Geometric Construction 3

Construct the astroid.

Construct the deltoid.

Construct the nephroid.

Construct the cardioid.

Construct an animation showing a rod of fixed length slides with its ends upon two fixed perpendicular lines.

What is the envelope formed by the rod?

Find the point of tangency.

Find the locus of a point carried by the rod. (A ladder standing on a smooth floor against a wall slides down. Along what curve does a cat sitting on the ladder move?)

A wooden triangle moves on a plane so that two vertices move along two fixed perpenducluar lines. How does the third vertex move?

This is known as Van Schooten's Locus Problem. See: Heinrich Doerrie: 100 Great Problems of Elementary Mathematics, p.214.

Construct the cardioid as the envelop of circles. Find the point of tangency.

Reference: Herman Baravalle, Dynamic Beauty of Geometrical Forms, Math. Scripta (1948) p.294.

Construct the nephroid as the envelope of circles. Find the point of tangency.

Two persons walk at constant speed around a circle. The ratio of their angular velocity is k ( k is not 0, 1 or -1). Find the envelope of all the straight lines joining them for k = 2, 3, -2, -3. Find the point of tangency.

References:
Martin Gardner: Wheels, Life and Other Mathematical Amusements, pp.1-9.
E.H. Lockwood, A Book of Curves, Cambridge University Press, 1962.
N.B. Vasilyer and V.L. Gutebmacher, Straight Lines and Curves, Mir, 1980.
Robert C. Yates, A Handbook on Curves and their Properties, Edwards, 1952.
This is related to the Residue Curves, see: Graphs of Linear Congruences, Math. Scripta (147) pp. 106-112. Also see pp. 114-115, p. 224, pp. 232-233 of the same issue.