with cusps . Proof Let be the circle with midpoint and radius . The diameter may lie on the x-axis (see diagram). The pencil of circles has equations: The envelope condition is One can easily check that the point of the nephroid is a solution of the system and hence a point of the envelope of the pencil of circles. Nephroid as envelope of a pencil of lines Similar to the generation of a cardioid as envelope of a pencil of lines the following procedure holds: Draw a circle, divide its perimeter into equal spaced parts with points (see diagram) and number them consecutively. Draw the chords: . (i.e.: The second point is moved by threefold velocity.) The envelope of these chords is a nephroid. Proof The following consideration uses trigonometric formulae for . In order to keep the calculations simple, the proof is given for the nephroid with cusps on the y-axis. Equation of the tangent: for the nephroid with parametric representation : Herefrom one determines the normal vector , at first. The equation of the tangent is: For one gets the cusps of the nephroid, where there is no tangent. For one can divide by to obtain Equation of the chord: to the circle with midpoint and radius : The equation of the chord containing the two points is: For the chord degenerates to a point. For one can divide by and gets the equation of the chord: The two angles are defined differently ( is one half of the rolling angle, is the parameter of the circle, whose chords are determined), for one gets the same line. Hence any chord from the circle above is tangent to the nephroid and the nephroid is the envelope of the chords of the circle. Nephroid as caustic of one half of a circle The considerations made in the previous section give a proof for the fact, that the caustic of one half of a circle is a nephroid. If in the plane parallel light rays meet a reflecting half of a circle (see diagram), then the reflected rays are tangent to a nephroid. Proof The circle may have the origin as midpoint (as in the previous section) and its radius is . The circle has the parametric representation The tangent at the circle point has normal vector . The reflected ray has the normal vector (see diagram) and containing circle