Draw several curves with the same rolling motion

The Cardioid is defined by these equations:

c(t) = RR * [ cos(t)+cos(2t)/2, sin(t)+sin(2t)/2 ]

Of course it is a special case of the Epicycloids with frequency = 2 and stick = 1. For small frequency values the rolling curves have individual names (cardioid, nephroid, deltoid, astroid).

The curves drawn with other stick lengths by the same rolling motion are in general called trochoids; here they are called limacons.

The demo shows at each moment two consecutive positions of the dots. Therefore the image does not look random but suggests a rotation of the "moving plane".

c(t) = RR * [ cos(t)+cos(2t)/2, sin(t)+sin(2t)/2 ]

Of course it is a special case of the Epicycloids with frequency = 2 and stick = 1. For small frequency values the rolling curves have individual names (cardioid, nephroid, deltoid, astroid).

The curves drawn with other stick lengths by the same rolling motion are in general called trochoids; here they are called limacons.

The demo shows at each moment two consecutive positions of the dots. Therefore the image does not look random but suggests a rotation of the "moving plane".

This page emphasizes that the same rolling motion allows to draw many curves,
one can choose any point of the plane attached

to the rolling wheel as drawing pen. This plane is emphasized by the square of random dots. The points near the rolling wheel

move almost on cardiods, all others on limacons. The dots near the stick move tangential to the curves.

The simplest rolling curve is the Cycloid.

Watch how the caustic rotates when the normals are tilted.

to the rolling wheel as drawing pen. This plane is emphasized by the square of random dots. The points near the rolling wheel

move almost on cardiods, all others on limacons. The dots near the stick move tangential to the curves.

The simplest rolling curve is the Cycloid.

Watch how the caustic rotates when the normals are tilted.