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Wikipedia / Nephroid

In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger one by a factor of one-half.

Name

Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Richard A. Proctor in 1878.

Strict definition

A nephroid is

Equations

Parametric

the starting point (see diagram) then one gets the parametric representation:

maps the unit circle to a nephroid.

Proof of the parametric representation

The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane (complex number. Hence the

  1. ,

  2. .

:

  1. .

Herefrom one gets

were used. See trigonometric functions.)

Implicit

into the equation

shows that this equation is an implicit representation of the curve.

Proof of the implicit representation

With

one gets

Orientation

If the cusps are on the y-axis the parametric representation is

and the implicit one:

Metric properties

For the nephroid above the

The proofs of these statements use suitable formulae on curves (arc length, area and radius of curvature) and the parametric representation above

and their derivatives

  1. Proof for the arc length:

  2. .

  3. Proof for the area:

  4. .

  5. Proof for the radius of curvature:

Construction

Nephroid as envelope of a pencil of circles

  • is a nephroid.

Proof

. The diameter may lie on the x-axis (see diagram). The pencil of circles has equations:

The envelope condition is

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:

  1. points (see diagram) and number them consecutively.

  2. . (i.e.: The second point is moved by threefold velocity.)

  3. The envelope of these chords is a nephroid.

Proof

The following consideration uses trigonometric formulae. 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

  1. :

, at first. is:

to obtain

Equation of the chord is:

and gets the equation of the chord:

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 Caustic

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 has the parametric representation

. Hence the reflected ray is part of the line with equation

which is tangent to the nephroid of the previous section at point

  1. (see above).

The evolute and involute of a nephroid

Evolute

The evolute of a curve is the locus of centers of curvature the evolute has the representation

the suitably oriented unit normal.

For a nephroid one gets:

  • The evolute of a nephroid is another nephroid half as large and rotated 90 degrees (see diagram).

Proof

The nephroid as shown in the picture has the parametric representation

the unit normal vector pointing to the center of curvature

  1. (see section above)

(s. section on metric properties). Hence the evolute has the representation:

which is a nephroid half as large and rotated 90 degrees (see diagram and section above)

Involute

Because the evolute of a nephroid is another nephroid, the involute of the nephroid is also another nephroid. The original nephroid in the image is the involute of the smaller nephroid.

Inversion of a nephroid

The inversion

maps the nephroid with equation

onto the curve of degree 6 with equation

  1. (see diagram) .

References

  • Arganbright, D., Practical Handbook of Spreadsheet Curves and Geometric Constructions, CRC Press, 1939, , p.54.

  • Borceux, F., A Differential Approach to Geometry: Geometric Trilogy III, Springer, 2014, , p.148.

  • Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961, , p.7.