1. Projective 3D geometry

2. Singular Value Decomposition

3. Homogeneous least-squares Span and null-space Closest rank r approximation Pseudo inverse

4. Projective 3D Geometry Points, lines, planes and quadrics Transformations П ∞, ω ∞ and Ω ∞

5. 3D points in R 3 in P 3 (4x4-1=15 dof) projective transformation 3D point

6. Planes Dual: points ↔ planes, lines ↔ lines 3D plane Euclidean representation Transformation

7. Planes from points (solve as right nullspace of ) Or implicitly from coplanarity condition

8. Points from planes (solve as right nullspace of ) Representing a plane by its span

9. Lines Example: X -axis (4dof) two points A and B two planes P and Q

10. Points, lines and planes

11. Plücker matrices Plücker matrix (4x4 skew-symmetric homogeneous matrix) 1.L has rank 2 2.4dof 3.generalization of 4. L independent of choice A and B 5.Transformation Example: x -axis

12. Plücker matrices Dual Plücker matrix L * Correspondence Join and incidence (plane through point and line) (point on line) (intersection point of plane and line) (line in plane) (coplanar lines)

13. Plücker line coordinates (Plücker internal constraint) (two lines intersect)

14. Quadrics and dual quadrics ( Q : 4x4 symmetric matrix) 1.9 d.o.f. 2.in general 9 points define quadric 3.det Q=0 ↔ degenerate quadric 4.pole – polar 5.(plane ∩ quadric)=conic 6.transformation 1.relation to quadric (non-degenerate) 2.transformation

15. Quadric classification RankSign.DiagonalEquationRealization 44(1,1,1,1)X 2 + Y 2 + Z 2 +1=0No real points 2(1,1,1,-1)X 2 + Y 2 + Z 2 =1Sphere 0(1,1,-1,-1)X 2 + Y 2 = Z 2 +1Hyperboloid (1S) 33(1,1,1,0)X 2 + Y 2 + Z 2 =0Single point 1(1,1,-1,0)X 2 + Y 2 = Z 2 Cone 22(1,1,0,0)X 2 + Y 2 = 0Single line 0(1,-1,0,0)X 2 = Y 2 Two planes 11(1,0,0,0)X 2 =0Single plane

16. Quadric classification Projectively equivalent to sphere: Ruled quadrics: hyperboloids of one sheet hyperboloid of two sheets paraboloid sphere ellipsoid Degenerate ruled quadrics: conetwo planes

17. Hierarchy of transformations Projective 15dof Affine 12dof Similarity 7dof Euclidean 6dof Intersection and tangency Parallellism of planes, Volume ratios, centroids, The plane at infinity π ∞ The absolute conic Ω ∞ Volume

18. Screw decomposition Any particular translation and rotation is equivalent to a rotation about a screw axis and a translation along the screw axis. screw axis // rotation axis

19. The plane at infinity The plane at infinity π  is a fixed plane under a projective transformation H iff H is an affinity 1.canical position 2.contains directions 3.two planes are parallel  line of intersection in π ∞ 4.line // line (or plane)  point of intersection in π ∞

20. The absolute conic The absolute conic Ω ∞ is a fixed conic under the projective transformation H iff H is a similarity The absolute conic Ω ∞ is a (point) conic on π . In a metric frame: or conic for directions: (with no real points) 1.Ω ∞ is only fixed as a set 2.Circle intersect Ω ∞ in two points 3.Spheres intersect π ∞ in Ω ∞

21. The absolute conic Euclidean : Projective: (orthogonality=conjugacy) plane normal

22. The absolute dual quadric The absolute conic Ω * ∞ is a fixed conic under the projective transformation H iff H is a similarity 1.8 dof 2.plane at infinity π ∞ is the nullvector of Ω ∞ 3.Angles: