Cissoid and Strophoid
drawn by different pens on the same moving plane
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select curve: -7 ... 7:
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.
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.