Homogeneous coordinate system

Homogeneous coordinates - a coordinate system used in projective geometry, similar to how Cartesian coordinates are used in Euclidean geometry .

Homogeneous coordinates have the property that the object they define does not change when all coordinates are multiplied by the same nonzero number. Because of this, the number of coordinates needed to represent the points is always one more than the dimension of the space in which these coordinates are used. For example, to represent a point on a line in one-dimensional space, 2 coordinates and 3 coordinates are needed to represent a point on a plane in two-dimensional space. In homogeneous coordinates, it is even possible to imagine points at infinity.

Introduced by Plücker as an analytical approach to the Gergonne – Poncelet principle of duality .

Content

Projective Geometry

The projective plane is usually defined as the set of lines in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ passing through the origin . Any such straight line is uniquely determined by a point that does not coincide with the origin $0$ ${\ displaystyle 0}$ . Let this line go through a point with coordinates $(a,b,c)$ ${\ displaystyle (a, b, c)}$ , then the homogeneous coordinates of the corresponding point on the projective plane are three numbers $(x_{1}:x_{2}:x_{3})$ ${\ displaystyle (x_ {1}: x_ {2}: x_ {3})}$ determined up to proportionality and such that all three coordinates cannot simultaneously be equal to zero ^[1] . For example, $(0:1:1)=(0:2:2).$ ${\ displaystyle (0: 1: 1) = (0: 2: 2).}$

From homogeneous coordinates to affine ones you can go as follows: in three-dimensional space, you can draw a plane that does not pass through the origin ; then the straight line passing through the origin is either parallel to this plane (in this case, the point is called "infinitely distant") or intersects it at a single point, then it can be associated with the coordinates of this point on the plane. For example, in a space with coordinates $(x_{1},x_{2},x_{3})$ ${\ displaystyle (x_ {1}, x_ {2}, x_ {3})}$ draw a plane $x_{3}=1$ ${\ displaystyle x_ {3} = 1}$ . Then the point with uniform coordinates $(a:b:c)$ ${\ displaystyle (a: b: c)}$ , if a $c\neq 0$ ${\ displaystyle c \ neq 0}$ corresponds to a point on the plane with coordinates $(a/c,b/c).$ ${\ displaystyle (a / c, b / c).}$ Conversely, a point with affine coordinates $(a,b)$ ${\ displaystyle (a, b)}$ in homogeneous coordinates will be written as $(a:b:1).$ ${\ displaystyle (a: b: 1).}$

Lines on the projective plane are planes in three-dimensional space passing through the origin. Such a plane can be defined by the equation $a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}=0$ ${\ displaystyle a_ {1} x_ {1} + a_ {2} x_ {2} + a_ {3} x_ {3} = 0}$ . It is easy to see that when multiplying $a_{1},a_{2},a_{3}$ ${\ displaystyle a_ {1}, a_ {2}, a_ {3}}$ the plane defined by the equation will not change by the same number. This means that each plane corresponds to uniform coordinates $(a_{1}:a_{2}:a_{3})$ ${\ displaystyle (a_ {1}: a_ {2}: a_ {3})}$ . A point recorded in homogeneous coordinates can be associated with a straight line, which is written in homogeneous coordinates in the same way. Thus, the lines on the projective plane form the "second projective plane", and this is the principle of projective duality .

Computational Geometry

In computational geometry, uniform coordinates are applied to computing operations on the Euclidean plane. The Euclidean plane is temporarily supplemented with the projective one, a homogeneous coordinate 1 is added to the Cartesian coordinates of the points, then operations are performed, then at the very end it is divided into a homogeneous coordinate to get Cartesian coordinates, and the infinitely distant points are processed separately. This approach makes it possible to quickly and accurately code operations with objects on the plane. A line passing through two points and a point at the intersection of two lines - both operations are encoded using a vector product . Also, often expanding the Euclidean plane to the projective one allows avoiding consideration of particular cases in intermediate constructions, for example, intersecting or parallel straight lines, and analyzing only at the very end.

Homogeneous integer coordinates generalize rational numbers . The third homogeneous coordinate serves as a common denominator for the first two coordinates, so all calculations can be performed without errors (in long arithmetic ).

Examples

Barycentric coordinates .

Sources

↑ Prasolov V.V., Tikhomirov V.N. Geometry . - M.: MCCNMO, 2007. ISBN 978-5-94057-267-1 ( Books by V.V. Prasolov )

[1] Prasolov V.V., Tikhomirov V.N. Geometry . - M.: MCCNMO, 2007. ISBN 978-5-94057-267-1 ( Books by V.V. Prasolov )