$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

5  Parameterization

(Relevant Section of the Textbook: 13.1 Vector Functions and Space Curves)

Definition 5.1 (Plane Curve) A plane curve is a function \(\vec{c}\colon\RR\to\RR^2\). Written in the coordinates \((x,y)\) of \(\RR^2\), a plane curve can be expressed using two coordinate functions \[\vec{c}(t)=(x(t),y(y))\]

We’ve seen examples of plane curves already, for instance parametric lines (like \(\ell(t)=(2t-1,3t+4)\)) and parametric circles, (like \(c(t)=(2\cos(t)+1,2\sin(t)-1)\)).

Definition 5.2 (Space Curve) A space curve is a function \(\vec{r}\colon\RR\to\RR^3\). Written in the coordinates \((x,y,z)\) of \(\RR^3\), a spade curve can be expressed using three coordinate functions \[\vec{c}(t)=(x(t),y(y),z(t))\]

We see curves like this in everyday life - watching a bird fly through the air we see its position changing in time, so its \(x,y\), and \(z\) components are all chaninging in time \((x(t),y(t),z(t))\). Likewise, watching your car driving on a GPS, we see the car’s position changing - its latitude and longitude are functions of time \(\mathrm{car}(t)=(\mathrm{lat}(t),\mathrm{long}(t))\). Indeed this is how we usually will use plane and space curves to trace out positions as a function of time. But it is also often useful to think of a curve as being traced out by a little arrow based at the origin (the vector picture, vs the position picture). This is particularly helpful when trying to build curves for yourself, as you can think about adding on terms, scalar multiplication, etc.

Here’s a graphing calculator for curves, so you can try making some of your own. Can you make this draw a circle in the \(yz\) plane?

Parametric curves are used in animation, physics, and engineering. In this visualization below, each little blob is animated using a parametric curve: this could trace out a school of fish swarming at sea, for instance.

5.1 Parameterization Tips

Studying the properties of parametric curves falls squarely within mathematics, and we will spend much time soon developing the calculus to do so. But creating parametric curves is more art than science: it really helps to build up some intution for a few basic examples, and then learn how to combine them and modify them to produce new and more interesting curves. I encourage you to follow along in the discussions below using the graphing calculator at the top of the page.

Example 5.1 (Parametrizing The Graph of a Function) If \(y=f(x)\) is a function, its graph consists of the \(y\) value \(f(x)\) whenever the \(x\)-value is \(x\). That means, the graph of \(f\) consists of the points \((x,f(x))\) in the plane. Expressed yet a third way, a parametric equation that traces out the graph is given by \[\vec{c}(t)=(t,f(t))\]

Recall that an implicit equation gives a relationship of \(y\) and \(x\) that are satisfied by an equation. Some of these express functions like \(y=x^2\), but others do not, for instance \(x^2+y^2=1\). Oftentimes given an implicit equation its desirable to parameterize it: to find a way to trace out the curve as a function of \(t\). There’s no single way to do this, and it often takes some trial and error. But some tips are below.

Example 5.2 (Parametrizing circles) The implicit equation for the unit circle is \(x^2+y^2=1\). Because the functions \(\cos t\) and \(\sin t\) satsify the equatio \[\cos^2(t)+\sin^2(t)=1\] We see that if \(x=\cos(t)\) and \(y=\sin(t)\) then \((x,y)\) must lie on the unit circle. Similarly, the equation \(f(t)=(r\cos(t),r\sin(t))\) parameterizes a circle of radius \(r\) centered at \((0,0)\), and \[f(t)=(r\cos(t)+h,r\sin(t)+k)\] parameterizes a circle of radius \(r\) centered at \((h,k)\).

Similar parameterizing an implicit plane curve by finding functions which satisfy the relations, one can parameterize curves that are the intersection of two known surfaces.

Example 5.3 Parameterize the twisted cubic, lying on \(y=x^2\) and \(z=x^3\). Here if \(x=t\) then we know \(t=t^2\) and \(z=t^3\): this fully specifes a point in 3-dimensional space, so we have our parameterization \[f(t)=(t,t^2,t^3)\]

Example 5.4 Parameterize the intersection of the cylinder \(x^2+y^2=3\) and the plane \(x+y+z=1\). On the cylinder, \(x\) and \(y\) both lie on a circle of radius \(\sqrt{3}\): so we can write \(x=\sqrt{3}\cos(t)\) and \(y=\sqrt{3}\sin(t)\). The cylinder equation doesn’t tell us anything about \(z\), so its no help there. But - we can solve the plane for \(z\) in terms of \(x\) and \(y\) to get \(z= 1-x-y\). Now we can plug in what we know \(x\) and \(y\) to be to get the parameterization:


Example 5.5 Intersection of \(4y=x^2+z^2\) and \(y=x\). We know already because \(y=x\) that our points in space are going to look like \((x,x,z)\). We can substitute this idea into the first equation to see that \(4y\) becoes \(4x\), and so \[x^2-4x+z^2=0\] This is the equation for a circle! We can find its center and radius by completing the square: \[x^2-4x=x^2-4x+4-4=(x-2x)^2-4\] So, this is the circle \[(x-2)^2+z^2=4\] Which is a circle of radius \(2\) centered at \((2,0)\). We can parameterize it as \(x=2\cos(t)+2\) and \(z=2\sin(t)\). So, in 3D along the plane \(y=x\) this becomes \[f(t)=\left(2\cos(t)+2,2\cos(t)+2,2\sin(t)\right)\]

5.1.1 Same Curve, Different Parameterizations

Different parameterizations can describe the same curve: since a parameterization is like an animation of the curve, you can think of this as tracing out the curve at different speeds.

Example 5.6 (Different Parameterizations of the Circle) All three of these parametric curves trace out the unit circle. \[(\cos t,\sin t)\] \[(\cos 2t, \sin 2t)\] \[(\cos t,-\sin t)\] The first traces it at unit speed, counterclockwise. The second at twice the speed in the same direction. And the third, at unit speed but backwards (clockwise).

Example 5.7 (Different Parameterizations of \(y^2=x^3\)) We can parameterize the implicit curve \(y^2=x^3\) in several ways: taking the square root of both sides gives \(y\) as a function of \(x\) (with a plus and minus component), \(y=\pm\sqrt{x^3}\) so one possible parameterization is \[\alpha(t)=(t,\pm\sqrt{t^3})\] This isn’t the nicest, as we have that \(\pm\) sign. Another thing we could do is take the cube root: this doesn’t cause any \(\pm\) ambiguity, and gives \(x\) as a function of \(y\), or \(x=\sqrt[3]{y^2}\), leading to the parametric curve \[\beta(t)=(\sqrt[3]{t^2},t)\] A third option is to find a function for \(x(t)\) where when we cube it, we get the same thing as if we squared the function we chose for \(y(t)\). This is of course tricker - but here one option is to take \(x=t^2\) and \(y=t^3\). Then \(x^3=t^6\) and \(y^2=t^6\) so \(x^3=y^2\) and our curve is \[\gamma(t)=(t^2,t^3)\]

5.1.2 New Curves from Old

Once we know a few parametric curves (circles, lines, some implicit curves, etc) - its easy to find more by modifying the ones we already know! Some of the simplest such tranfsormations we’ve already used in the case of circles, scaling and translation.

Theorem 5.1 (Scaling a Parametric Curve) If \(f(t)=(x(t),y(t))\) is a parametric curve, then \(rf(t)=(rx(t),ry(t))\) is a curve where all the coordinates are \(r\) times a big.

Theorem 5.2 (Translating a Parametric Curve) If \(f(t)=(x(t),y(t))\) is a parametric curve, then \(f(t)+(a,b)=(x(t)+a,y(t)+b)\) is the result of shifting the curve over by \((a,b)\).

Of course, more interesting transformations are also possible - and it’s easiest to see this through a couple examples!

5.2 Case Study: Spirals

We will make and understand a collection of spirals starting with the basic equation of the unit circle \[(\cos(t),\sin(t))\]

Example 5.8 (Archimedean Spiral) The archimedean spiral is the curve that rotates about the origin at unit speed, but after rotating angle \(t\), lies not at unit distane (like a circle) but at distance \(t\) from the origin. To parameterize, we multiply the circle by \(t\): \[\gamma(t)=(t\cos(t),t\sin(t))\]

Example 5.9 (Logarithmic Spiral) The logarithmic spiral moves away from the origin exponentially fast, instead of linearly. This has radius at time \(t\) equal to \(e^t\), so \[\gamma(t)=(e^t\cos(t),e^t\sin(t))\]

Different functions \(r(t)\) for the radius multiplied by the circle give spirals that move outwards (or inwards) at different speeds. Try making some of these in the graphing calculator above!

Exercise 5.1 (Whirlpool) Can you make a spiral that rotates about the origin at unit speed, but whose radius asymptotes to 2, never getting any larger?

Example 5.10 (Helix) A helix is a curve where \(x,y\) travel around a circle, and \(z\) increases at unit speed. For example, the unit helix is \[\gamma(t)=(\cos(t),\sin(t),t)\]

Example 5.11 (Slinky-Like Helix) What if we want a helix like curve to move vertically at an uneven rate? Replace the \(z\) component with a more interesting function of \(t\)! For instance, if \(z=e^t\) then the curve bunches up as \(t\to-\infty\) along the \(xy\) plane: \[\gamma(t)=(\cos(t),\sin(t),e^t)\]

Example 5.12 (Spiral On a Cone) The surface \(z=\sqrt{x^2+y^2}\) traces out a cone - the height is equal to the radius! How can we draw a spiral on the surface of the cone? Well, if we know what we want the spiral to do in its \(x\) and \(y\) components, we can calculate the \(z\) component using the formual above! For instance, given the archimedean spiral \((t\cos(t),t\sin(t))\) we see \(z=\sqrt{(t\cos t)^2+(t\sin t)^2}=t\). Thus, the curve is \[\gamma(t)=(t\cos t,t\sin t, t)\]

5.3 Videos

A recap of parametric curves

Parametric curves and elimination of parameters: