In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, the trisectrix of Maclaurin is a
cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation
:
applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equ ...
notable for its
trisectrix
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not c ...
property, meaning it can be used to
trisect an angle. It can be defined as
locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of the point of
intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a
sectrix of Maclaurin. The curve is named after
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
who investigated the curve in 1742.
Equations
Let two lines rotate about the points
and
so that when the line rotating about
has angle
with the ''x'' axis, the rotating about
has angle
. Let
be the point of intersection, then the angle formed by the lines at
is
. By the
law of sines,
:
so the equation in
polar coordinates is (up to translation and rotation)
:
.
The curve is therefore a member of the
Conchoid of de Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze..
The curves are defined by the polar equation
:r=\sec\theta+a\cos\theta \,.
In cart ...
family.
In
Cartesian coordinate
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 ...
s the equation of this curve is
:
.
If the
origin is moved to (''a'', 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes
:
making it an example of a limacon with a loop.
The trisection property
Given an angle
, draw a ray from
whose angle with the
-axis is
. Draw a ray from the origin to the point where the first ray intersects the curve. Then, by the construction of the curve, the angle between the second ray and the
-axis is
.
Notable points and features
The curve has an
x-intercept at
and a
double point at the origin. The vertical line
is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at
. As a nodal cubic, it is of
genus zero.
Relationship to other curves
The trisectrix of Maclaurin can be defined from
conic sections
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
in three ways. Specifically:
* It is the
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when a ...
with respect to the unit circle of the
hyperbola
::
.
* It is
cissoid of the circle
::
:and the line
relative to the origin.
* It is the
pedal
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control ...
with respect to the origin of the
parabola
::
.
In addition:
* The inverse with respect to the point
is the
Limaçon trisectrix
In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or ep ...
.
* The trisectrix of Maclaurin is related to the
Folium of Descartes by
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
.
References
*
*
"Trisectrix of Maclaurin" at MacTutor's Famous Curves Index''Maclaurin Trisectrix''at mathcurve.com
External links
{{commonscat, Maclaurin's Trisectrix
Plane curves