Parametric equation

Parametric equations involve continuous functions of some parameters (let’s use $t$), $x=f(t)$, $y=g(t)$. We describe a curve in the plane with these functions: $(f(t),g(t))$.

For example, the unit circle can be described by:

$$r=\{\begin{array}{l}x=\cos (t)\\ y=\sin (t)\end{array}$$

We can also extend this to ${\mathbb{R}}^3$. Parameterisation is an important tool in multivariable calculus, especially with line integrals and surface integrals.

Common parameterisations

Important to remember common ones. We’re looking for computationally minimal mappings. Occasionally we find that the easiest mappings are not computationally efficient, but are reliable.

For a line in $\mathbb{R}$ or $\mathbb{C}$, we parameterise with:

$$z(t)=t{z}_2+(1-t)z_1$$

In 2D

For a circle, we use:

$$x=r\cos \theta$$ $$y=r\sin \theta$$

For an ellipse, we slightly modify this procedure, where we take:

$$x=a\cos \theta$$ $$y=b\sin \theta$$

then we substitute into our equation. We let the $a$ and $b$ coefficients equal each other (such that we can factor out the trigonometric functions), find $a$ in terms of $b$ (or the other way) then substitute until we find the other variable.

In 3D

For cylinders, we parameterise with cylindrical coordinates, where $u=\theta$ and $v=z$. The cross product of the tangent vectors is $|{\vec{t}}_u\times {\vec{t}}_v|=a$.

$$x^2+y^2=z,z\in [0,1]$$ $$x=a\cos \theta ,y=a\sin \theta ,z=z$$

For a hemisphere, we use spherical coordinates, where $u=\phi$ and $v=\theta$. $|{\vec{t}}_u\times {\vec{t}}_v|=a^2\sin \phi$.

$$x=a\sin \phi \cos \theta ,y=a\sin \phi \sin \theta ,z=a\cos \phi$$

For paraboloids, we use cylindrical coordinates, where $u=r$ and $v=\theta$. $|{\vec{t}}_u\times {\vec{t}}_v|=r\sqrt{1+4r^2}$

$$x=r\cos \theta ,y=r\sin \theta ,z=r^2$$

For a cone with maximum radius $a$,¹ we use cylindrical coordinates, where $u=\theta$ and $v=z$. $|{\vec{t}}_u\times {\vec{t}}_v|=\sqrt{2}r$

$$x=\frac{az}{h}\cos \theta ,y=\frac{az}{h}\sin \theta ,z=z$$

For an ellipsoid, we parameterise with spherical coordinates.

$$x=a\sin \phi \cos \theta$$ $$y=b\sin \phi \sin \theta$$ $$z=c\cos \phi$$

Footnotes

1. If this is difficult to remember, derive it on the spot by taking traces. ↩