In
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
, the circular points at infinity (also called cyclic points or isotropic points) are two special
points at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
in the
complex projective plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2, ...
that are contained in the
complexification of every real
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
.
Coordinates
A point of the complex projective plane may be described in terms of
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. ...
, being a triple of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s , where two triples describe the same point of the plane when the coordinates of one triple are the same as those of the other aside from being multiplied by the same nonzero factor. In this system, the points at infinity may be chosen as those whose ''z''-coordinate is zero. The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates
: and .
Trilinear coordinates
Let ''A''. ''B''. ''C'' be the measures of the vertex angles of the reference triangle ABC. Then the
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
of the circular points at infinity in the plane of the reference triangle are as given below:
:
or, equivalently,
:
or, again equivalently,
:
where
.
Complexified circles
A real circle, defined by its center point (''x''
0,''y''
0) and radius ''r'' (all three of which are
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s) may be described as the set of real solutions to the equation
:
Converting this into a
homogeneous equation and taking the set of all complex-number solutions gives the complexification of the circle. The two circular points have their name because they lie on the complexification of every real circle. More generally, both points satisfy the homogeneous equations of the type
:
The case where the coefficients are all real gives the equation of a general circle (of the
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
). In general, 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 cu ...
that passes through these two points is called
circular.
Additional properties
The circular points at infinity are the
points at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
of the
isotropic lines.
They are
invariant under
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s and
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s of the plane.
The concept of
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
can be defined using the circular points,
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
and
cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...
:
:The angle between two lines is the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their intersection to the circular points, divided by 2i.
Sommerville configures two lines on the origin as
Denoting the circular points as ω and ω′, he obtains the cross ratio
:
so that
:
Imaginary transformation
The transformation
is called the imaginary transformation by
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
. He notes that the equation
becomes
under the transformation, which
changes the imaginary circular points ''x'' : ''y'' = ± i, ''z'' = 0, into the real infinitely distant points ''x''’ : ''y''’ = ± 1, ''z'' = 0, which are the points at infinity in the two directions that make an angle of 45° with the axes. Thus all circles are transformed into conics which go through these two real infinitely distant points, i.e. into equilateral hyperbolas whose asymptotes make an angle of 45° with the axes.Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
, translators E.R. Hendrick & C.A. Noble (1939) 908''Elementary Mathematics from an Advanced Standpoint – Geometry'', third edition, pages 119, 120
References
* Pierre Samuel (1988) ''Projective Geometry'', Springer, section 1.6;
* Semple and Kneebone (1952) ''Algebraic projective geometry'', Oxford, section II-8.
{{DEFAULTSORT:Circular Points At Infinity
Projective geometry
Complex manifolds
Infinity