Jules Bloomenthal & Jon Rokne "Homogeneous Coordinates"

Jules Bloomenthal and Jon Rokne (Department of Computer Science, The University of Calgary)
Homogeneous Coordinates
http://portal.acm.org/citation.cfm?id=205426
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.3319
http://www.springerlink.com/content/p356406661505622/

homog-coords.pdf

Introduction

http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29
무게중심좌표 http://100.naver.com/100.nhn?docid=64206

http://mathworld.wolfram.com/BarycentricCoordinates.html

Barycentric coordinates are homogeneous, so

(1)

for .

http://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
(d:m) are the Plücker coordinates of L.

Although neither d nor m alone is sufficient to determine L, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between x and y. That is, the coordinates

(d:m) = (d₁:d₂:d₃:m₁:m₂:m₃)

may be considered homogeneous coordinates for L, in the sense that all pairs (λd:λm), for λ ≠ 0, can be produced by points on L and only L, and any such pair determines a unique line so long as d is not zero and d•m = 0.

http://en.wikipedia.org/wiki/Ray_tracing_%28graphics%29
technique for generating an image by tracing the path of light through pixels in an image plane

http://en.wikipedia.org/wiki/Projective_space

http://mathworld.wolfram.com/GrassmannCoordinates.html

Plücker embedding = Grassmann coordinates
http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding

Projective Plane

http://en.wikipedia.org/wiki/Point_at_infinity
http://en.wikipedia.org/wiki/Hyperplane_at_infinity
The real projective plane  By Harold Scott Macdonald Coxeter

http://en.wikipedia.org/wiki/Projective_plane
"A projectivity is any conceivable invertible linear transform of homogeneous coordinates."

A projective transformation in P² space is an invertible mapping of points in P² to points in P² that maps lines to lines. A P² projectivity has the equation

x′ = Hx

where H is an invertible 3 × 3 matrix.

http://mathworld.wolfram.com/ProjectivePlane.html

http://vision.stanford.edu/~birch/projective/

ideal line = line at infinity
http://en.wikipedia.org/wiki/Line_at_infinity

http://en.wikipedia.org/wiki/Linear_perspective
http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm

Quadrilateral Perspective, drawing in perspective, parallel, oblique and integrated perspectives
by Yvonne Tessuto Tavares

AERIAL PARALLEL PERSPECTIVE (2 VANISHING POINTS)

PARALLEL PERSPECTIVE - AERIAL VIEW GEOMETRIC STRUCTURE

PARALLEL PERSPECTIVE - AERIAL VIEW WITH A VIEW FROM BOTTOM TO TOP

The mapping from planes and lines through the center of projection to lines and points on the projective plane is the transformation of the usual Euclidean space into projective space.

A projective space is not a vector space in the same manner as the Euclidean space.

Riesenfeld, R. F. 1981. Homogeneous Coordinates and Projective Planes in Computer Graphics. IEEE Comput. Graph. Appl. 1, 1 (Jan. 1981), 50-55. DOI= http://dx.doi.org/10.1109/MCG.1981.1673814

Homogeneous Coordinates and Projective Planes in Computer G.pdf

Unification of the translation, scaling and rotation of geometric objects
: "All affine transformations are matrix multiplication."


Affine Transformations

Homogeneous Lines

Conics
"matrix of the second degree curve"
http://en.wikipedia.org/wiki/Ellipse
http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
http://en.wikipedia.org/wiki/Conic_section

Rational Curves
: extended parametric curve (control points + basis functions)

The use of homogeneous coordinates not only produces polynomials of fixed degree, it also provides a method for consistent manipulation of the Euclidean space.

Perspective Projection
perspective divide

A loss of depth information is due to the linear dependence of the third and fourth columns of the matrix.

Introducing a second non-zero term, e.g. -1, into the third column does not affect x’ and y’, but z’ becomes D-D/z. The purpose of this additional term is to compress the Euclidean space z Î [1, ¥] to z’ Î [0, D].

Perspective Space

"The homogeneous perspective transformation transforms Euclidean points to new homogeneous points."

perspective space (of the transformed points) vs. object space

The perspective matrix is invertible whereas the perspective-projection matrix is singular.

http://en.wikipedia.org/wiki/Viewing_frustum
http://en.wikipedia.org/wiki/Frustum




Perspective Transformation

Homogeneous Clipping