In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are...
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Coordinate system (redirect from Coordinates)
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...
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In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action...
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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...
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Transformation matrix (redirect from Homogeneous transformation matrix)
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...
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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...
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Barycentric coordinate system (redirect from Areal coordinates)
barycentric coordinates that are called barycentric coordinates. In this case the above defined coordinates are called homogeneous barycentric coordinates. With...
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Real projective plane (section Homogeneous coordinates)
origin (a projective "line") is called the line at infinity. (See § Homogeneous coordinates below.) In topology, the name real projective plane is applied...
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Projective line (section Homogeneous coordinates)
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}]}...
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The mapping from 3D coordinates of points in space to 2D image coordinates can also be represented in homogeneous coordinates. Let x {\displaystyle...
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{\displaystyle \mathbf {x} } be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let y {\displaystyle \mathbf {y}...
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Fano plane (section Homogeneous coordinates)
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...
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coordinate system is a homogeneous coordinate system in the graphics pipeline that is used for clipping. Objects' coordinates are transformed via a projection...
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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...
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homogeneous coordinates as w = (wx, wy, wz, 1). To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates)...
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points are represented using homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py...
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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...
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Conic section (redirect from Conic Sections in Polar Coordinates)
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...
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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)...
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Camera resectioning (section Homogeneous coordinates)
position in world coordinates. In both cases, they are represented in homogeneous coordinates (i.e. they have an additional last component, which is initially...
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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...
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Plücker embedding (redirect from Grassmann coordinates)
Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian G r ( k , V ) {\displaystyle \mathbf...
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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...
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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...
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included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract...
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the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular...
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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 )...
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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...
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{\displaystyle \mathbf {v} } , each homogeneous vector p {\displaystyle \mathbf {p} } (written in homogeneous coordinates) can be multiplied by this translation...
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ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice...
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