### 6.5 Parameter Representation of Surfaces: $Math content$

Using the parameter representation it is possible to describe very complicated surfaces in space. The functions $Math content$ displayed in the three function windows of the simulation map the $Math content$-plane into the space described by $Math content$. If there are periodic functions of the parameters among $Math content$ closed or self penetrating surfaces in space are created.

From the formula for the first surface in the list of functions you realize, that the parameter $Math content$ periodically modulates the value $Math content$ of the $Math content$-function: $Math content$. For $Math content$ the modulation factor is equal to $Math content$. The parameter $Math content$ determines the amplitude of the modulation; $Math content$ fixes a reasonable initial value. The remaining parameters $Math content$ and $Math content$ are not used in this example; please observe for the individual functions which quantities are modulated by a term containing $Math content$.

The scale for the $Math content$,$Math content$ and $Math content$-axes is adjusted in such a way, that the interval $Math content$ is covered. The range of the parameters $Math content$ and $Math content$ is from $Math content$ to $Math content$, such that the simple trigonometric functions like $Math content$ run through a full period in the parameter interval.

Via clicking at the selection window the preset functions are called.

With the sliders $Math content$ you can change the parameters of the spatial surfaces also during the animation. Via editing the corresponding formulas you can also switch the animation to other quantities.

You can edit the formulas In the formula window or enter formulas from scratch. Do not forget to press the Enter-key after this.

Some elementary surfaces were already covered by the basic functions $Math content$; thus you may compare the formulas in both representations.

Since $Math content$ and $Math content$ are scaled by $Math content$ ($Math content$), there always appears a factor of $Math content$, when $Math content$ and $Math content$ are directly connected with $Math content$, i.e. outside of periodic functions. A factor $Math content$ shows, that the quantity that is multiplied by it is modulated in the animation. Reset returns the value of $Math content$ to $Math content$.

The following functions are preset in the selection windows (for the sake of clarity we have left out the multiplication sign * in the simulation syntax).

tilting plane x = p/pi; y = q/pi; z = cos(vt)(a/pi-0.6)p

hyperbolic plane x = p/pi ; y = q/pi ; z = cos(vt)pq/piˆ2

cylinder x = cos(vt)acos(p) ; y = bsin(p); z = cq/(2pi)

Möbius band x = acos(p)(1+q/(2pi)cos(p/2));
y = 2bsin(p)(1+q/(2pi)cos(p/2)); z = cq/(pi)sin(p/2t)

sphere x = cos(vt)acos(p)abs(cos(q));
y = cos(vt)asin(p)abs(cos(q)); z = cos(vt)asin(q)

ellipsoid x = acos(p)abs(cos(q)); y = cos(vt)bsin(p)abs(cos(q));
z = csin(q)

double cone x = a/pi(1+qcos(p)); y = cos(vt)b/pi(1+qsin(p));
z = cq/pi

torus x = (a+cos(vt)bcos(q))sin(p);
y = (c+cos(vt)bcos(q))cos(p); z = bsin(q)

8-torus x = (a+bcos$Math content$(q))sin(p);
y = ((cos(vt)ˆ2)c+bcos(q))(cos(p))$Math content$; z = 0.6bsin(q)

mouth x = (cos(vt)c+bcos(q))cos$Math content$(p); y = (a+bcos(q))sin(p);
z = bsin(q)

boat_1 x = (c+bcos(q))cos$Math content$(p); y = (a+bcos(q))sin(p);
z = cos(vt)bcos(q)

boat_2 x = (c+bcos(q))cos$Math content$(p); y = (a+bcos(q))sin(p);
z = cos(vt)bcos$Math content$(q)

The formulas of the simulation contain additional fixed numbers, that guarantee a reasonable size of the graphs when opening them.

Using the parameter representation aesthetically very pleasing spatial surfaces can be created, that can be used as inspiration for design and construction, such that the playful element is not short changed. The simulation file may now be opened to show the following, interactive graphic in Fig6.10 of a torus.

The handling of the simulation in Fig.6.10 is analogue to that for the previous $Math content$-presentations. Details and suggestions for experiments are given on the description pages.