famous space curves in geometry history?

Question

For an university assignment I have to visualize some curves in 3 dimensional space.

Until now I've implemented Bézier, helix and conical spiral.

Could you give me some advice about some famous curves in geometry history?

Answer 1

Space-filling curves, in particular the Hilbert curve (in 3D):

Answer 2

How about Viviani's curve? It is the intersection of a sphere with a cylinder. Viviani was a student (or disciple) of Gallileo. Wikipedia has a beautiful image:

Answer 3

I'd mention the curve of Archytas, the space curve formed by the intersection of a horn torus and a cylinder:

The parametric equations for the curve of Archytas are

$$\begin{align*}x&=\frac{\cos^4 t}{(1+\sin^2 t)^2}\\y&=\frac{\cos\,t\sin\,2t}{(1+\sin^2 t)^2}\\z&=\frac{\sin\,2t}{\sqrt 2(1+\sin^2 t)}\end{align*}$$

Archytas of Tarentum used this curve to help him solve the Delian problem.

Answer 4

Though is it is not 3D, the Clothoid or Cornu Spiral is an amazing curve. It surely can be made 3D by adding a simple extra parameter $z(t)=t$. It has infinite length but converges to two points in the plane. It has several applications in optics and road engineering, for example. An it looks quite nice:

I found a 3D plot too:

Answer 5

Let $t \in [0,2\pi]$. And let $r(t)=\frac{1}{2-\sin (2t)}[\cos(3t),\sin(3t),\cos(2t)]$. Then $r(t)$ parametrizes a 3-dimensional curve with no three-tangent plane (I must admit I'm not sure of the English terminology here) - that is, every tangent plane of the curve meets the curve in no more than two points.

(I have no clue how to prove such things, however)

Answer 6

If I may be allowed to give another example, there is the spherical epitrochoid, the locus of a circle rolling on another circle. It was studied by Johann Bernoulli and Franz Reuleaux in the context of the kinematics of gear systems.

Letting $p$ be the radius of the fixed circle ("pitch circle"), $r$ the radius of the rolling circle, $hr$ the distance of the tracing point from the center of the rolling circle, and $\omega$ the inclination of the rolling circle from the x-y coordinate plane, the parametric equations for the spherical epitrochoid are:

$$\begin{align*}x&=(p+r\cos\;\omega)\cos\;t-h r\left(\cos\;\omega\cos\left(\frac{p}{r}t\right)\cos\;t-\sin\left(\frac{p}{r}t\right)\sin\;t\right)\\y&=(p+r \cos\;\omega)\sin\;t-h r\left(\cos\;\omega\cos\left(\frac{p}{r}t\right)\sin\;t+\sin\left(\frac{p}{r}t\right)\cos\;t\right)\\z&=r\sin\;\omega\left(1-h\cos\left(\frac{p}{r}t\right)\right)\end{align*}$$

Here is the case $p=1$, $r=\frac15$, $h=1$, $\omega=\frac{2\pi}{3}$:

Here is the case $p=1$, $r=\frac15$, $h=1$, $\omega=\frac{\pi}{6}$:

Answer 7

wow! thanks all, everyone got very interesting answer i've found also slinky that is quite funny cause it is also a toy