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Introduction to Algebra & Rigid-Body Motion

This lecture introduces algebra and explores the concept of rigid-body motion in 3D vision. Topics covered include Euclidean space, points and vectors, cross products, singular value decomposition, and velocity transformations.

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Introduction to Algebra & Rigid-Body Motion

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  1. Lecture 6 Introduction to Algebra & Rigid-Body Motion Allen Y. Yang September 18th, 2006 Invitation to 3D vision

  2. Outline • Euclidean space • Points and Vectors • Cross products • Singular value decomposition (SVD) • Rigid-body motion • Euclidean transformation • Representation • Canonical exponential coordinates • Velocity transformations Invitation to 3D vision

  3. Points and vectors are different! Bound vector & free vector: Euclidean space Invitation to 3D vision

  4. The set of all free vectors, V, forms a linear space over the field R. (points don’t) Closed under “+” and “*” V is completely determined by a basis, B: Change of basis: Linear space Invitation to 3D vision

  5. Change of basis Summary: Invitation to 3D vision

  6. Cross product between two vectors: Cross product • Properties: • Pop quiz: • Homework: Invitation to 3D vision

  7. Rank Pop Quiz: R is a rotation matrix, T is nontrivial. rank( )=? Invitation to 3D vision

  8. Singular Value Decomposition (SVD) Invitation to 3D vision

  9. Fixed-Rank Approximation Invitation to 3D vision

  10. A Geometric Interpretation Invitation to 3D vision

  11. To describe an object movement, one should specify the trajectory of all points on the object. For rigid-body objects, it is sufficient to specify the motion of one point, and the local coordinate axes attached at it. Rigid-Body Motion Invitation to 3D vision

  12. Rigid-body motions preserve distances, angles, and orientations. Goal: finding representation of SE(3). Translation T Rotation R Rigid-Body Motion Invitation to 3D vision

  13. Rotation Orthogonal change of coordinates Collect coordinates of one reference frame relative to the other into a matrix R Invitation to 3D vision

  14. Translation T has 3 DOF . Rotation R has 3 DOF. Can be specified by three space angles. Summary: R in SO(3) has 3 DOF. g in SE(3) has 6 DOF. Homogeneous representation Degree of Freedom (DOF) Invitation to 3D vision

  15. Points Vectors Transformation Representation Homogeneous representation (summary) Invitation to 3D vision

  16. Canonical Exponential Coordinates Invitation to 3D vision

  17. One such solution: Yet the solution is NOT unique! when w is a unit vector. Multiplication: Canonical Exponential Coordinates Invitation to 3D vision

  18. Canonical exponential coordinates for rigid-body motions. Similar to rotation: (twist) Hence, Canonical Exponential Coordinates Invitation to 3D vision

  19. Canonical Exponential Coordinates Twist coordinates Velocity transformations Given Invitation to 3D vision

  20. Summary Invitation to 3D vision

  21. We will prove this if we have time Invitation to 3D vision

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