6.5 Parameter Representation of Surfaces: $x=f_x\left(p,q\right);y=f_y\left(p,q\right)z=f_z\left(p,q\right)$

Using the parameter representation it is possible to describe very complicated surfaces in space. The functions $f_x,f_y,f_z$ displayed in the three function windows of the simulation map the $pq$-plane into the space described by $x,y,z$. If there are periodic functions of the parameters among $f_x,f_y,f_z$ 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 $v$ periodically modulates the value $z_i$ of the $z$-function: $z=z_{i}acos\left(vt\right)$. For $t=0$ the modulation factor is equal to $1$. The parameter $a$ determines the amplitude of the modulation; $a-0.6$ fixes a reasonable initial value. The remaining parameters $b$ and $c$ are not used in this example; please observe for the individual functions which quantities are modulated by a term containing $cosvt$.

The scale for the $x$,$y$ and $z$-axes is adjusted in such a way, that the interval $-1\le x,y,z\le +1$ is covered. The range of the parameters $p$ and $q$ is from $-\pi$ to $+\pi$, such that the simple trigonometric functions like $cosp$ run through a full period in the parameter interval.

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

With the sliders $a,b,c$ 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 $z=f\left(x,y\right)$; thus you may compare the formulas in both representations.

Since $p$ and $q$ are scaled by $pi$ ($\pi$), there always appears a factor of $1∕pi$, when $p$ and $q$ are directly connected with $x,y,z$, i.e. outside of periodic functions. A factor $cosvt$ shows, that the quantity that is multiplied by it is modulated in the animation. Reset returns the value of $cosvt$ to $1$.

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${}^2$(q))sin(p);
y = ((cos(vt)ˆ2)c+bcos(q))(cos(p))${}^2$; z = 0.6bsin(q)

mouth x = (cos(vt)c+bcos(q))cos${}^3$(p); y = (a+bcos(q))sin(p);
z = bsin(q)

boat_1 x = (c+bcos(q))cos${}^3$(p); y = (a+bcos(q))sin(p);
z = cos(vt)bcos(q)

boat_2 x = (c+bcos(q))cos${}^3$(p); y = (a+bcos(q))sin(p);
z = cos(vt)bcos${}^2$(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.

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Figure 6.10: Function plotter for animated $3D$– parameter surfaces; in the picture a torus is shown, whose dimensions can be changed with sliders. This animation also contains the Möbius band, that was shown at the beginning of the book, in a simpler form.

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The handling of the simulation in Fig.6.10 is analogue to that for the previous $3D$-presentations. Details and suggestions for experiments are given on the description pages.