Tangents to Hypocycloids To generate a hypocycloid by its tangents, use the following inst...

Tangents to Hypocycloids To generate a hypocycloid by its tangents, use the following instructions in Sketchpad. For a specific value of n that is useful, take n = 3. Its value can be changed to obtain other hypocycloids.

[1] Construct a small circle AB, and any point C on that circle.

[2] Select B, C, and circle AB. Choose Arc On Circle under CONSTRUCT, then Arc Angle under MEASURE.

[3] Calculate the quantity x = (1 -n) • (Arc angle ) displaying it on screen.

[4] Select x and choose Mark Angle Measurement under TRANSFORM. Select A and choose Mark Center "A." Select B and rotate about A through the angle xc to yield the point B'. (Choose Rotate By Marked Angle.)

[5] Construct line and put a tracer on it (choose Trace under DISPLAY). Try moving C to examine its motion. Hide all points except C. Select C, then circle AB, and choose Animate under DISPLAY, checking Once (around the circle) and Normally. Does this produce the tangents to a deltoid?

Try this for several other values of n, say n = 4 and 5. The value n = 10 yields a rather spectacular display.