### About

#
Animated *Möbius strip*

##
and other *3D* strips

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 = f*

_{x}( p, q , t ); y = f_{y}( p, q, t ); z = f_{z}( p, q, t )
The coordinate functions *f _{x}, f_{y,}, f_{z }*,
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

*f*contain periodic functions of the variables

_{x}, f_{y,}, f_{z}*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**:

*f _{x} = cosp (1+q/2π cos(a/2 p-vt)*

*f _{y} = sinp (1+q/2π cos(a/2 p-vt) *

*f _{z }= 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.

# Visualization alternatives

**Rotation: **Mark any point within the tripod using the mouse
pointer and *draw* (while the left mouse key stays pressed).

**Shift: ** *Draw* while the **Strg** key is pressed.

**Zoom: ***Draw* while the **Shift **key is
pressed.

**Correction: **To return to the default projection activate **Reset .**

**Show coordinates:** Mark a point on the surface while pressing the **Alt**
key. When you *draw*, a cutting plane will pass through the
surface. Depending on orientation, different cuts can be evaluated.

**Camera- Inspector: **Press the * right* mouse key. A
context menu will appear. Choose

*Elements option/ drawing 3D panel/ Camera.*The

*will appear. It will stay visible until it is deactivated. It offers the following options:*

**Camera Inspector**

**Perspective:** Distant lines appear shorter than near ones.

**No perspective: ** No perspective distortion.

**Planar xy or yz or yx **: One looks onto the
respective planes.

**Other options: **Degree and angle of perspective can be defined.

**Optimizing ** **parameters.** The spatial impression can be
optimized by adjusting parameters. The optimum will be different for
different projections.

The context menu also offers programs for producing a **picture** or a**
video.**

These are the predefined functions for the bands (as code in *InitPage*)

x_functions = "cos(p)*(1+q/(2*pi)*cos(a/2*p.v*t))";

y_functions = "sin(p)*(1+q/(2*pi)*cos(a/2*p-v*t))";

z_functions = "b*q/(pi)*sin(a/2*p-v*t)";

pi = π

a number of half twists

a = 1.0 common Moebius strip.

b defines height;

p and q range *-π* to *+π *

**E1: **Run the animation and rotate the *3D* frame. Follow
the band around its periphery with the mouse pointer to confirm that it
has a single surface, yet two sides.

**E2:** Choose *a* = *0*. You will see a truncated,
reversing cone. With *b = 0* a breathing ring is displayed.

**E3: **Reset**. **Vary *a* from 1 to a slightly higher
value. The band will be cut. When will it be closed again?

**E4: **When do you get Möbius type strips, when normal bands?

**E5: **Change signs in the formulas and observe the results.

**E6: **Make more drastic changes in the formulas, trying to still
create bands.

**E7: **Consider beforehand what kind of band will be created by
a certain change.

**E8:** Work with different modulation terms.

**E9: **Delete the modulation term and use the now free
parameter *v* in the formulas.

**E10:** Load the *Camera Inspector *(use the right mouse key,
search in the uppermost line of the appearing context menu). Test the
various projections. Change *a* and interpret the results.

**E11:** Can you construct formulas for a band with a knot?

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH- Foundation; Fremont Teng; Loo Kang Wee

### end faq

### Sample Learning Goals

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### For Teachers

## Eye Catcher 3D Surface JavaScript Simulation Applet HTML5

### Instructions

#### Control Panel

#### Analytical Curve Function Fieldboxes

#### Toggling Full Screen

#### Play/Pause and Reset Buttons

Research

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### Video

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### Version:

### Other Resources

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