In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a cuspidal cubic or semicubical parabola is an
algebraic plane 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 ...
that has an
implicit equation
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit functi ...
of the form
:
(with ) in some
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
Solving for leads to the ''explicit form''
:
which imply that every
real
Real may refer to:
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* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
point satisfies . The exponent explains the term ''semicubical parabola''. (A
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
can be described by the equation .)
Solving the implicit equation for yields a second ''explicit form''
:
The
parametric equation
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 ...
:
can also be deduced from the implicit equation by putting
[.]
The semicubical parabolas have a
cuspidal singularity; hence the name of ''cuspidal cubic''.
The arc length of the curve was calculated by the English mathematician
William Neile
William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was ...
and published in 1657 (see
section History).
Properties of semicubical parabolas
Similarity
Any semicubical parabola
is
similar to the ''semicubical unit parabola''
''Proof:'' The similarity
(uniform scaling) maps the semicubical parabola
onto the curve
with
Singularity
The parametric representation
is ''
regular except'' at point At point
the curve has a ''
singularity'' (cusp). The ''proof'' follows from the tangent vector Only for
this vector has zero length.
Tangents
Differentiating the ''semicubical unit parabola''
one gets at point
of the ''upper'' branch the equation of the tangent:
:
This tangent intersects the ''lower'' branch at exactly one further point with coordinates
:
(Proving this statement one should use the fact, that the tangent meets the curve at
twice.)
Arclength
Determining the
arclength of a curve
one has to solve the integral
For the semicubical parabola
one gets
:
(The integral can be solved by the
substitution
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* Substitution (poetry), a variation in poetic scansion
* "Substitution" (song), a 2009 song by Silversun Pi ...
''Example:'' For (semicubical unit parabola) and which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073.
Evolute of the unit parabola
The
evolute of the ''parabola'' is a semicubical parabola shifted by 1/2 along the ''x''-axis:
Polar coordinates
In order to get the representation of the semicubical parabola
in polar coordinates, one determines the intersection point of the line
with the curve. For
there is one point different from the origin:
This point has distance
from the origin. With
and
( see
List of identities) one gets
[August Pein: ''Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten '',p. 10]
:
Relation between a semicubical parabola and a cubic function
Mapping the semicubical parabola
by the
projective map
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
(involutoric perspectivity with axis
and center yields
hence the ''cubic function''
The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis.
This property can be derived, too, if one represents the semicubical parabola by ''
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
'': In equation (A) the replacement
(the line at infinity has equation and the multiplication by
is performed. One gets the equation of the curve
*in ''homogeneous coordinates'':
Choosing line
as line at infinity and introducing
yields the (affine) curve
Isochrone curve
An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. In this way it is related to the
tautochrone curve
A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ...
, for which particles at different starting points always take equal time to reach the bottom, and the
brachistochrone curve
In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...
, the curve that minimizes the time it takes for a falling particle to travel from its start to its end.
History
The semicubical parabola was discovered in 1657 by
William Neile
William Neile (7 December 1637 – 24 August 1670) was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was ...
who computed its
arc length
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* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
. Although the lengths of some other non-algebraic curves including the
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
and
cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
had already been computed (that is, those curves had been ''rectified''), the semicubical parabola was the first
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 c ...
(excluding the
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
and
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
) to be rectified.
References
*August Pein: ''Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten '', 1875
Dissertation Clifford A. Pickover: ''The Length of Neile's Semicubical Parabola''
External links
*
{{DEFAULTSORT:Semicubical Parabola
Plane curves