This function plotter displays animated surfaces in 3D that may be closed or intersecting. Its predefined function presents open or closed twisted bands, among them the famous Möbius strip.
x = fx( p, q , t ); y = fy( p, q, t ); z = fz( p, q, t )
The coordinate functions fx, fy,, fz , which in the simulation are shown in three editable text fields, map the points of the plane of variables p, q unambiguously into a surface in space x y z . If fx, fy,, fz contain periodic functions of the variables p and q, closed or intersecting surfaces may be produced.
The functions may contain three constants a, b, v that can be changed by sliders. In the predefined functions v is used to animate the surfaces by oscillating one or more of the coordinates via time dependent terms.
At start of the simulation you will see the projection of a closed band in space, viewed under perspective distortion. It is embedded into an x y z tripod, and is accompanied by the x y- plane z = 0. This plane can be deactivated by its check box.
The band is a Möbius strip, which is a band twisted by one half turn. It has only one surface: when one walks around its axis in the xy-plane, one passes both sides of the strip without needing to penetrate the surface.
The 3 coordinate functions contain periodic modulation terms, controlled by time t and the parameter v, which is defined by slider v:
fx = cosp (1+q/2π cos(a/2 p-vt)
fy = sinp (1+q/2π cos(a/2 p-vt)
fz = b q/2π sin(a/2 p-vt)
The height of the band is defined by slider b.
Slider a defines the number of half turns of the band. For an integer the band is closed. a = 1 results in the Möbius strip. With a > 0 an even integers create normal bands with 2 surfaces and a/2 full twists. With a an uneven integer Möbius-similar bands are created with a half twists.
If a is not an integer, the band is not closed. By varying a a closed band can be cut open and closed again for a different number of half twists. Slider a is adjusted in such a way that it jumps to the next integer, when a is close to one.
Play starts the animation program by increasing t linear in time, starting from zero. The periodic modulation generates a band that is rotating around its symmetry axis. Slider v defines the speed of rotation. Pause freezes the animation at any orientation, Reset recreates the initial parameters and projection.
Scales of x, y z extend from -1 to +1 . The x y-plane crosses the z-axis at z = 0.
The range of variables p and q extend from -π to +π, creating full cycles in the periodic terms.
The orientation of the tripod in space can be changed by drawing with the mouse.
Other ways of visualization are described in the next page.
You can edit the formulas in the formula fields or input completely different ones. Don not forget to hit the ENTER key after every change.