Using this parameter representation very complicated curves (paths) in space can be described. The functions ${f}_{x},{f}_{y},{f}_{z}$ ,that are displayed in the three function windows, map the interval covered by the only parameter $t$ uniquely to a curve $x\left(t\right),y\left(t\right),z\left(t\right)$in space. If ${f}_{x},{f}_{y},{f}_{z}$ contain periodic functions of the parameters, closed ore self-intersecting space curves are created.

For the simulation in Fig6.11 the one-dimensional parameter $t$ is interpreted as time. This parameter is repeatedly incremented by a constant time-step, such that the curve starting at the origin grows accordingly, until one of the coordinates becomes larger than $2$ and leaves the range of the figure and the animation stops.

The blue path marker is connected to the origin with a vector. The vector and the $xy$-pane can be switched on and off with the option switch.

The program calculates the functions in time-steps of $\Delta t=p\times 0.1$ milliseconds. Thus animation speed can be set With the slider $p$. For $p=0$ the picture is static.

With the sliders $a,b,c$ up to three constants in the parameter functions can adjusted between 0 and 1. The sliders actually determine the product of the constant with 100, such that that the constant as well as the ratio of two of these constants. This leads to closed orbits in the case of oscillation plots. In the second example the irrational number $\sqrt{2}$ is added to the rational number $c$, which results in the orbit not being closed. This shows who you can in general create orbits that are not closed. You may increase the animation speed to recognized this quickly. For the detailed observation the projection settings of the camera inspector are useful. In the $xy$-plane one sees the corresponding plane orbits, i.e. plane Lissajou-figures.

Choose after the first animation the constants $a,b,c$ such, that the range of coordinates is fully used. Many plots become graphically interesting only if the constants $a,b,c$ are chosen differently. The default value for all of them is $0.5$, to the show the basic functions during the first run.

You can edit the formulas or enter new ones from scratch.

The scale has been chosen in such a way for all three axes, that the range $-1$ to $+1$ is available. The $xy$-plane is intersected by $z$-axis in the middle of the $z$-vectors. maximum and minimum values are marked on the $z$-axis via a red and green point respectively.

With the sliders $a,b,c$ you may, even during the animation, change the parameters of the space curves. With suitable entries of time-dependent functions you can also switch the animation to other quantities.

The handling of the simulation is otherwise again analogous to that of the previous $3D$- presentations. Details are given on the description pages.

There are however two keys for starting the simulation with slightly different functions:

Start starts the simulation and erases all the curves that are present.

Play does not delete previous curves , continues for equal parameters with the simulation and superimposes old an new curves for change parameters or changed function types.

Stop stops as second functionality of the Play-button the simulation, that is continued with Play.

Clear deletes all curves.

Reset a b c resets $a,b,c$ to the default values.

This simulation also gives ample opportunities for creative and playful experiments. The following picture shows the simulation. Es shows to interleaved orbits, one of which with the hyperbolic envelope is already closed, while the one with the envelope in the shape of a torus is still open.

End of chapter 6