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In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, homogeneous coordinates or projective coordinates, introduced by
August Ferdinand Möbius
August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.
Early life and education
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on hi ...
in his 1827 work , are a
system of coordinates used 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, ...
, just as
Cartesian coordinates
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 ...
are used in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. They have the advantage that the coordinates of points, including
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 ...
, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and 3D
computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
, where they allow
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 generall ...
s and, in general,
projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
s to be easily represented by a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
.
If homogeneous coordinates of a point are multiplied by a non-zero
scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
Introduction
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 ...
can be thought of as the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
with additional points added, which are called
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 ...
, and are considered to lie on a new line, the
line at infinity
In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...
. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point on the Euclidean plane, for any non-zero real number ''Z'', the triple is called a ''set of homogeneous coordinates'' for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, is such a system of homogeneous coordinates for the point .
For example, the Cartesian point can be represented in homogeneous coordinates as or . The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.
The equation of a line through the origin may be written where ''n'' and ''m'' are not both 0. In
parametric form this can be written . Let ''Z'' = 1/''t'', so the coordinates of a point on the line may be written . In homogeneous coordinates this becomes . In the limit, as ''t'' approaches infinity, in other words, as the point moves away from the origin, ''Z'' approaches 0 and the homogeneous coordinates of the point become . Thus we define as the homogeneous coordinates of the point at infinity corresponding to the direction of the line . As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates.
To summarize:
*Any point in the projective plane is represented by a triple , called homogeneous coordinates or projective coordinates of the point, where ''X'', ''Y'' and ''Z'' are not all 0.
*The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
*Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant.
*When ''Z'' is not 0 the point represented is the point in the Euclidean plane.
*When ''Z'' is 0 the point represented is a point at infinity.
The triple is omitted and does not represent any point. The
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
of the Euclidean plane is represented by .
Notation
Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. The use of colons instead of commas, for example (''x'':''y'':''z'') instead of , emphasizes that the coordinates are to be considered ratios. Square brackets, as in emphasize that multiple sets of coordinates are associated with a single point. Some authors use a combination of colons and square brackets, as in
'x'':''y'':''z''
Other dimensions
The discussion in the preceding section applies analogously to projective spaces other than the plane. So the points on the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
may be represented by pairs of coordinates , not both zero. In this case, the point at infinity is . Similarly the points in projective ''n''-space are represented by (''n'' + 1)-tuples.
Other projective spaces
The use of
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 ...
s gives homogeneous coordinates of points in the classical case of the real projective spaces, however any
field may be used, in particular, the
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 fo ...
s may be used for
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. For example, the
complex projective line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers p ...
uses two homogeneous complex coordinates and is known as the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
. Other fields, including
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s, can be used.
Homogeneous coordinates for projective spaces can also be created with elements from a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
(a skew field). However, in this case, care must be taken to account for the fact that multiplication may not be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
.
For the general
ring ''A'', a
projective line over ''A'' can be defined with homogeneous factors acting on the left and the
projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
acting on the right.
Alternative definition
Another definition of the real projective plane can be given in terms of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es. For non-zero elements of R
3, define to mean there is a non-zero ''λ'' so that . Then ~ is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
and the projective plane can be defined as the equivalence classes of If is one of the elements of the equivalence class ''p'' then these are taken to be homogeneous coordinates of ''p''.
Lines in this space are defined to be sets of solutions of equations of the form where not all of ''a'', ''b'' and ''c'' are zero. Satisfaction of the condition depends only on the equivalence class of so the equation defines a set of points in the projective plane. The mapping defines an inclusion from the Euclidean plane to the projective plane and the complement of the image is the set of points with . The equation is an equation of a line in the projective plane (
see definition of a line in the projective plane), and is called the ''line at infinity''.
The equivalence classes, ''p'', are the lines through the origin with the origin removed. The origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in R
3 that pass through the origin and the coordinates of a non-zero element of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane.
Again, this discussion applies analogously to other dimensions. So the projective space of dimension ''n'' can be defined as the set of lines through the origin in R
''n''+1.
Homogeneity
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say , does not determine a function defined on points as with Cartesian coordinates. But a condition defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
. Specifically, suppose there is a ''k'' such that
If a set of coordinates represents the same point as then it can be written for some non-zero value of λ. Then
A
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of degree ''k'' can be turned into a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
by replacing ''x'' with ''x''/''z'', ''y'' with ''y''/''z'' and multiplying by ''z
k'', in other words by defining
The resulting function ''f'' is a polynomial, so it makes sense to extend its domain to triples where . The process can be reversed by setting , or
The equation can then be thought of as the homogeneous form of and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line is
Line coordinates and duality
The equation of a line in the projective plane may be given as where ''s'', ''t'' and ''u'' are constants. Each triple determines a line, the line determined is unchanged if it is multiplied by a non-zero scalar, and at least one of ''s'', ''t'' and ''u'' must be non-zero. So the triple may be taken to be homogeneous coordinates of a line in the projective plane, that is
line coordinates as opposed to point coordinates. If in ''sx'' + ''ty'' + ''uz'' = 0 the letters ''s'', ''t'' and ''u'' are taken as variables and ''x'', ''y'' and ''z'' are taken as constants then the equation becomes an equation of a set of lines in the space of all lines in the plane. Geometrically it represents the set of lines that pass through the point and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may be given sets of four homogeneous coordinates, and so on for higher dimensions.
The same relation, , may be regarded as either the equation of a line or the equation of a point. In general, there is no difference either algebraically or logically between homogeneous coordinates of points and lines. So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are equivalent except for interpretation. This leads to the concept of ''duality'' in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will also be a theorem. Analogously, the theory of points in projective 3-space is dual to the theory of planes in projective 3-space, and so on for higher dimensions.
Plücker coordinates
Assigning coordinates to lines in projective 3-space is more complicated since it would seem that a total of 8 coordinates, either the coordinates of two points which lie on the line or two planes whose intersection is the line, are required. A useful method, due to
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the dis ...
, creates a set of six coordinates as the determinants from the homogeneous coordinates of two points and on the line. The
Plücker embedding is the generalization of this to create homogeneous coordinates of elements of any dimension ''m'' in a projective space of dimension ''n''.
Application to Bézout's theorem
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
predicts that the number of points of intersection of two curves is equal to the product of their degrees (assuming an algebraically closed field and with certain conventions followed for counting intersection multiplicities). Bézout's theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates are used to locate the point of intersection in this case. Similarly, Bézout's theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them. For example, and have only one point of intersection in the finite (affine) plane. To find the other point of intersection, convert the equations into homogeneous form, and . This produces and, assuming not all of ''x'', ''y'' and ''z'' are 0, the solutions are and . This first solution is the point in Cartesian coordinates, the finite point of intersection. The second solution gives the homogeneous coordinates which corresponds to the direction of the ''y''-axis. For the equations and there are no finite points of intersection. Converting the equations into homogeneous form gives and . Solving produces the equation which has a double root at . From the original equation, , so since at least one coordinate must be non-zero. Therefore, is the point of intersection counted with multiplicity 2 in agreement with the theorem.
Circular points
The homogeneous form for the equation of a circle in the real or complex projective plane is . The intersection of this curve with the line at infinity can be found by setting . This produces the equation which has two solutions over the complex numbers, giving rise to the points with homogeneous coordinates and in the complex projective plane. These points are called the
circular points at infinity In projective geometry, 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 of every real circle.
Coordi ...
and can be regarded as the common points of intersection of all circles. This can be generalized to curves of higher order as
circular algebraic curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation ''F''(''x'', ''y'') = 0, where ''F'' is a polynomial with real coefficients and the highest-order terms of ''F'' form a polynomia ...
s.
Change of coordinate systems
Just as the selection of axes in the Cartesian coordinate system is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore, it is useful to know how the different systems are related to each other.
Let be homogeneous coordinates of a point in the projective plane. A fixed matrix
with nonzero
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
, defines a new system of coordinates by the equation
Multiplication of by a scalar results in the multiplication of by the same scalar, and ''X'', ''Y'' and ''Z'' cannot be all 0 unless ''x'', ''y'' and ''z'' are all zero since ''A'' is nonsingular. So are a new system of homogeneous coordinates for the same point of the projective plane.
Barycentric coordinates
Möbius's original formulation of homogeneous coordinates specified the position of a point as the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
(or barycenter) of a system of three point masses placed at the vertices of a fixed triangle. Points within the triangle are represented by positive masses and points outside the triangle are represented by allowing negative masses. Multiplying the masses in the system by a scalar does not affect the center of mass, so this is a special case of a system of homogeneous coordinates.
Trilinear coordinates
Let ''l'', ''m'', ''n'' be three lines in the plane and define a set of coordinates ''X'', ''Y'' and ''Z'' of a point ''p'' as the signed distances from ''p'' to these three lines. These are called the ''trilinear coordinates'' of ''p'' with respect to the triangle whose vertices are the pairwise intersections of the lines. Strictly speaking these are not homogeneous, since the values of ''X'', ''Y'' and ''Z'' are determined exactly, not just up to proportionality. There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of to represent the same point. More generally, ''X'', ''Y'' and ''Z'' can be defined as constants ''p'', ''r'' and ''q'' times the distances to ''l'', ''m'' and ''n'', resulting in a different system of homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity.
Use in computer graphics and computer vision
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
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 ...
,
scaling and
perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple and efficient processing. By contrast, using Cartesian coordinates, translations and perspective projection cannot be expressed as matrix multiplications, though other operations can. Modern
OpenGL
OpenGL (Open Graphics Library) is a cross-language, cross-platform application programming interface (API) for rendering 2D and 3D vector graphics. The API is typically used to interact with a graphics processing unit (GPU), to achieve hardwa ...
and
Direct3D
Direct3D is a graphics application programming interface (API) for Microsoft Windows. Part of DirectX, Direct3D is used to render three-dimensional graphics in applications where performance is important, such as games. Direct3D uses hardware ...
graphics card
A graphics card (also called a video card, display card, graphics adapter, VGA card/VGA, video adapter, display adapter, or mistakenly GPU) is an expansion card which generates a feed of output images to a display device, such as a computer mo ...
s take advantage of homogeneous coordinates to implement a
vertex shader
In computer graphics, a shader is a computer program that calculates the appropriate levels of light, darkness, and color during the rendering of a 3D scene - a process known as ''shading''. Shaders have evolved to perform a variety of speci ...
efficiently using
vector processor
In computing, a vector processor or array processor is a central processing unit (CPU) that implements an instruction set where its instructions are designed to operate efficiently and effectively on large one-dimensional arrays of data calle ...
s with 4-element registers.
For example, in perspective projection, a position in space is associated with the line from it to a fixed point called the ''center of projection''. The point is then mapped to a plane by finding the point of intersection of that plane and the line. This produces an accurate representation of how a three-dimensional object appears to the eye. In the simplest situation, the center of projection is the origin and points are mapped to the plane , working for the moment in Cartesian coordinates. For a given point in space, , the point where the line and the plane intersect is . Dropping the now superfluous ''z'' coordinate, this becomes . In homogeneous coordinates, the point is represented by and the point it maps to on the plane is represented by , so projection can be represented in matrix form as
Matrices representing other geometric transformations can be combined with this and each other by matrix multiplication. As a result, any perspective projection of space can be represented as a single matrix.
Notes
References
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Further reading
*
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External links
* Jules Bloomenthal and Jon Rokne, Homogeneous coordinate
{{Webarchive, url=https://web.archive.org/web/20210226225843/http://www.unchainedgeometry.com/jbloom/pdf/homog-coords.pdf , date=2021-02-26
* Ching-Kuang Shene, Homogeneous coordinate
Wolfram MathWorld
Linear algebra
Projective geometry
1827 in science