Computer Graphics. Summer Elif Tosun

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1 Computer Graphics Summer 07 Elif Tosun

2 E.T. 06 Announcements Please Register ASAP!! Office Hours Mailing List Book Suggestions/Ideas Hw 0(setup)

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5 Grading Weekly/Bi-weekly assignments (~8) Mini project Extra credits Programming Skills Important! -- Style Guide! Late hws -- not accepted extension? - request one at least 3 days before deadline

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7 E.T. 06 Lecture 1 Intro to Graphics Geometry Review

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10 Entertainment Animation Applications Pixar

11 Entertainment Animation, Games Applications

12 Applications Entertainment Animation, Games, Special Effects

13 Entertainment CAD/CAM Conceptual Design Applications

14 Applications Entertainment CAD/CAM Conceptual Design, Simulation

15 Applications Entertainment CAD/CAM Conceptual Design, Simulation, Architectural Design

16 Entertainment CAD/CAM Scientific Visualization vector fields, Applications

17 Entertainment Applications CAD/CAM Scientific Visualization vector fields, weather simulation

18 Entertainment Applications CAD/CAM Scientific Visualization vector fields, weather simulation, molecular representations

19 Entertainment Applications CAD/CAM Scientific Visualization vector fields, weather simulation, molecular representations, mathematical functions

20 Entertainment Applications CAD/CAM Scientific Visualization Medicine Visible Human,MRI scans

21 Applications Entertainment CAD/CAM Scientific Visualization Medicine System Visualization!""!#$%.7%8)9:

22 E.T. 06 Graphics Modeling and Animation Rendering Image Processing user interaction, virtual reality, 3d scanning...

23 E.T. 06 Modeling Create the environment using - primitives (shapes) points, lines, curves, surfaces...

24 E.T. 06 Modeling Create the environment using - primitives (shapes) points, lines, curves, surfaces... - attributes (appearance) color, texture, lighting

25 E.T. 06 Modeling Create the environment using - primitives (shapes) points, lines, curves, surfaces... - attributes (appearance) color, texture, lighting - geometric transformations affine, non-affine view direction, position..

26 E.T. 06 Make them move parameters how shapes, attributes, positions change Physically based Animation graphics.ucsd.edu/~henrik/papers/fire/

27 E.T. 06 Make them move parameters how shapes, attributes, positions change Physically based Autonomous motion planning Animation Mark Overmars

28 E.T. 06 Make them move parameters how shapes, attributes, positions change Physically based Autonomous motion planning Motion Capture Animation

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32 .'(/'-)(0 123*)4*567*'/8%*)9:;7<'%;)02<%=-,=707<),( >9=)-)47;8%:*'%<-)7;%7(/%'--,-%<,%0'<%=)4<:-'* & E.T. 06 Boivin&Gagalowicz

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34 Graphics APIs Application Program Interface software interface that provides a model for how an application program can access system functionality Java3D graphics toolkit + user interface toolkit OpenGL, Direct3D GLUT

35 Geometry Review

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40 E.T. 06 Dot Product (w. v) = dot product of two vectors w and v

41 E.T. 06 Dot Product Used to compute projections

42 E.T. 06 Dot Product Used to compute projections angles "

43 E.T. 06 Dot Product Used to compute projections angles lengths

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49 E.T. 06 Determinants A scalar assigned to a square matrix, a measure Useful in analysis and solution of systems of equations Each matrix ==> system of equations

50 E.T. 06 A = a 11 a a 1n a 21 a a 2n. a n1 a n2... a nn x = x 1 x 2. x n Ax = 0 a 11 x 1 + a 12 x a 1n x n = 0 a 21 x 1 + a 22 x a 2n x n = 0. a n1 x 1 + a n2 x a nn x n = 0

51 mxn nx1 x = 0 m<n : under determined no unique solution x = 0 m>n : over determined E.T. 06 x = 0 m=n : square existence of a non-trivial solution depends on the determinant

52 E.T. 06 For a square matrix Non-trivial solution to Ax = 0 exists iff det = 0 Has an inverse iff det!= 0. (unique trivial soln) det = 0 : singular(infinitely many solutions)

53 Computation of Determinant Cofactor Expansion : write an nxn determinant in terms of (n-1)x(n-1) determinants Minors and Cofactors Minor Mij = (n-1)x(n-1) submatrix acquired by removing row i, column j. Cofactor k ij = ( 1) i+j det(m ij ) E.T

54 Computation of Determinant Cofactor Expansion : write an nxn determinant in terms of (n-1)x(n-1) determinants Minors and Cofactors Minor Mij = (n-1)x(n-1) submatrix acquired by removing row i, column j. Cofactor k ij = ( 1) i+j det(m ij ) E.T. 06 Row Cofactor Theorem : For any row i of an nxn matrix A n det(a) = a ij k ij j=1

55 E.T. 06 Computation of Determinant 1x1 2x2 a 11 = a 11 a 11 a 12 a 21 a 22 = a 11a 22 a 12 a 21 3x3 a 11 a 12 a 13 a 21 a 22 a 23 = a 11 a 22 a 23 a 32 a 33 a 31 a 32 a 33 a 12 a 21 a 33 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 = a 11 (a 22 a 33 a 23 a 32 ) a 12 (a 21 a 33 a 23 a 31 ) + a 13 (a 21 a 32 a 22 a 31 )

56 Back to the Cross Product... v w = (v x e x + v y e y + v z e z ) (w x e x + w y e y + w z e z ) =... = (v y w z v z w y )e x + (v z w x v x w z )e y + (v x w y v y w x )e z = e x e y e z v x v y v z w x w y w z ey E.T. 06 right handed coord sys ez ex

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58 E.T. 06 More Properties Scalar Triple Product a (b c) = a x (b y c z b z c y ) + a y (b z c x b x c z ) + a z (b x c y b y c x ) = a x a y a z b x b y b z c x c y c z a (b c) = b (c a) = c (a b)

59 E.T. 06 More Properties Triple Vector Product!! a (b c) = (a c)b (a b)c Four - Vector Product (a b) (c d) = (a c)(b d) (a d)(b c)

60 E.T. 06 Outer (Tensor) Product Given two vectors a,b ( a. b ) = scalar ( a x b) = vector ( a b) = matrix a b = a T b = a 1 ba 12 ca 13 (b 1, b 2, b 3 ) = a 1b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3

61 E.T. 06 Basis From a Vector Given a vector a, find vectors u, v, w s.t. u in same direction as a and {u, v, w} form an orthonormal basis u = a/ a need a vector t not collinear with u: set t=u, then change smallest component of t to 1 then w = (t x u)/ t x u and v = u x w

62 E.T. 06 Basis From a Vector Given a vector a, find vectors u, v, w s.t. u in same direction as a and {u, v, w} form an orthonormal basis u = a/ a need a vector t not collinear with u: set t=u, then change smallest component of t to 1 then w = (t x u)/ t x u and v = u x w Note that this is NOT unique!

Lecture 2. Determinants. Ax = 0. a 11 x 1 + a 12 x a 1n x n = 0 a 21 x 1 + a 22 x a 2n x n = 0

Lecture 2. Determinants. Ax = 0. a 11 x 1 + a 12 x a 1n x n = 0 a 21 x 1 + a 22 x a 2n x n = 0 A = a 11 a 12... a 1n a 21 a 22... a 2n. a n1 a n2... a nn x = x 1 x 2. x n Lecture 2 Math Review 2 Introduction to OpenGL Ax = 0 a 11 x 1 + a 12 x 2 +... + a 1n x n = 0 a 21 x 1 + a 22 x 2 +... + a 2n