Hypotrochoid 3/4
(1/4 · cos(u) + h · cos(u/3), 0, 1/4 · sin(u) − h · sin(u/3))
Open in the app The dials and keys named below are the app's.
What it draws
A circle of radius b = 3/4 rolls inside a fixed circle of radius 1. A pen h from its centre draws the curve, flat in the floor's plane.
The first term of each expression is the rolling circle's centre, running round at radius 1 − b = 1/4. The second is the pen about it, turning the other way at (1 − b)/b = 1/3 of the speed.
Four lobes
After 3 turns of u the pen has turned once the other way, and the curve closes. That is why the preset opens with Turns of u at 3. The two counts add to the symmetry, 3 + 1 = 4.
At the opening h = 0.5 the curve swings between 0.25 and 0.75 from the centre, in four lobes that cross one another.
From star to square
At h = b the pen is on the rim and stops each time the rim meets the fixed circle: the curve becomes a four-pointed star with its points at radius 1.
Far past that, the pen's own circle of radius h dominates. It turns once in the 3 turns of u. The centre's small circle turns 3 times the other way, and the difference of 4 adds a fourfold wobble. At h = 2.3 the radius only swings between 2.05 and 2.55, and the curve bends the same way everywhere: a rounded square.
Try
- Replace h with 3/4 in both expressions. The pen is on the rim, and the lobes close into a four-pointed star.
- Drag h to 2.3. The rounded square.
- Drag h to 1.2. Between the two, the sides of the square bend inward between its corners.
- Drag h to 0.1. The centre's small circle, taken three times with a slight wobble.