Circular points at infinity
Encyclopedia
In projective geometry
, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are
and (1: −i: 0).
Here the coordinates are homogeneous coordinates (x: y: z); so that the line at infinity
is defined by z = 0. These points at infinity are called circular points at infinity because they lie on the complexification
of every real circle. In other words, 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 general, an algebraic curve that passes through these two points is called circular
.
The circular points at infinity are the points at infinity of the isotropic line
s.
The circular points are invariant
under translation
and rotation
.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, the circular points at infinity in the complex projective plane (also called cyclic points or isotropic points) are
and (1: −i: 0).
Here the coordinates are homogeneous coordinates (x: y: z); so that the line at infinity
Line at infinity
In geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...
is defined by z = 0. These points at infinity are called circular points at infinity because they lie on the complexification
Complexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
of every real circle. In other words, 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
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
). In general, an algebraic curve that passes through these two points is called circular
Circular algebraic curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F = 0, where F is a polynomial with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2...
.
The circular points at infinity are the points at infinity of the isotropic line
Isotropic line
An isotropic line or null line is a line in the complex projective plane with slope i or -i.-Equation:All isotropic lines have equations of the following form:y = ix + az or y = -ix + az,...
s.
The circular points are invariant
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
under translation
Translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
and rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
.