement at a fixed speed and this example, due to the quadratic terms, describes accelerating movement along the shape.

Example 6.1.6.

A very important example is the unit circle, which can be described as \(\gamma(t) = (\cos t, \sin t)\) for \(t \in [0,2\pi]\text{.}\) This curve traces the circle counter clockwise and starts at \((1,0)\text{.}\) I could change the parameter domain. If \(t \in [0,8\pi]\text{,}\) then there are four periods of \(\sin t\) and \(\cos t\text{,}\) so this curve traces the same circle four times. A parametric curve can trace over itself any number of times.

Example 6.1.7.

Figure 6.1.8. \(\gamma(t) = (\cos 3t, \sin 5t)\)

Consider the curve \(\gamma(t) = (\cos 3t, \sin 5t)\) for \(t \in [0, 2\pi]\text{.}\) This curve gives a lovely pattern, perhaps reminiscent of spirographs (if those happened to be a part of your childhood experience). The curve is shown in Figure 6.1.8. It is a good example of the fact that a parametric curve can have many self-intersections.

Example 6.1.9.

Figure 6.1.10. The Cycloid for \(t \in [0,4\pi]\)

The next example is called the cycloid. ‘cycliod’ is a greek word; many curves have history going back hundreds or thousands of years in geometry and have greek or latin names reflecting that history. The cycloid is the curve \(\gamma(t) = (a(t-\sin t), a(1-\cos t))\) for \(t \in [0,\infty]\text{.}\) It describes the movement of a point at the edge of a wheel as the wheel rolls.

A spiral is a parameteric curve of the form \(\gamma(t) = (f(t) \cos t, f(t) \sin t)\) for \(f(t)\) a positive monotonic function. \(f(t)\) represents the change in radius while the cosine and sine function show (counterclockwise) rotation around the origin.

Example 6.1.11.

Figure 6.1.12. The Archimedian Spiral

The Archemidean spiral is the curve \(\gamma(t) = (t \cos t, t \sin t)\) with \(t \in [0,\infty]\text{.}\) It has linear radius growth, so the arms of the spiral are equally spaced.

Example 6.1.13.

Figure 6.1.14. The Hyperbolic Spiral

The hyperbolic spiral is an inward spiral, (it has decreasing radius). Its expression is \(\gamma(t) = (\frac{\cos t}{t}, \frac{\sin t}{t})\) for \(t \in (0,\infty)\text{.}\)

Example 6.1.15.

Figure 6.1.16. The Logarithmic Spiral

The last spiral in these examples is the logarithmic spiral, which has exponential growth in radius. Its form is \(\gamma(t) = (e^{t} \cos t, e^{t} \sin t)\) for \(t \in \RR\text{.}\)

The logarithmic spiral shows up frequently, both in mathematics and (approximately) in the natural world. Examples of approximate logarithmic spirals include nautilus shells and spiral galaxies (though the degree to which these natural spirals really fit the logarithmic shape is the subject of some debate.)

Example 6.1.17.

Figure 6.1.18. The Cardiod

A final example, which is not a spiral, is the cardiod. It has the form \(\gamma(t) = ((1-\sin t) \cos t, (1-\sin t) \sin t\) for \(t \in [0, 2\pi]\text{.}\) The name comes from its vaguely heart-shaped path.

Subsection 6.1.3 Reparametrization

A parametric curve is more than just a shape in \(\RR^2\text{.}\) The curve also records the start point, end point, direction, and rate of movement along the shape. All that information depends on the parametrization: the way in which position depends on the parameter.

I might like to change the direction or speed along a given curve while keeping the same shape. I can do that by changing the parametrization. Consider a curve \(\gamma(t) = (x(t), y(t))\) and a substitution \(t = g(u)\) for some monotonic function \(g\text{.}\) I can replace \(t\) in the curve with \(g(u)\) to get \(\gamma(u) = (x(g(u)),y(g(u)))\text{.}\) This is called reparametrization. Often it is conventional (though slighly confusing) to just write \(t(u)\) instead of \(t = g(u)\text{.}\)

The circle \(\gamma(t) = (\cos t, \sin t)\) for \(t \in [0,2\pi]\) can be reparametrized in a variety of ways. \(t(u) = u^2\) gives the new circle \(\gamma(u) = (\cos u^2, \sin u^2)\) with \(u \in [0, \sqrt{2\pi}]\) . Similarly, \(t(u) = \sqrt{u}\) gives the new circle \(\gamma(u) = (\cos \sqrt{u}, \sin \sqrt{u})\) with \(u \in [0, (2\pi)^2]\text{.}\) Each reparametrization doesn’t change the shape, but does change the rate of movement around the circle. Notice that I need to adjust the bounds of the new parametrizations if I want to cover the same distance.

Each curve shape in \(\RR^2\) has many (infinitely many) parametrizations. Any monotonic function \(t = g(u)\) can give a valid reparametrization, and there are infinitely many monotonic functions. This is what makes parametric curves much richer than loci: each shape also has an infinite variety of movement possibilities along the shape.

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