1. Projective Geometric Computing Speaker: Ambjörn Naeve, Ph.D., Senior researcher Affiliation: Centre for user-oriented IT-Design (CID) Dept. of Numerical Analysis and Computing Science Royal Institute of Technology (KTH) 100 44 Stockholm, Sweden Research: Ambjörn is presently investigating principles for the development of interactive learning environments with a focus on mathematics. email-address: amb@nada.kth.se web-sites: www.nada.kth.se/cid cid.nada.kth.se/il

2. Projective Drawing Board (PDB) PDB is an interactive program for doing plane projective geometry that will be used to illustrate this lecture. PDB has been developed by Harald Winroth and Ambjörn Naeve as a part of Harald’s doctoral thesis work at the Computational Vision and Active Perception (CVAP) laboratory at KTH. PDB is avaliable as freeware under Linux. www.nada.kth.se/~amb/pdb-dist/linux/pdb2.5.tar.gz

5. Geometric algebra in n-dim Euclidean space A k-blade = blade of grade k : Geometric algebra: has 2 n dimensions. Underlying vector space V n with ON-basis e 1,...,e n. A multivector is a sum of k-vectors: A k-vector is a sum of k-blades: are linearly independent. Hence: Note: the grade of a blade is the dimension of the subspace that it spans.

6. Blades correspond to geometric objects 1 line segments 2 surface regions 3 3-dim regions length area volume k k-dim regions k-volume... blade of grade equivalence class of directed equal orientation and

7. Some equivalence operations on 2-blades a ^ ⁄ a b = ⁄ aa ^ aa ^ = a b = bb ^ b ^ ⁄ b = b ^ a b a R b R ⁄ abb R a R ⁄ = b a ⁄ a b

8. The dual of a multivector x : A pseudoscalar: A unit pseudoscalar: The bracket of P : Pseudoscalars and duality Def: A n-blade in G n is called a pseudoscalar. is a (n-k)-blade.Note: Notation: If A is a k-blade, then

9. The subspace of a blade To every non-zero m-blade Fact: there corresponds a m-dim subspace If is an ON-basis for then and iffor i = 1,…,m, Fact: with..

10. Dual subspaces orthogonal complements If A is a non-zero m-blade Fact:. We can WLOG choose an ON-basis for V n Proof: such that We then have which implies that. and.

11. The join and the meet of two blades Def: The join of blades A and B is a common dividend of lowest grade. Def: The meet of blades A and B is a common divisor of greatest grade. such that we say that A is a dividend of B and B is a divisor of A. The join and meet provide a representation in geometric algebra of the lattice algebra of subspaces of V n. Given blades A and B, if there exists a blade C Def:

12. Join of two blades sum of their subspaces For two blades A and B with we can define:. Def: and In this case:. a b Example:

13. Meet of blades intersection of subspaces. then: Def: If blades and In this case:. Note:The meet product is related to the outer product by duality:

14. Dualisation: ~ Dual outer product: ⁄ ⁄ ~~ ~ Dual outer product ~

15. Example:,, Hence:

16. eye 1-dim subspace through the eye point at infinity horizon 1-d subspace parallell to the ground plane ground plane point Projective geometry - historical perspective parallell lines line at infinity

17. A point p is a 1-dim subspace n-dimensional projective space P n = the set of non-zero subspaces of V n+1. A line l is a 2-dim subspace (spanned by a 1-blade a ). (spanned by a 2-blade B 2 ).. Let B denote the set of non-zero blades of the geometric algebra G(V n+1 ).. B ΠB ΠB. Hence we have the mapping

18. a b The projective plane P 2 Line:

19. The intersection of two lines in P 2 A B { non-zero 2-blades in G 3 }.

20. p q r Collinear points The points p, q, r are collinear. if and only if

21. Q P R Concurrent lines The lines P, Q, R are concurrent. if and only if

22. Desargues’ configuration

23. Desargues’ configuration (cont.) Leads to: = 0 if and only if p,q,r are collinear P,Q,R are concurrent

24. p, q, r are collinear iff P, Q, R are concurrent. Desargues’ theorem P Q R B A C A’ C’ B’ L

25. Let be five given points in Consider the second degree polynomial given by. It is obvious that and easy to verify that Pascal’s theorem. Hence: must be the equation of the conic on the 5 given points..

26. Verifying that p ( a´ ) = 0 same point same line

27. The three points of intersection of opposite sides of a hexagon inscribed in a conic are collinear. Pascal’s theorem (cont.) Geometric formulation:. Hence, a sixth point lies on this conic if and only if which Blaise Pascal discovered in 1640, at the age of 16. This is a property of the hexagrammum mysticum,

28. Pascal’s theorem (cont.) 1 2 3 4 5 6 ⁄ 1 4 = ⁄ 2 5 = ⁄ 3 6 =

29. P Q R R’ P’ Q’ L Pascal’s theorem (cont.) The three points of intersection of opposite sides of a hexagon inscribed in a conic are collinear.

30. P Q R P’ Q’ R’ L If the conic degenerates into two straight lines, Pappus’ theorem emerges as a special case of Pascal’s. Pappus’ theorem (ca 350 A.D.)

31. Outermorphisms A mapping is called an outermorphism if (i) (ii) (iii) (iv) Def: f is linear.

32. The induced outermorphism denote a linear mapping.Let T induces an outermorphism given by and linear extension., Fact: T maps the blades of V Interpretation: T maps the vectors of V. in accordance with how

33. denote a symmetric linear map,Let The corresponding quadric (hyper)surface Q in. The polar of the k-blade A with respect to Q Polarization with respect to a quadric in P n which means that. Def: is the (n+1-k)-blade defined by is given by.,

34. Polarization (cont.) For a blade A we have Note: In this case and polarization becomes identical to dualization.. Fact: (i) (ii) If A is tangent to Q then Pol Q ( A ) is tangent to Q. If x is a point on Q, then Pol Q ( x ) is the hyperplane Especially: which is tangent to Q at the point x.

38. Polarity with respect to a conic The polar of the join of x and y is the meet of the polars of x and y = =

40. i.e. the point y lies on the polar of the point x Hence: Polar reciprocity Letrepresent two points in. Then we have from the symmetry of : ~. ~ if and only if x lies on the polar of y.

41. 1 4 2 5 3 6 6 3 2 5 1 4 Brianchon’s theorem Pascal line (1640) Brianchon point (1810)

42. The dual map Let be linear, and assume that GG f GG ~~ GG ~ ~ ~ Def: The dual map is the linear map given by Note:.

43. Polarizing a quadric with respect to another Fact: Letand be symmetric outermorphisms, be the corresponding two quadrics. Polarizing the multivectors of the quadric P with respect to the quadric Q gives a quadric R and we have with equation., and let

44. P Q R Q : P : R : Polarizing the quadric with respect to the quadric generates the quadric The reciprocal quadric

45. Cartesian-Affine-Projective relationships x x * x ΠVnVn V n+1 VnVn y Πy * ΠVnVn x * * y * y e1e1 enen VnVn Cartesian space x * The affine part of Projective Space: Πy Πwell defined V n+1 e n+1 Affine space A n x * ΠA n VnVn VnVn

46. The intersection of two lines in the plane A B C D P a,b,c,da,b,c,d * ( ) A,B,C,DA,B,C,D p p ⁄ ⁄ = (AB)B) ⁄ (C(CD)D) P d c b a Affine plane e3e3 Cartesian plane R2R2

47. The intersection of two lines (cont.) 0 = contain e 3 does not

48. The intersection of two lines (cont.) In the same way we get Hence we can write p as: contain e 3 does not contain e 3 does not

49. r t = t(s+ds)t1t1 u u1u1 = -t(p-m)t = - t 1 (p-m 1 )t 1 m(s+ds) = m(s) = s M lim ds Æ0 ⁄ ⁄ (m(m u)u) ⁄ (m1(m1 u1)u1) = = m(s). p q m m1m1 in-caustic out-caustic curvature centre of the mirror Reflection in a plane-curve mirror

50. Reflection in a plane-curve mirror (cont.) Making use of the intersection formula deduced earlier we get and introducingfor the unit mirror normal This is an expression of Tschirnhausen’s reflection law.

51. Reflection in a plane-curve mirror (cont.) u r p q m v Tschirnhausens reflection formula

52. Hestenes, D. & Ziegler, R., Projective Geometry with Clifford Algebra, Acta Applicandae Mathematicae 23, pp. 25-63, 1991. Naeve, A. & Svensson, L., Geo-Metric-Affine-Projective Unification, Sommer (ed.), Geometric Computations with Clifford Algebra, Chapter 5, pp.99-119, Springer, 2000. Winroth, H., Dynamic Projective Geometry, TRITA-NA-99/01, ISSN 0348-2953, ISRN KTH/NA/R--99/01--SE, Dissertation, The Computational Vision and Active Perception Laboratory, Dept. of Numerical Analysis and Computing Science, KTH, Stockholm, March 1999. References: