Other operations. ▫ Intersection, basis orthogonalization, etc. • Linear Algebra is
the standard framework. 2. Introduction to Geometric Algebra (2013.1) ...
Introduction to Geometric Algebra Lecture 1 Why Geometric Algebra? Professor Leandro Augusto Frata Fernandes
[email protected]
Lecture notes available in http://www.ic.uff.br/~laffernandes/teaching/2013.1/topicos_ag
Geometric Problems
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Geometric data Lines, planes, circles, spheres, etc.
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Transformations Rotation, translation, scaling, etc.
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Other operations Intersection, basis orthogonalization, etc.
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Linear Algebra is the standard framework Introduction to Geometric Algebra (2013.1)
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Using Vectors to Encode Geometric Data
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Directions
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Points
Introduction to Geometric Algebra (2013.1)
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Using Vectors to Encode Geometric Data
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Directions
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Points
Drawback
HOMEWORK
The semantic difference between a direction vector and a point vector is not encoded in the vector type itself. R. N. Goldman (1985), Illicit expressions in vector algebra, ACM Trans. Graph., vol. 4, no. 3, pp. 223–243.
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Assembling Geometric Data
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Straight lines Point vector Direction vector
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Planes Normal vector Distance from origin
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Spheres Center point Radius Introduction to Geometric Algebra (2013.1)
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Assembling Geometric Data
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Straight lines Point vector Direction vector
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Planes Normal vector Distance from origin
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Spheres Center point Radius Introduction to Geometric Algebra (2013.1)
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Assembling Geometric Data
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Straight lines Point vector Direction vector
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Planes Normal vector Distance from origin
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Spheres Center point Radius Introduction to Geometric Algebra (2013.1)
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Assembling Geometric Data
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Straight lines Point vector Direction vector
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Planes Normal vector Distance from origin
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Spheres Center point Radius
Drawback The factorization of geometric elements prevents their use as computing primitives.
Introduction to Geometric Algebra (2013.1)
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Intersection of Two Geometric Elements
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A specialized equation for each pair of elements Straight line × Straight line Straight line × Plane Straight line × Sphere Plane × Sphere etc.
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Special cases must be handled explicitly Straight line × Circle may return o Empty set o One point o Points pair Introduction to Geometric Algebra (2013.1)
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Intersection of Two Geometric Elements
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A specialized equation for each pair of elements Straight line × Straight line Straight line × Plane Straight line × Sphere Plane × Sphere etc.
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Linear Algebra Extension
Plücker Coordinates Special cases must be handled explicitly An alternative Straight line × Circle may returnto represent flat geometric elements
o Empty set o One point o Points pair
Points, lines and planes as elementary types Allow the development of more general solution Not fully compatible with transformation matrices
J. Stolfi (1991) Oriented projective geometry. Academic Press Professional, Inc.
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Using Matrices do Encode Transformations Projective Affine Similitude Linear Rigid / Euclidean Scaling Identity Translation
Isotropic Scaling
Rotation
Reflection Shear
Perspective Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Rigid Body / Euclidean Transforms
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Preserves distances Preserves angles Rigid / Euclidean Identity Translation
Rotation
Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Similitudes / Similarity Transforms
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Preservers angles Similitudes Rigid / Euclidean Identity Translation
Isotropic Scaling
Rotation
Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Linear Transformations
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L(p + q) = L(p) + L(q) L(α p) = α L(p) Similitudes Linear Rigid / Euclidean Scaling Identity Translation
Isotropic Scaling
Rotation
Reflection Shear
Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Affine Transformations
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Preserves parallel lines Affine Similitudes Linear Rigid / Euclidean Scaling Identity Translation
Isotropic Scaling
Rotation
Reflection Shear
Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Projective Transformations
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Preserves lines
Projective Affine
Similitudes Linear Rigid / Euclidean Scaling Identity Translation
Isotropic Scaling
Rotation
Reflection Shear
Perspective Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Using Matrices do Encode Transformations Projective Affine Similitude Linear Rigid / Euclidean Scaling Identity Translation
Isotropic Scaling
Reflection
Rotation
Shear
Perspective Adapted from MIT Course 6.837, Computer Graphics, Fall 2003. Introduction to Geometric Algebra (2013.1)
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Drawbacks of Transformation Matrices
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Non-uniform scaling affects point vectors and normal vectors differently
HOMEWORK
90°
> 90°
1.5 X A. H. Barr (1984), Global and local deformations of solid primitives. Introduction to Geometric Algebra (2013.1) ACM SIGGRAPH Comput. Graph., 18(3), pp. 21-30.
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Drawbacks of Transformation Matrices
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Non-uniform scaling affects point vectors and normal vectors differently
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Rotation matrices Not suitable for interpolation Encode the rotation axis and angle in a costly way
Introduction to Geometric Algebra (2013.1)
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Drawbacks of Transformation Matrices
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Non-uniform scaling affects point vectors and normal vectors differently
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Rotation matrices Linear Algebra Extension
Not suitable for interpolation
Quaternion
Encode the rotation axis and angle in a costly way
Represent and interpolate rotations consistently
Can be combined with isotropic scaling Not well connected with other transformations Not compatible with Plücker coordinates Defined only in 3-D W. R. Hamilton (1844) On a new species of imaginary quantities connected with the theory of quaternions. In Proc. of the Royal Irish Acad., vol. 2, pp. 424-434.
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Linear Algebra
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Standard language for geometric problems Well-known limitations Aggregates different formalisms to obtain complete solutions Matrices Plücker coordinates Quaternion
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Jumping back and forth between formalisms requires custom and ad hoc conversions Introduction to Geometric Algebra (2013.1)
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Geometric Algebra
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High-level framework for geometric operations Geometric elements as primitives for computation Naturally generalizes and integrates Plücker coordinates Quaternion Complex numbers
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Extends the same solution to Higher dimensions All kinds of geometric elements Introduction to Geometric Algebra (2013.1)
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Motivational Example 1. Create the circle through points c1, c2 and c3
2. Create a straight line L through points a1 and a2
3. Rotate the circle around the line and show n rotation steps
4. Create a plane through point p and with normal vector n
Dual plane
Adapted from L. Dorst, D. Fontijine, S. Mann. Geometric algebra for computer science. Morgan Kaufmann Publishers, 2007.
5. Reflect the whole situation with the line and the circlers in the plane
Introduction to Geometric Algebra (2013.1)
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Motivational Example 1. Create the circle through points c1, c2 and c3
2. Create a straight line L through points a1 and a2
3. Rotate the circle around the line and show n rotation steps
The reflected line becomes a circle.
4. Create a plane spherethrough throughpoint pointppand and with normal center cvector n
The reflected rotation becomes a scaled rotation around the circle.
Dual sphere plane
Adapted from L. Dorst, D. Fontijine, S. Mann. Geometric algebra for computer science. Morgan Kaufmann Publishers, 2007.
The only thing that is different is that the plane was changed by the sphere.
5. Reflect the whole situation with the line and the circlers in the plane sphere
Introduction to Geometric Algebra (2013.1)
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Why I never heard about GA before?
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Geometric algebra is a “new” formalism
Paul Dirac (1902-1984) William R. Hamilton (1805-1865)
Hermann G. Grassmann (1809-1877)
William K. Clifford (1845-1879)
1840s
1877
1878
David O. Hestenes (1933-) Wolfgang E. Pauli (1900-1958)
1980s, 1920s 2001
D. Hestenes (2001) Old wine in new bottles..., in Geometric algebra with applications in science and engineering, chapter 1, pp. 3-17, Birkhäuser, Boston.
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Differences Between Algebras
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Clifford algebra Developed in nongeometric directions Permits us to construct elements by a universal addition Arbitrary multivectors may be important
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Geometric algebra The geometrically significant part of Clifford algebra Only permits exclusively multiplicative constructions o The only elements that can be added are scalars, vectors, pseudovectors, and pseudoscalars
Only blades and versors are important Introduction to Geometric Algebra (2013.1)
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