“homogeneous coordinates”的概念、定义、翻译、参考文献-科学参考

homogeneous coordinates

Mathematics

Coordinates used in projective spaces. In the real projective plane ℙ² = (ℝ³–{0})/ℝ* the vectors (x,y,z) and (cx,cy,cz), where c is non-zero, represent the same point. Then [x:y:z] are the homogeneous coordinates for this point, so that [x:y:z]=[cx:cy:cz]. Points of the complex projective line have coordinates of the form [1:z] or [0:1], so that [0:1] equates with ∞ in the extended complex plane.

Computer

A coordinate system that algebraically treats all points in the projective plane (both Euclidean and ideal) equally. For example, the standard homogeneous coordinates [p₁,p₂,p₃] of a point P in the projective plane are of the form [x,y,1] if P is a point in the Euclidean plane z=1 whose Cartesian coordinates are (x,y,1), or are of the form [a,b,0] if P is the ideal point—the point at infinity—associated to all lines in the Euclidean plane z=1 with direction numbers a,b,0. Homogeneous coordinates are so called because they treat Euclidean and ideal points in the same way.
Homogeneous coordinates are widely used in computer graphics because they enable affine and projective transformations to be described as matrix manipulations in a coherent way.

单词	homogeneous coordinates
释义	homogeneous coordinates Mathematics Coordinates used in projective spaces. In the real projective plane ℙ² = (ℝ³–{0})/ℝ* the vectors (x,y,z) and (cx,cy,cz), where c is non-zero, represent the same point. Then [x:y:z] are the homogeneous coordinates for this point, so that [x:y:z]=[cx:cy:cz]. Points of the complex projective line have coordinates of the form [1:z] or [0:1], so that [0:1] equates with ∞ in the extended complex plane. Computer A coordinate system that algebraically treats all points in the projective plane (both Euclidean and ideal) equally. For example, the standard homogeneous coordinates [p₁,p₂,p₃] of a point P in the projective plane are of the form [x,y,1] if P is a point in the Euclidean plane z=1 whose Cartesian coordinates are (x,y,1), or are of the form [a,b,0] if P is the ideal point—the point at infinity—associated to all lines in the Euclidean plane z=1 with direction numbers a,b,0. Homogeneous coordinates are so called because they treat Euclidean and ideal points in the same way. Homogeneous coordinates are widely used in computer graphics because they enable affine and projective transformations to be described as matrix manipulations in a coherent way.
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