In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a nephroid () is a specific
plane curve. It is a type of
epicycloid
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equat ...
in which the smaller circle's radius differs from the larger 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
Richard Anthony Proctor (23 March 1837 – 12 September 1888) was an English astronomer. He is best remembered for having produced one of the earliest maps of Mars in 1867 from 27 drawings by the English observer William Rutter Dawes. His map w ...
in 1878.
Strict definition
A nephroid is
* an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of
degree 6.
* an
epicycloid
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equat ...
with two
cusps
* a plane simple closed curve = a
Jordan curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
Equations
Parametric
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
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
:
:
:
The complex map
maps the unit circle to a nephroid
Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath
/ref>
=Proof of the parametric representation
=
The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. 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
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
.)
Implicit
Inserting and 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
* arclength is
*area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
and
*radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
is
The proofs of these statements use suitable formulae on curves (arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
, area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
and radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
) and the parametric representation above
:
:
and their derivatives
:
:
;Proof for the arc length:
: .
;Proof for the area:
: .
;Proof for the radius of curvature:
:
Construction
* 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
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equat ...
.
Nephroid as envelope of a pencil of circles
*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 lines
Similar to the generation of a cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single Cusp (singularity), cusp. It is also a ty ...
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
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
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 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).
The evolute and involute of a nephroid
Evolute
The evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...
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 above)
Involute
Because the evolute of a nephroid is another nephroid, the involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or ...
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
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
:
across the circle with midpoint and radius maps the nephroid with equation
:
onto the curve of degree 6 with equation
: (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.
External links
{{Commonscat, Nephroid
Mathworld: nephroid
Roulettes (curve)