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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, pr ...
, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
of every real
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 con ...
.


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 numbers , 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: :-1 : \cos C - i\sin C : \cos B + i\sin B,\qquad -1 : \cos C + i\sin C : \cos B - i\sin B or, equivalently, :\cos C + i\sin C : -1 :\cos A - i\sin A, \qquad\cos C - i\sin C : -1 :\cos A + i\sin A or, again equivalently, :\cos B + i\sin B : \cos A - i\sin A : -1, \qquad \cos B-i\sin B : \cos A+i\sin A: -1, where i=\sqrt.


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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s) may be described as the set of real solutions to the equation :(x-x_0)^2+(y-y_0)^2=r^2. Converting this into a
homogeneous equation In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...
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 :Ax^2 + Ay^2 + 2B_1xz + 2B_2yz - Cz^2 = 0. 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 is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
). 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 ...
that passes through these two points is called circular.


Additional properties

The circular points at infinity are the points at infinity of the
isotropic line In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, an ...
s. They are
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
under
translation Translation is the communication of the Meaning (linguistic), 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 ...
s and
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s of the plane. The concept of
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
can be defined using the circular points,
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
and cross-ratio:
Duncan Sommerville Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...
(1914
Elements of Non-Euclidean Geometry
page 157, link from
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
Historical Math Collection
:The angle between two lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their intersection to the circular points. Sommerville configures two lines on the origin as u : y = x \tan \theta, \quad u' : y = x \tan \theta ' . Denoting the circular points as ''ω'' and ω''′, he obtains the cross ratio :(u u' , \omega \omega ') = \frac \div \frac , so that :\phi = \theta ' - \theta = \tfrac \log (u u', \omega \omega ') .


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