### About

#
*3D* function plotter

##
*x = f*_{x}( p, q ); y = f_{y}( p, q ); z = f_{z}(
p, q)

_{x}( p, q ); y = f

_{y}( p, q ); z = f

_{z}( p, q)

This function plotter displays animated *3D* surfaces described by
three coordinate functions of two variables *p, q *for the spatial
coordinates *x, y, z. *

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

*pq*unambiguously into a surface in space

*xyz*. If

*f*contain periodic functions of the parameters

_{x }, f_{y ,}, f_{z}*p*and

*q,*closed or intersecting surfaces may be produced.

The functions may contain four constants *a, b, c, 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 the start of the simulation you will see the projection of a plane 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.**

Other predefined surfaces in space can by selected in the **ComboBox.**

The formula for the plane (*pi ≡ π*) shows that *v *modulates
the *z*-function periodically: *z = cos(vt)(a - 0.6)p.* For*
t = 0 *the modulation factor is 1; *a* determines the degree of
modulation; *(a- 0.6)* defines a reasonable initial value. The
other variable constants *b, c* are not used with the plane.

**Play** starts the animation, with time* t* starting at *0, *as
indicated in the* t *number field. With *cos(t)* fluctuating
periodically, the plane oscillates in space. Slider *a *defines the
base orientation. **Pause** freezes the animation at any spatial
position. **Reset** leads back to the initial conditions.

The range of *p *and *q* is *∓ π* . Scaling of
all three axes *x, y, z* has a range of *∓ 1*. The *xy-*plane
cuts the *z* axis at the center of the *z*- arrow. As
variables *p* and *q* change in the range *-π* to *+π,
*a periodic function as *cos(p)* completes a full period in the
variable plane.

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.

Predefined functions are selected in the **ComboBox** with a mouse
click. Constants *a,b,c *can be varied by sliders while the
animation is running. By editing the formulas you can change the terms
that are animated. You can input new parametric formulas to create your
own surfaces. Do not forget to press the *ENTER *key after a change!

Touching a surface with the mouse pointer lets its color filling disappear; the wire mesh of calculation will be pronouncedly visible.

# Visualization alternatives

**Rotation: **Mark any point within the tripod by 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 choose another
surface, and then the oldone anew

**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.**

**Predefined functions **

As p and scale in *pi* (*π*), for all terms where *p*
and *q* enter directly into formulas for *x ,y, z *(e.g __not__
in periodic function of the variables), a factor *1/pi* (1/π)
appears. A factor *cos(v*t)* indicates that the associated term is
animated. **Reset** redefines *t = 0 *and hence *cos(vt) *= *1.*

Fixed numbers in the formulas are used to define a reasonable scaling at
the start of the simulation for uniform default values of parameters *a,
b, c* (*= 0.5*).

x_function = p/pi

y_function = q/pi

z_function = cos(v*t)*(a/pi-0.6)*p

Plane

x_function = p/pi

y_function = q/pi

z_function = cos(v*t)*p*q/pi^2

Saddle

x_function = cos(v*t)*a*cos(p)

y_function = b*sin(p)

z_function = c*q/(2*pi)

Cylinder

x_function = a*cos(p)*(1+q/(2*pi)*cos(p/2))

y_function = 2*b*sin(p)*(1+q/(2*pi)*cos(p/2))

z_function = c*q/(pi)*sin(p/2*t)

Möbius strip

x_function = cos(v*t)*a*cos(p)*abs(cos(q))

y_function = cos(v*t)*a*sin(p)*abs(cos(q))

z_function = cos(v*t)*a*sin(q)

Sphere

x_function = a*cos(p)*abs(cos(q))

y_function = cos(v*t)*b*sin(p)*abs(cos(q))

z_function = c*sin(q)

Ellipsoid

x_functio n= a/pi*q*cos(p)*cos(v*t)");

y_function = b/pi*q*sin(p)*cos(v*t)");

z_function = c*q/pi");

Double cone

x_function = (a+0.6*cos(v*t)*b*cos(q))*sin(p)

y_function = (c+0.6*cos(v*t)*b*cos(q))*cos(p)

z_function = 0.6*b*sin(q)

Torus

x_function = 2*(a+0.3*b*cos(q))*sin(p)*cos(p)

y_function = 2*((cos(v*t)^2)*c+0.3*b*cos(q))*cos(p)*cos(p)*cos(p)

z_function = 0.6*b*sin(q)

Torus-8

x_function = (cos(v*t)*c+0.3*b*cos(q))*cos(p)*cos(p)*cos(p)

y_function = (a+0.3*b*cos(q))*sin(p)

z_function = b*0.3*sin(q)

Mouth

x_function = (0.4*c+0.4*b*cos(q))*cos(p)*cos(p)*cos(p)

y_function = (2*a+0.4*b*cos(q))*sin(p)

z_function = cos(v*t)*0.4*b*cos(q)

Boat_1

x_function = (0.4*c+0.4*b*cos(q))*cos(p)*cos(p)*cos(p)

y_function = (2*a+0.4*b*cos(q))*sin(p)

z_function = cos(v*t)*0.4*b*cos(q)*cos(q)

Boat_2

**E1: **Test the different surfaces without a change of
parameters. Rotate the frames and train your *3D* perception of
these functions.

**E2:** Study the formulas and develop a sense for the relation
between formulas and surface.

**E3: **Rotate, tilt and zoom the *3D* projection to optimize
the visual impression of the surface . Use the Camera Inspector, too
(see *Visualization page*).

**E4: **Vary parameters and study the influence on the
appearance of the surface.

**E5: **Change signs (+/-) in the formulas and study the effect.

**E6: **Vary** **the** **grade of power functions in the
formulas and study the resulting surfaces.

**E7: **Edit the formulas freehandedly and consider in advance
how that should influence the surface.

**E8:** Animate different parameters, or two at the same time.

**E9: **Delete the animation term and use* p* as a free
fourth parameter in your own formula.

**E10: **Reflect in using *3D* surfaces created this way in *design
of 3D objects*. The last three predefined functions are intended as
encouragement (mouth and boat). Remember that numerical machines could
directly use the formulas for control.

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Credits

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

### Sample Learning Goals

[text]

### For Teachers

## Parametric Surface 3D Simulator JavaScript Simulation Applet HTML5

### Instructions

#### Object Combo Box

#### Sliders

#### Analytical Surface Equation Boxes

#### Show xy-plane Check Box

#### Toggling Full Screen

#### Play/Pause and Reset Buttons

Research

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### Video

[text]

### Version:

### Other Resources

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### end faq

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- Details
- Written by Fremont
- Parent Category: Interactive Resources
- Category: Mathematics
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