### 6.6 Parameter representation of curves, space paths $Math content$

Using this parameter representation very complicated curves (paths) in space can be described. The functions $Math content$ ,that are displayed in the three function windows, map the interval covered by the only parameter $Math content$ uniquely to a curve $Math content$in space. If $Math content$ contain periodic functions of the parameters, closed ore self-intersecting space curves are created.

For the simulation in Fig6.11 the one-dimensional parameter $Math content$ 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 $Math content$ 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 $Math content$-pane can be switched on and off with the option switch.

The program calculates the functions in time-steps of $Math content$ milliseconds. Thus animation speed can be set With the slider $Math content$. For $Math content$ the picture is static.

With the sliders $Math content$ 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 $Math content$ is added to the rational number $Math content$, 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 $Math content$-plane one sees the corresponding plane orbits, i.e. plane Lissajou-figures.

Choose after the first animation the constants $Math content$ such, that the range of coordinates is fully used. Many plots become graphically interesting only if the constants $Math content$ are chosen differently. The default value for all of them is $Math content$, 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 $Math content$ to $Math content$ is available. The $Math content$-plane is intersected by $Math content$-axis in the middle of the $Math content$-vectors. maximum and minimum values are marked on the $Math content$-axis via a red and green point respectively.

With the sliders $Math content$ 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 $Math content$- 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 $Math content$ 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