Cissoid and Strophoid
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.