Hypotrochoid 7/13
(6/13 · cos(u) + h · cos(6u/7), 0, 6/13 · sin(u) − h · sin(6u/7))
Open in the app The dials and keys named below are the app's.
What it draws
A circle of radius b = 7/13 rolls without slipping inside a fixed circle of radius 1. A pen fixed to the rolling circle, h from its centre, traces the curve. It lies flat in the floor's plane, so y is 0.
The first term of each expression is the rolling circle's centre. It runs round at radius 1 − b = 6/13 ≈ 0.46, once per turn of u. The second term is the pen about that centre. Rolling makes it turn the other way, (1 − b)/b = 6/7 as fast.
Thirteen petals
After 7 turns of u the pen has turned exactly 6 times the other way, and the curve closes. That is why the preset opens with Turns of u at 7. The two counts add to the curve's symmetry, 7 + 6 = 13, so there are 13 petals.
At the opening h = 0.4 the curve swings between 0.06 and 0.86 from the centre. The pen sits inside the rolling circle, so it never stops and each petal is rounded.
At h = b the pen is on the rim. The rim touches the fixed circle at rest each time it rolls past, so the pen stops there and the petals sharpen into 13 cusps, at radius 1. Past b the pen swings back faster than the centre moves forward, and each cusp opens into a loop.
History
Curves traced by a rolling circle are roulettes, studied in the 17th century for the shapes of gear teeth. The British engineer Denys Fisher sold them as a toy in 1965: Spirograph, a toothed wheel rolling inside a toothed ring with a pen in one of its holes.
Try
- Drag h to 0.1. The pen stays near the centre, and the petals thin into a ring between 0.36 and 0.56.
- Replace h with 7/13 in both expressions. The pen is on the rim, and the curve becomes a 13-pointed star.
- Drag h to 1.2. The pen is outside the rolling circle, and 13 loops weave between 0.74 and 1.66.
- Set Turns of u to 1. One pass of the centre draws only about two petals' worth of path, still open.