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Coordinate system used in projective geometry
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are
Homogeneous_coordinates
Coordinate system that is defined by points instead of vectors
unity. Barycentric coordinates were introduced by August Möbius in 1827. They are special homogeneous coordinates. Barycentric coordinates are strongly related
Barycentric_coordinate_system
Method for specifying point positions
cylindrical coordinates (r, z) to polar coordinates (ρ, φ) giving a triple (ρ, θ, φ). A point in the plane may be represented in homogeneous coordinates by a
Coordinate_system
Curve from a cone intersecting a plane
One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic
Conic_section
Method of assigning coordinates to every line in projective 3-space
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective
Plücker_coordinates
Compact non-orientable two-dimensional manifold
origin (a projective "line") is called the line at infinity. (See § Homogeneous coordinates below.) In topology, the name real projective plane is applied
Real_projective_plane
Coordinates used to specify position of a line
a homogeneous function then φ(l, m, n) = 0 represents a curve in the dual space given in homogeneous coordinates, and may be called the homogeneous tangential
Line_coordinates
Geometric transformation
points are represented using homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py
Scaling_(geometry)
Coordinate system based on distances from a triangle's sidelines
the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular
Trilinear_coordinates
Line with a point at infinity added
projective line P1(K) may be represented by an equivalence class of homogeneous coordinates, which take the form of a pair [ x 1 : x 2 ] {\displaystyle [x_{1}:x_{2}]}
Projective_line
Central object in linear algebra; mapping vectors to vectors
perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be used. The matrix to rotate an angle θ about any axis defined
Transformation_matrix
Family of elliptic curves used in cryptography
C {\displaystyle C=x(1-y),D=C} In the context of cryptography, homogeneous coordinates are used to prevent field inversions that appear in the affine
Edwards_curve
Isomorphism of projective spaces in geometry
may thus be represented by the coordinates of any nonzero point of this line, which are thus called homogeneous coordinates of the projective point. Given
Homography
Coordinate system used in computer graphics
coordinate system is a homogeneous coordinate system in the graphics pipeline that is used for clipping. Objects' coordinates are transformed via a projection
Clip_coordinates
Generalized scaling operation in geometry
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and
Homothety
Metric on a complex projective space endowed with Hermitian form
point in CPn with homogeneous coordinates [ Z 0 : ⋯ : Z n ] {\displaystyle [Z_{0}:\dots :Z_{n}]} , there is a unique set of n coordinates ( z 1 , … , z n
Fubini–Study_metric
Topological space in group theory
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere as one moves through it, with movement given by the action
Homogeneous_space
Process of estimating the parameters of a pinhole camera model
position in world coordinates. In both cases, they are represented in homogeneous coordinates (i.e. they have an additional last component, which is initially
Camera_resectioning
Model of 3D points projected onto planar image via a lens-less aperture
The mapping from 3D coordinates of points in space to 2D image coordinates can also be represented in homogeneous coordinates. Let x {\displaystyle
Pinhole_camera_model
Common point(s) shared by two lines in Euclidean geometry
The mapping from 3D to 2D coordinates is (x′, y′) = (x/w, y/w). We can convert 2D points to homogeneous coordinates by defining them as (x, y, 1)
Line–line_intersection
ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice
Homogeneous_coordinate_ring
Invariant in projective geometry
that contains them. If four collinear points are represented in homogeneous coordinates by vectors α , β , γ , δ {\displaystyle \alpha ,\beta ,\gamma
Cross-ratio
Geometry with 7 points and 7 lines
plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to
Fano_plane
Geometry theorem
parallelity A c ∥ C a {\displaystyle \;Ac\parallel Ca\;} . Choose homogeneous coordinates with C = ( 1 , 0 , 0 ) , c = ( 0 , 1 , 0 ) , X = ( 0 , 0 , 1 )
Pappus's_hexagon_theorem
Polynomial whose nonzero terms all have the same degree
which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element
Homogeneous_polynomial
Computer vision geometry concept
{\displaystyle \mathbf {x} } be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let y {\displaystyle \mathbf {y}
Camera_matrix
Geometric concept of a 2D space with "points at infinity" adjoined
embeds into KP2 via the map which sends affine (non-homogeneous) coordinates to homogeneous coordinates, ( x 1 , x 2 ) ↦ ( 1 , x 1 , x 2 ) . {\displaystyle
Projective_plane
Euclidean space without distance and angles
system in an (n−1)-dimensional space are barycentric coordinates and affine "homogeneous" coordinates (1, x1, … , xn−1). In the latter case the x0 coordinate
Affine_space
Type of geometry
included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract
Projective_geometry
Polynomial equation whose integer solutions are sought
a n ) {\displaystyle \left(a_{1},\ldots ,a_{n}\right)} are the homogeneous coordinates of a rational point of the hypersurface defined by Q. Conversely
Diophantine_equation
Number of intersection points of algebraic curves and hypersurfaces
projective coordinates by a homogeneous polynomial p ( x , y , t ) {\displaystyle p(x,y,t)} of degree n, the substitution of y provides a homogeneous polynomial
Bézout's_theorem
Embedding of a Grassmannian into projective space
Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian G r ( k , V ) {\displaystyle \mathrm
Plücker_embedding
Means of projecting three-dimensional objects in two dimensions
it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as P = [ 1 0 0 0 0 1 0 0 0
Orthographic_projection
Completion of the usual space with "points at infinity"
projective coordinates or homogeneous coordinates of a point p(v) on a frame (p(e0), ..., p(en+1)) with en+1 = e0 + ... + en are the coordinates of v on
Projective_space
Simple curve of Euclidean geometry
}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.} In homogeneous coordinates, each conic section with the equation of a circle has the form
Circle
Rational function of the form (az + b)/(cz + d)
[z1:z2] are homogeneous coordinates on CP1; the point [1:0] corresponds to the point ∞ of the Riemann sphere. By using homogeneous coordinates, many calculations
Möbius_transformation
Geometric transformation that preserves lines but not angles nor the origin
(specifically, a shear transformation). The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean
Affine_transformation
homogeneous coordinates, being a triple of complex numbers (x : y : z), where two triples describe the same point of the plane when the coordinates of
Circular_points_at_infinity
written as coordinate triples, are the homogeneous coordinates of the given point, called point coordinates. With respect to this basis, the solution
Incidence_(geometry)
Riemannian manifold with SU(n) holonomy
variety consisting of all of the zeros of a homogeneous quintic polynomial in the homogeneous coordinates of the C P 4 {\displaystyle \mathbb {CP} ^{4}}
Calabi–Yau_manifold
Computer-based generation of digital images
homogeneous coordinates as w = (wx, wy, wz, 1). To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates)
2D_computer_graphics
Planar movement within a Euclidean space without rotation
fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication:
Translation_(geometry)
Mathematical concept in polynomial theory
, α n {\displaystyle \alpha _{1},\ldots ,\alpha _{n}} are the homogeneous coordinates of a common zero of P 1 , … , P n − 1 . {\displaystyle P_{1},\ldots
Resultant
Type of topological space
sphere. Real projective spaces are smooth manifolds. On Sn, in homogeneous coordinates, (x1, ..., xn+1), consider the subset Ui with xi ≠ 0. Each Ui is
Real_projective_space
Spline function
rational B-splines (NURBS). NURBS are essentially B-splines in homogeneous coordinates. Like B-splines, they are defined by their order, and a knot vector
B-spline
German mathematician and astronomer (1790–1868)
tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates into projective geometry. He is recognized for the introduction
August_Ferdinand_Möbius
Function with a multiplicative scaling behaviour
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied
Homogeneous_function
Element of an exterior algebra
of coordinates for lines, planes and hyperplanes that have properties similar to the homogeneous coordinates of points, called Grassmann coordinates. Points
Multivector
German mathematician and physicist (1801–1868)
the n × k {\displaystyle n\times k} matrix of homogeneous coordinates, also known as Plücker coordinates, apply. The embedding of the Grassmannian G r
Julius_Plücker
Concept in projective geometry
vector space (with a companion antiautomorphism) and conversely. Homogeneous coordinates may be used to give an algebraic description of dualities. To simplify
Duality_(projective_geometry)
3D space. 4D vector A common datatype in graphics code, holding homogeneous coordinates or RGBA data, or simply a 3D vector with unused W to benefit from
Glossary_of_computer_graphics
4-component vector data type in computer science
science, a 4D vector is a 4-component vector data type. Uses include homogeneous coordinates for 3-dimensional space in computer graphics, and red green blue
4D_vector
Design technique
_{y}+\mathbf {e} _{y}.\end{aligned}}} Or, in matrix form using homogeneous coordinates, the system [ f x f y f w ] = [ 1 0 e x e z 0 1 e y e z 0 0 1 e
3D_projection
Type of differential equation
characteristic form Q(ζ) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis
Partial_differential_equation
Four-dimensional number system
called rotors, can be very useful for applications involving homogeneous coordinates. But it is only in 3D that the number of basis bivectors equals
Quaternion
The latter space has a distinguished system of homogeneous coordinates, given by the Plücker coordinates. An effective algebraic cycle in P n − 1 {\displaystyle
Chow_variety
Class of mathematical expression
JSTOR 20876802 Wegman, Edward J.; Said, Yasmin H. (2010), "Natural homogeneous coordinates", Wiley Interdisciplinary Reviews: Computational Statistics, 2
Division_by_zero
Type of geometric transformation
coordinates on the blowup, we can write down equations for the above incidence correspondence. Give P 2 {\displaystyle \mathbf {P} ^{2}} homogeneous coordinates
Blowing_up
Relation between algebraic varieties and polynomial ideals
0 on S {\displaystyle f=0{\text{ on }}S} we mean: for every homogeneous coordinates ( a 0 : ⋯ : a n ) {\displaystyle (a_{0}:\cdots :a_{n})} of a point
Hilbert's_Nullstellensatz
Algebraic variety in a projective space
x n {\displaystyle x_{0},\dots ,x_{n}} are referred to as the homogeneous coordinates of the point. A projective variety is, by definition, a closed
Projective_variety
Polynomial with all terms of degree two
variables has important applications to algebraic topology. Using homogeneous coordinates, a non-zero quadratic form in n variables defines an (n − 2)-dimensional
Quadratic_form
Matrix representing a Euclidean rotation
for a unit quaternion, we find that nonzero quaternions act as homogeneous coordinates for 3 × 3 rotation matrices. The Cayley transform, discussed earlier
Rotation_matrix
Curve defined as zeros of polynomials
set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables P(x, y, z). Every affine algebraic
Algebraic_curve
Concept in geometry and topology
terms of lower order in X and Y. More formally, we should use homogeneous coordinates [ X : Y : Z ] {\displaystyle [X:Y:Z]} and note that the line at
Line_at_infinity
Mathematical object studied in the field of algebraic geometry
k[x0, ..., xn] be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in Pn in homogeneous coordinates. However, because
Algebraic_variety
Mathematical space
vectors ( W 1 , … , W k ) {\displaystyle (W_{1},\dots ,W_{k})} . The homogeneous coordinates of the element w ∈ G r k ( V ) {\displaystyle w\in \mathbf {Gr}
Grassmannian
\nu :\mathbf {P} ^{1}\to \mathbf {P} ^{n}} which assigns to the homogeneous coordinates [S : T] the value ν : [ S : T ] ↦ [ S n : S n − 1 T : S n − 2 T
Rational_normal_curve
Mathematical idealization of the trace left by a moving point
simplifies to a homogeneous polynomial g(u, v, w) of degree d. The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points
Curve
Method of determining a point in 3D space
′ {\displaystyle \mathbf {y} '_{1},\mathbf {y} '_{2}} are the homogeneous coordinates of the detected image points and C 1 , C 2 {\displaystyle \mathbf
Triangulation (computer vision)
Triangulation_(computer_vision)
a real point is a point in the complex projective plane with homogeneous coordinates (x,y,z) for which there exists a nonzero complex number λ such
Real_point
In mathematics, straight line touching a plane curve without crossing it
converting to homogeneous coordinates. Specifically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree
Tangent
Algebraic curve
the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n + Y n = Z n . {\displaystyle
Fermat_curve
In computer graphics, removal of lines outside the view area/volume
leads to a O(lg N) run-time complexity. This algorithm is based on homogeneous coordinates and duality. It can be used for line or line-segment clipping against
Line_clipping
Austrian mathematician
{\frac {a}{k}}={\frac {v}{c}}} or with a Lorentz boost by using homogeneous coordinates: ( x , y , x ′ , y ′ ) = ( x 1 x 0 , x 2 x 0 , x 1 ′
Gustav_von_Escherich
Term in mathematics
\cdots ,X_{n}),1\leq i\leq n-m,} in the homogeneous coordinates Xj, which generate all other homogeneous polynomials that vanish on V. Geometrically, each
Complete_intersection
Class of mathematical functions
an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the ℘ {\displaystyle \wp } -function and its derivative
Weierstrass_elliptic_function
Problem-solving technique in geometry
used as early as 1827 by August Ferdinand Möbius in his theory of homogeneous coordinates. The theory of mass points is defined according to the following
Mass_point_geometry
Parameterization of a rotation into a unit vector and angle
{v} |,r)\,,} where |v| is the Euclidean norm of the 3-vector v. Homogeneous coordinates Pseudovector Rotations without a matrix Screw theory, a representation
Axis–angle_representation
Existence of a line through two points
using real numbers for the coordinates of their points (Cartesian coordinates for the Euclidean plane and homogeneous coordinates for the projective plane)
Sylvester–Gallai_theorem
Concept in mathematics
\mathbf {P} ^{m}-\{y_{0}=0\}} is a morphism, where yi are the homogeneous coordinates. Note the target space is the affine space Am through the identification
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
Algebraic structure designed for geometry
procedure has some similarities to the procedure for working with homogeneous coordinates in projective geometry, and in this case allows the modeling of
Geometric_algebra
Algorithmic processing of digitally-represented images
For example, 2-dimensional coordinates only permit rotation about the origin (0, 0). But 3-dimensional homogeneous coordinates can be used to first translate
Digital_image_processing
Method of representing curves and surfaces in computer graphics
point is a regular non-homogenous coordinate [no 'w'] rather than a homogeneous coordinate. That is equivalent to having weight "1" at each control point;
Non-uniform_rational_B-spline
Combinatorics problem proposed by Thomas Penyngton Kirkman
Conwell. The Galois field GF(2) with two elements is used with four homogeneous coordinates to form PG(3,2) which has 15 points, 3 points to a line, 7 points
Kirkman's_schoolgirl_problem
{\displaystyle \lambda } and consists of the points in the plane whose homogeneous coordinates ( x : y : z ) {\displaystyle (x:y:z)} satisfy the equation. The
Hesse_pencil
is not the same. It's possible to express the above line coordinates as homogeneous coordinates z = [ sin ( θ + ε R 2 ) : cos ( θ + ε R 2 ) ] {\displaystyle
Laguerre_transformations
Generalizations of codimension-1 subvarieties of algebraic varieties
n-space with the homogeneous coordinates x0, ..., xn. Let U = {x0 ≠ 0}. Then U is isomorphic to the affine n-space with the coordinates yi = xi/x0. Let
Divisor_(algebraic_geometry)
Particular mapping that projects a sphere onto a plane
zeros of a non-singular quadratic form f(x0, ..., xn+1) in the homogeneous coordinates xi. Fix any point Q on S and a hyperplane E in Pn+1 not containing
Stereographic_projection
Map in projective geometry
^{n}\times \mathbb {P} ^{m}\to \mathbb {P} ^{(n+1)(m+1)-1}\ } given in homogeneous coordinates by σ ( [ X 0 : X 1 : ⋯ : X n ] , [ Y 0 : Y 1 : ⋯ : Y m ] ) = [
Segre_embedding
Straight line that only contains one real point
complex projective plane P2(C) where points are represented by three homogeneous coordinates ( x 1 , x 2 , x 3 ) , x i ∈ C . {\displaystyle (x_{1},\ x_{2}
Imaginary_line_(mathematics)
The group of K-rational points of an abelian variety is a finitely-generated abelian group
question of how many digits are required to write down a set of homogeneous coordinates. For an abelian variety, there is no a priori preferred representation
Mordell–Weil_theorem
Family of geometric objects with a common property
the same circle; thus, these quadruples may be considered to be homogeneous coordinates for the space of circles. Straight lines may also be represented
Pencil_(geometry)
Upper bound in coding theory
with homogeneous coordinates. Form the ( N + 1 ) × m {\displaystyle (N+1)\times m} matrix G {\displaystyle G} whose columns are the homogeneous coordinates
Singleton_bound
Locus of the zeros of a polynomial of degree two
then the ray misses the quadric surface. Alternatively, using homogeneous coordinates, one may represent the ray as r = p 0 + d t {\displaystyle {\mathbf
Quadric
Line along which a quadratic form applied to any two points' displacement is zero
are represented by homogeneous coordinates ( x 1 , x 2 , x 3 ) {\displaystyle (x_{1},x_{2},x_{3})} and lines by homogeneous coordinates ( a 1 , a 2 , a 3
Isotropic_line
Branch of mathematics
line have the same set of coordinates, up to the multiplication by an element of k. This defines the homogeneous coordinates of a point of Pn as a sequence
Algebraic_geometry
Mathematical operation
its projective line have homogeneous coordinates written U [ a , b ] {\displaystyle U[a,b]} to indicate that the homogeneous factor multiplies on the
Cayley_transform
Rational surface in 5-dimensional projective space
[x^{2}:y^{2}:z^{2}:yz:xz:xy]} where [ x : ⋯ ] {\displaystyle [x:\cdots ]} denotes homogeneous coordinates. The map ν {\displaystyle \nu } is known as the Veronese embedding
Veronese_surface
Metric based on the exact solution of Einstein's field equations of general relativity
coordinates: c 2 d τ 2 = c 2 d t 2 − d x 2 − d y 2 − d z 2 {\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}} An isotropic, homogeneous
Friedmann–Lemaître–Robertson–Walker metric
Friedmann–Lemaître–Robertson–Walker_metric
Method of data interpolation and smoothing
are in 2 dimensions ( D = 2 {\displaystyle D=2} ). One can use homogeneous coordinates for the point-set where a point y i {\displaystyle y_{i}} is represented
Thin_plate_spline
HOMOGENEOUS COORDINATES
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HOMOGENEOUS COORDINATES