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Cycloid Curves

Among the famous planar curves is the cycloid. A cycloid is defined as the trace of a point on a disk when this disk rolls along a line. The disk is not allowed to slide. In the case of a regular cycloid, the generating point is on the rim of the disk. A shorter distance between the generating point and the center of the rolling disk will result in a curtate cycloid, a longer distance then the radius of the disk will give a prolate cycloide. These curves are being looked at more closely in this mathematical experiment.

The Experiments Options

The shape of the cycloid is determined by two parameters, the first being the radius r of the rolling disk and second the distance d between the generating point and the center of the disk. The parametrisation of the cycloid curve is the following:

c(t) = ( rt - d sin(t), r - d cos(t) )

We normalized the radius r of the disk to 1, since only the relation between the distance d to the radius r is relevant for the shape of the cycloid curve. The distance can be set using the slider Distance d or by entering a number in the corresponding text field. If d equals 1 the point is laying on the rim of the disk,  for smaller values the point is within the disk and accordingly outside for larger values. This will determine the shape of the cycloid curve. If the distance is exactly the radius of the disk, then the curve looks like several arcs joined side by side forming cusps. If the point is within the disk, the cycloid is smoothened at the cusps and a generating point outside the disk will replace the cusps with loops.
The two sliders Cycloid Discr and Cycloid Length control the number of points of the cycloid curve, that are calculated and the total length of the cycloid respectively. A larger number of calculated point shows a better approximation of the cycloid curve.

Summary
Cycloid Discr Number of calculated points of the cycloid curve
Cycloid Length Total length of the cycloid
Distance d Controls the distance from the generating point to the center of the rolling disk. Determines the shape of the cycloid curve.

Additional Options

There are three more checkboxes below the sliders. The first of those toggles the rolling disk, if unchecked the disk is not visible. The second checkbox does the same thing for the cycloid curve. More interesting is the third checkbox, that shows the surface of rotation of the cycloid curve if checked.


Since the display shows a top-down view of the XY-plane, only a projection of the surface of rotation is visible. To get a better look at the surface, you can change to a perspective view. To change to this view, open the camera control panel in the main menu via Inspector - Camera or by pressing CTRL-g. In this panel you can change the view with the radio boxes at the top of the panel. To get a perspective 3D view, activate the Perspective box.

User Activity
  1. Enable Show Surface checkbox
  2. Press CTRL-g to open camera panel
  3. Enable Perspective radio box

Summary
Show surface of rotation Activate the Show Surface checkbox in the main window
Change camera view In the main menu, Inspector - Camera or CTRL-g, then choose the view at the top of the camera info panel.
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