Conchoids of Nicomedes
Draw different conchoids with different pens on the same moving plane
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The Concoid is over 2000 years old. It was geometrically defined,
formulas came much later. We use this parametrization:
c(t) = RR * [ tan(t) + k*sin(t), 1 + k•cos(t) ]
The parameter k specifies different pen positions on the same drawing
mechanism. The drawing mecahnism is a line through the origin with a
fixed point P on this line (big yellow dot) moving on the yellow line,
called "directrix" of the conchoid. k is the (signed) distance from
P to the drawing pen (big magenta dot) on the moving line.
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.