drawn by different pens on the same moving plane

The Cissoid of Diocles is over 2000 years old. It was geometrically defined,
formulas came much later. We use Newton's mechanical generation, which also draws
the modern Strophoid. It gives the following parametrization:

c(t) = bb * [sin(t)*(1/(1+cos(t))-k), 1-k*cos(t) ]

The parameter k specifies different pen positions on Newton's drawing mechanism, a so called carpenter's square. One leg of this tool passes through the origin, the other endpoint moves on the straight yellow line (called directrix). k is the signed distance of the pen (magenta dot) from this endpoint (yellow dot).

The caustic of the normals of the Cissoid (select = 2) is a parabola. A parabola is also obtained by inverting the Cissoid in a circle around the cusp.

c(t) = bb * [sin(t)*(1/(1+cos(t))-k), 1-k*cos(t) ]

The parameter k specifies different pen positions on Newton's drawing mechanism, a so called carpenter's square. One leg of this tool passes through the origin, the other endpoint moves on the straight yellow line (called directrix). k is the signed distance of the pen (magenta dot) from this endpoint (yellow dot).

The caustic of the normals of the Cissoid (select = 2) is a parabola. A parabola is also obtained by inverting the Cissoid in a circle around the cusp.

All such mechanical constructions of curves also come with a
tangent construction. Imagine that a so called

moving plane is attached to the drawing mechanism. A square of random dots can be added to emphasize the

rotating motion of the plane. The big green dot is, at each moment, the fixed point of this momentary rotation.

For any pen position the segment from the currently drawn point to the current center of rotation is orthogonal

to the current tangent. One may think of this segment as a generalized radius.

The random dots are shown in two consecutive positions to bring out the rotation pattern.

moving plane is attached to the drawing mechanism. A square of random dots can be added to emphasize the

rotating motion of the plane. The big green dot is, at each moment, the fixed point of this momentary rotation.

For any pen position the segment from the currently drawn point to the current center of rotation is orthogonal

to the current tangent. One may think of this segment as a generalized radius.

The random dots are shown in two consecutive positions to bring out the rotation pattern.