The Nephroid is defined by these equations:

c(t) = RR * [ cos(t)+cos(3t)/3, sin(t)+sin(3t)/3 ]

Of course it is a special case of the Epicycloids with frequency = 3 and stick = 1. For small frequency values the cycloids have individual names (cardioid, nephroid, deltoid, astroid).

The curves drawn with other stick lengths by the same rolling motion are called trochoids. If a square of random dots is attached to the rolling circle (see demo), then all these random dots move on cycloids or trochoids.

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(3t)/3, sin(t)+sin(3t)/3 ]

Of course it is a special case of the Epicycloids with frequency = 3 and stick = 1. For small frequency values the cycloids have individual names (cardioid, nephroid, deltoid, astroid).

The curves drawn with other stick lengths by the same rolling motion are called trochoids. If a square of random dots is attached to the rolling circle (see demo), then all these random dots move on cycloids or trochoids.

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".

"Rolling" means that the point of the rolling circle that touches the
"street"-circle has velocity zero.

A motion with a point at rest is a rotation around this rest point. Applied to the drawing pen on the stick

this means: the curve tangent at the point just drawn is orthogonal to its connection to its rest point.

Compare Cycloid Watch how the caustic rotates when the normals are tilted.

A motion with a point at rest is a rotation around this rest point. Applied to the drawing pen on the stick

this means: the curve tangent at the point just drawn is orthogonal to its connection to its rest point.

Compare Cycloid Watch how the caustic rotates when the normals are tilted.