The Conchoid is a particular locus formed by the endpoints of segments of a fixed length that have midpoints on a curve as the lines containing the segments rotate around a fixed point. When the curve containing the midpoints is a line, the Conchoid is the Conchoid of Nicomedes. Here, the exploration that is proposed is to contruct the Conchoid of Nicomedes geometrically and to examine its variations.
A second level of problem would be to derive an algebraic equation for the Conchoid of Nichodemes and explore the effect of variations in parameters contained in the equation. One such equation is
. Polar equations and parametric equations may be simpler.
Who was Nichomedes?
Conchoid of Nicodemes. Consider a set of line segments of a given length. All the segments lie on lines that pass through a fixed point P and the segments all are bisected by a line not through P. What is the locus of the extremities (ends) of the line segments? Note that the lines all pass through P but P may be outside of the segment that is marked off on the line.
One GSP implementation/animation is found HERE. Other constructions are possible. Yours might be better.
Suggested construction:
Construct a line segment AB and select its midpoint.
Construct a line k and select a random point Q on the line.
Construct a point P not on the line.
Construct line PQ
Using Q as the midpoint, mark off a segment A'B' congruent to segment AB on line PQ.
Trace the locus of A' and B' as Q is moved along line k. (i.e. animate Q on k)
See what changes as the distance from point P to the line of midpoints is varied. This is an important part of the exploration in order to see the variation in the curves.
For example, here are a few.
Construct the conchoid if the curve for the midpoints is a circle.
Construct the conchoid if the curve for the midpoints is a parabola.