Geometric Construction 4

Construct an animation displaying all possible pairs of orthogonal tangents of a cardioid.

Construct an animation displaying all possible pairs of orthogonal tangents of a nephroid.

Construct an animation displaying all possible pairs of orthogonal tangents of a deltoid.

Construct an animation displaying all possible pairs of orthogonal tangents of an astroid.

Given a cardioid, construct an animation displaying all possible cardioids sharing the same cusp and orthogonal to it.

Illustrate the principle of the cardioid condensor invented by Siedentoff.

Reference: C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, pp. 259-260.

Two points of the cardioid have mutually orthogonal tangents if their parameters differ by 60 degrees. Base on this fact, construct the six points where the tangents are either parallel or mutually orthogonal.

Reference: C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, pp. 260-282.

If the cardioid be pivoted at the cusp and rotated with constant angular velocity, a pin, constrained to a fixed straight line and bearing on the cardioid, will move with simple harmonic motion.

Reference: R.C. Yates, A Handbook on Curves and Their Properties, p. 6.

Construct a cardioid tangent to a nephroid internally and yet still be able to make a 360-degree turn within the nephroid.

Construct a nephroid tangent to a cardioid externally and yet still be able to make a 360-degree turn in the exterior of the cardioid.

Construct a deltoid tangent to an astroid internally and yet still be able to make a 360-degree turn within the astroid.

Construct an astroid tangent to a deltoid externally and yet still be able to make a 360-degree turn in the exterior of the deltoid.

JavaSketchpad file