On the Algebraic and Geometric Foundations of Computer Graphics

1
On the Algebraic and Geometric Foundations of
Computer Graphics

• Review of paper by Ron Goldman
• Presented by Gail Banaszkiewicz

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Whats it all about?

• Ron Goldman says

Only Grassmann space supports all the algebra
and geometry needed for contemporary computer
graphics.
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Outline

• Possible ambient spaces for computer graphics
• Vector spaces
• Affine spaces
• Projective spaces
• Grassmann spaces
• Applications of Grassmann spaces
• Critique of paper

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What spaces need to support

• Transformations (rotation, reflection, scaling,
  translation, orthogonal projection, perspective
  projection)
• Natural geometry
• Points and vectors
• Complete algebra
• Standard curves and surfaces

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Vector Spaces

• Addition, subtraction, and scalar multiplication
  are defined
• Linear transformations include rotation,
  reflection, scaling, and orthogonal projection
• Vectors only no points

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Affine Spaces

• A vector space with no origin
• Points have fixed position
• Vectors have direction and length
• Points can be added to vectors
• Vectors can be added to vectors
• Vectors can be subtracted from vectors

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Affine Spaces

• Affine combinations allow for limited algebra of
  points
• Well defined Midpoint (PQ)/2 between points P
  and Q
• Meaningless (PQ)/3
• Affine transformations
• Maps points to points
• Maps vectors to vectors
• Preserves affine combinations

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Affine Spaces

• Some algebra for points and vectors
• Affine transformations include rotation,
  reflection, scaling, orthogonal projection, and
  translation
• Affine space includes curves, surfaces, and
  solids
• Algebra not complete
• Perspective projection is not an affine
  transformation

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Projective Space

• Contains affine points and points at infinity
• Vectors represent points at infinity, no matter
  how they are scaled
• Homogeneous coordinate distinguishes between
  points at infinity and affine points
• v, 0 cv, 0 c ? 0
• P, 1 cP, c c ? 0

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Projective Space

• Completes geometry of affine space
• Projective transformations include translation,
  rotation, scaling, and perspective
• Unnatural geometry
• No vectors, no notion of orientation or distance
• No standard curves or spaces

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Grassmann Space
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Grassmann Space

• Can be derived from physical mechanics
• locations (points)
• forces (vectors)
• objects to act on (mass)

• Let (mP, m) be the mass point located at P
• Vectors have zero mass, (v, 0)

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Grassmann Space

• To add mass points, sum coordinates
• To scale mass points, multiply the mass and leave
  the point alone
• Scale and add vectors like normal
• To add a mass point to a vector
• (mP, m) (v, 0) (mP v, m)
• Grassmann Space 4-D Vector Space

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Grassmann Space

• Affine transformations are well defined
• Perspective projection is well defined
• Geometry coincides with natural world
• Can construct standard curves and surfaces
• ??

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Application Pseudoperspective

• Important transformation is to map viewing
  frustum to a rectangular box
• Traditional way is to derive matrix involves
  solving 15 homogeneous equations with 16 unknowns

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Application Pseudoperspective

• Affine and Projective spaces also fail to solve
  the problem
• Grassmann space alone allows for simple and
  elegant formulas for the pseudoperspective
  transformation

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Other Applications

• Shading algorithms and texture mapping via linear
  interpolation
• Overcrowns (bulges) and morphing
• new!

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Critique

• Introduction
• Projective spaces and homogeneous coordinates
  are incompatible with much of the algebra and a
  good deal of the geometry currently in actual use
  in computer graphics.
• Graphics practitioners are getting correct
  results by performing calculations for which they
  have no mathematical foundations.

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Critique

• Conclusion
• from these observations it follows that many
  of the formulas, even most of the computations,
  are identical in both frameworks Grassmann and
  projective spaces. The good news is that
  programs and code need not change only their
  high-level interpretation is different.
• So there is not much new to learn, but as we
  have seen there is much to gain.