Hypocycloid

The hypocycloid is the path traced by a moving point (a "light") on the rim of a wheel rolling without slipping along the inside a bounding circle. This mathlet extends that by allowing the moving point to be along a radial line at some factor times the radius. In this applet, the bounding circle always has radius 1. The parametric formulas for the hypocycloid are \[ x(\theta) = (1-a)\cdot \cos(a\theta) + ar\cdot \cos\left((1-a)\cdot\theta\right)\] \[ y(\theta) = (1-a)\cdot \sin(a\theta) - ar\cdot \sin\left((1-a)\cdot\theta\right)\] Here \(a\) is the radius of the wheel; \(r\) is the radial factor mentioned above; \(\theta\) is a parameter whose geometric meaning is described below,

There are two color schemes in the applet one with a dark background and one with a white background. In describing the applets we will use the colors that go with dark background and put those from the white background in parentheses.

The graphing window displays a wheel of radius \(a\) in orange (brown). And the bounding circle in gray (gray). The segment containing the centers of the two circles and their point of tangency is in green (purple). The radial segment from the center of the wheel to the moving point ("light") is in cyan (blue). The "light" is in yellow (tangerine) as is the trajectory it traces out. The "light" is at a distance \(ra\) from the wheel center.

If \(a \lt 1\) then the wheel is inside the bounding circle.
If \(a \gt 1\) then the bounding circle is inside the wheel.
If \(a \lt 0\) then the wheel rolls around the outside of the bounding circle. In this the case trace is called a hypercycloid.

The parameter \(\theta\) represents the angle between the radius from the center of the wheel to the point of tangency (green (purple) line) and the radius from the center of the wheel to the light (cyan (blue) line). Positive \(\theta\) measures the clockwise rotation of the cyan (blue) line from the green (purple) line.

Sliders
All of the sliders can be controlled by the arrow keys once they have the focus. The up/right arrows increase the value and the down/left arrows decrease the value.

The \(\theta\, a, r\) slides control the values of their respective parameters.

For the \(a, r\) sliders, the shift and alt (option) keys modify the increment. The smallest increment is with no modifier key pressed. Then, in increasing increments, use shift, alt, shift-alt.

For the \(\theta\) slider, the shift and alt (option) keys also modify the increment. Play with the different combinations to see what they do.

The speed of animation is controlled by the speed slider. If the speed is negative, \(\theta\) runs backwards.

The zoom slider zooms the graphing window.

Animating the rolling wheel
Animation of the rolling wheel is controlled by the button just below the \(\theta\)-slider. Originally, the button will show [>>], but this will change depending on the speed and animation state.

When the animation is on, the button will change to [||] and clicking it will stop the animation.

speed > 0: The animate button will show [>>] when the animation is off. Clicking it will start the animation. If \(\theta\) reaches its maximum value, the animation will stop and the button changes to [|<]. Clicking this rewinds \(\theta\) to \(\theta = 0\).

speed < 0: The animate button will show [<<] when the animation is off. Clicking it will start the animation. If \(\theta\) reaches 0, the animation will stop and the button changes to [>|]. Clicking this 'rewinds' \(\theta\) to its maximum value.

speed = 0: The button shows [--] when the animation is off. Clicking this will 'start the animation', but, of course, nothing will move.

Using your mouse to move the \(\theta\) slider automatically stops the animation.

Toggles (checkboxes)
Select the toggle [trace] to cause the light to leave yellow (tangerine) trace.

Select the toggle [velocity vector] to display the velocity vector in pink (maroon).

Select the toggle [Show wheel] to display the wheel.

xy readout
Roll the cursor over the graphing window to create crosshairs and a readout of the cursor position.