Nephroid

Summary

In geometry, a nephroid (from the Greek ὁ νεφρός ho nephros) is a specific plane curve whose name means 'kidney-shaped' (compare nephrology).

generation of a nephroid by a rolling circle

NameEdit

Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Proctor in 1878.[1]


Strict definitionEdit

A nephroid is


EquationsEdit

 
Nephroid: definition

ParametricEdit

If the small circle has radius  , the fixed circle has midpoint   and radius  , the rolling angle of the small circle is   and point   the starting point (see diagram) then one gets the parametric representation:

 
 

The complex map   maps the unit circle to a nephroid[2]


proof of the parametric representation

The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane. The movement of the small circle can be split into two rotations. In the complex plane a rotation of a point   around point   (origin) by an angle   can be performed by the multiplication of point   (complex number) by  . Hence the

rotation   around point   by angle   is   ,
rotation   around point   by angle   is  .

A point   of the nephroid is generated by the rotation of point   by   and the subsequent rotation with  :

 .

Herefrom one gets

 

(The formulae   were used. See trigonometric functions.)

ImplicitEdit

Inserting   and   into the equation

  •  

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


proof of the implicit representation

With

 

one gets

 

orientationEdit

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

 

and the implicit one:

 

Metric propertiesEdit

For the nephroid above the

  • arclength is  
  • area   and
  • radius of curvature is  

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

 
 
proof for the arc length
  .
proof for the area
  .
proof for the radius of curvature
 
 
Nephroid as envelope of a pencil of circles

ConstructionEdit

  • It can be generated by rolling a circle with radius   on the outside of a fixed circle with radius  . Hence, a nephroid is an epicycloid.


Nephroid as envelope of a pencil of circlesEdit

  • Let be   a circle and   points of a diameter  , then the envelope of the pencil of circles, which have midpoints on   and are touching   is a nephroid 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 linesEdit

 
nephroid: tangents as chords of a circle, principle
 
nephroid: tangents as chords of a circle

Similar to the generation of a cardioid as envelope of a pencil of lines the following procedure holds:

  1. Draw a circle, divide its perimeter into equal spaced parts with   points (see diagram) and number them consecutively.
  2. Draw the chords:  . (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 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 Edit

 
nephroid as caustic of a circle: principle
 
nephroide 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 point  . Hence the reflected ray is part of the line with equation

 

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

  (see above).
 
Nephroid caustic at bottom of tea cup

The evolute and involute of a nephroidEdit

 
nephroid and its evolute
magenta: point with osculating circle and center of curvature

EvoluteEdit

The evolute of a curve is the locus of centers of curvature. In detail: For a curve   with radius of curvature   the evolute has the representation

 

with   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

  (see section above)

and the radius of curvature   (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 § Equations above)

InvoluteEdit

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 (green) of a nephroid (red) across the blue circle

Inversion of a nephroidEdit

The inversion

 

across the circle with midpoint   and radius   maps the nephroid with equation

 

onto the curve of degree 6 with equation

  (see diagram) .
 
A nephroid in daily life: a caustic of the reflection of light off the inside of a cylinder.

ReferencesEdit

  1. ^ Weisstein, Eric W. "Nephroid". MathWorld.
  2. ^ Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath
  • Arganbright, D., Practical Handbook of Spreadsheet Curves and Geometric Constructions, CRC Press, 1939, ISBN 0-8493-8938-0, p. 54.
  • Borceux, F., A Differential Approach to Geometry: Geometric Trilogy III, Springer, 2014, ISBN 978-3-319-01735-8, p. 148.
  • Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961, ISBN 978-0-521-0-5585-7, p. 7.

External linksEdit

  • Mathworld: nephroid
  • Xahlee: nephroid