3D Coordinate Transformations. Tuesday September 8 th 2015

Transcription

1 3D Coordinate Transformations Tuesday September 8 th 25

2 CS 4 Ross Beveridge & Bruce Draper Questions / Practice (from last week I messed up!) Write a matrix to rotate a set of 2D points about the origin by 45 degrees, scale it by a factor of 2 and translate it down four pixels and to the right pixel. 9/9/5 2

3 Ross Beveridge & Bruce Draper, CSU, 23 Homogeneous Coordinates in 3D Same basic idea as for 2D. Now transformations are 4x4 matrices. 3D points represented as 4 element vectors. Let us consider the standard transformation Translation Scaling Rotation Shear 9/9/5 3

4 Translation and Scaling 9/9/5 Ross Beveridge & Bruce Draper, CSU, 23 4 To transform a Point A into translated Point B. = T t x t y t z = A x y z t x t y t z x y z + x t x + y t y + z t z = S s x s y s z = A x y z s x s y s z x y z = s x x s y y s z z = To transform a Point A into scaled Point B.

5 Ross Beveridge & Bruce Draper, CSU, 23 Rotation Part : Euler Angles Rotations in 3D are more complex, because we now have three orthogonal axes that we can rotate around: These are called the Euler Angles y z x Trust me: Any rotation around any axis can be expressed as a sequence of three rotations about the x, y and z axes. 9/9/5 5

6 Ross Beveridge & Bruce Draper, CSU, 23 Rotation about Each Axis About the X axis Rx = cos ( α ) - sin ( α ) sin ( α ) cos ( α ) About the Y axis Ry = cos ( β ) - sin ( β ) sin ( β ) cos ( β ) About the Z axis Rz = cos ( γ ) - sin ( γ ) sin ( γ ) cos ( γ ) 9/9/5 6

7 To rotate around any axis, start with the empty template: The axis you rotate around won t change: Euler Angle Matrices - Pattern 9/9/5 Ross Beveridge & Bruce Draper, CSU, 23 7 (Z) (X) (Y)

8 Ross Beveridge & Bruce Draper, CSU, 23 Remembering. Fill the remaining slots with the entries of the 2D rotation matrix: % " # " cos x cos( x) sin( x) % sin x # sin( x) cos( x) " & cos( x) sin( x) (Z) sin( x) cos( x) " # (X) # ( ) sin( x) ( ) cos( x) & % (Y) & cos( x) sin( x) sin x ( ) cos( x) % & 9/9/5 8

9 Ross Beveridge & Bruce Draper, CSU, 23 Composition of Rotations Rotate about x, then y, then z. R = cos ( γ ) - sin ( γ ) sin ( γ ) cos ( γ ) cos ( β) - sin ( β) sin ( β) cos ( β) cos ( α) - sin ( α) sin ( α ) cos ( α) cos ( γ ) cos ( β) - sin ( γ ) cos ( α ) - cos ( γ ) sin ( β ) sin ( α ) sin ( γ ) sin ( α ) - cos ( γ ) sin ( β ) cos ( α) R = sin ( γ ) cos ( β ) cos ( γ ) cos ( α ) + sin ( γ ) sin ( β ) sin ( α) - cos ( γ ) sin ( α ) + sin ( γ ) sin ( β ) cos ( α) - sin ( β ) cos ( β ) sin ( α ) cos ( β ) cos ( α) Order maders, consider z, then y, then x. R = cos ( γ ) cos ( β ) - sin ( γ ) cos ( β ) - sin ( β ) cos ( γ ) sin ( β ) sin ( α ) - sin ( γ ) cos ( α ) - sin ( γ ) sin ( β ) sin ( α ) + cos ( γ ) cos ( α ) - cos ( β ) sin ( α ) - cos ( γ ) sin ( β ) cos ( α ) - sin ( γ ) sin ( α ) sin ( γ ) sin ( β ) cos ( α ) + cos ( γ ) sin ( α ) cos ( β ) cos ( α ) Please do not memorize these! 9/9/5 9

10 Ross Beveridge & Bruce Draper, CSU, 23 Nice Flash Illustration 9/9/5

11 Ross Beveridge & Bruce Draper, CSU, 23 But Don t use Euler angles! Euler angles are the oldest and most widely known 3D angle system, but Not widely used in graphics Not very intuitive We will specify rotations differently, First we need Truly understand Rotation as Projection! Cross products and what they give us. 9/9/5

12 Ross Beveridge & Bruce Draper, CSU, 23 Three forms of the dot product General: A & B are arbitrary vectors A B = A B cos θ ( ) Projective: A is a unit vector A B = B cos θ ( ) = B proj onto A Angle: A & B both unit vectors A B = cos θ ( ) 9/9/5 2

13 Ross Beveridge & Bruce Draper, CSU, 23 Rotation, Think About Projection A space ship leaves Earth in the direcon (4,3). How far has it traveled when it passes closest to Mars? Earth is at posion (, ). Mars is at posion (5, ). 9/9/5 3

14 Ross Beveridge & Bruce Draper, CSU, 23 Projection Do the Math. P = 5 d = P ˆ U = = = = ( ) +( 3 5 )! U = 4 3 How far has it traveled when it passes closest to Mars? ˆ U = d = P ˆ U It has traveled units. (this is a smaller universe) 9/9/5 4

15 Ross Beveridge & Bruce Draper, CSU, 23 Spaceship is a Ruse - Rotate Mars! P = 5 P! = = u v ˆ U P ˆ V P = M P! U =!V P ˆ V = ˆ U = = v = P ˆ V u = P ˆ U = = 5 9/9/5 5

16 Ross Beveridge & Bruce Draper, CSU, 23 Another View of the Same 9/9/5 6

17 Ross Beveridge & Bruce Draper, CSU, 23 Learn from this Example! Rotation is projection onto new axes. Axes expressed as vectors. Unit length, mutually orthogonal. Matrix R contains one axes per row. R R T =I (pronounced orthonormal ) Rotated points are written as P =R P Finally, be able to picture projections 9/9/5 7

18 Copyright Bruce Draper & Ross Beveridge, 2 Cross-Product The cross product of two (3D) vectors is a new 3D vector that is perpendicular to both of the original vectors. A x B A B Right-hand Rule 9/6/ 8

19 More Formally Copyright Bruce Draper & Ross Beveridge, 2 V = V uv V 2 2 sinθ Where u is a unit vector perpendicular to both V and V 2, as determined by the right-hand rule If V=(V x, V y, V z ) then: V V 2 = ( V y V 2z V z V 2y,V z V 2x V x V 2z,V x V 2y V y V ) 2x 9/6/ 9

20 It s easier to compute by hand than it looks V V 2 = det x y z V x V y V z V 2x V 2y V 2z # % % % & ( ( ( = z y x z y x z y x z y x V V V V V V z y x V V V V V V z y x V V V V 2 = V y V 2z V z V 2y, V z V 2x V x V 2z, V x V 2y V y V 2x ( ) Axes Blue products minus red products.don t despair! 9/6/ Copyright Bruce Draper & Ross Beveridge, 2 2

21 Why introduce the cross-product now? Copyright Bruce Draper & Ross Beveridge, 2. To ensure the polygon vertices are coplanar 2. To help us specify 3D rotations 9/6/ 2

22 Copyright Bruce Draper & Ross Beveridge, 2 Coplanarity B Every polygon has at least 3 vertices: A,B,C A C 3 points define a plane, so the st 3 points are coplanar! But what plane? 9/6/ 22

23 Copyright Bruce Draper & Ross Beveridge, 2 Coplanarity (II) B AB lies in the plane AB = (B A) A C BC lies in the plane BC = (C B) The plane normal N = AB BC 9/6/ 23

24 Copyright Bruce Draper & Ross Beveridge, 2 Coplanarity (III) A B C A 4 th point D is coplanar with AB&C iff CD is perpendicular to N CD = (D C) CD N = ± round-off error The same for additional points 9/6/ 24

25 Copyright Bruce Draper & Ross Beveridge, 2 (Back to) 3D Rotation Remember the geometry of matrix multiplication: y axis au + bv a = cu+ dv c b d (u, v) u v v (a, b) (c, d) (c,d) ( u u,v) x axis 9/6/ 25

26 3D Rotation In P = MP, the points in P are projected onto the rows of M. In a rotation matrix: The rows are unit Otherwise it scales the data The rows are orthogonal Otherwise it shears the data To specify a rotation matrix, just specify the (orthogonal, unit) basis vectors of the new coordinate system! 26

27 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle Rotation In the special case that the axis is the Z axis, no problem: " # cos θ sin θ ( ) sin( θ) ( ) cos( θ) Where θ is the magnitude of the rotation But what about all the other possible axes? % & 9/6/ 27

28 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (II) General rule: if there is a special-case coordinate system that makes life easy, adopt that coordinate system! In this case:. Rotate data to make Z the angle of rotation 2. Rotate about Z 3. Apply the inverse of the original rotation 9/6/ 28

29 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (III) How do we rotate the data to make the angle of rotation Z? Multiplication is projection onto the rows of M If M is orthonormal, it is a rotation matrix Magnitude of every row is Dot product of every pair of rows is If the third row is the axis of rotation, Z becomes the axis of rotation! 9/6/ 29

30 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (IV) Step : normalize the axis of rotation Write the normalized axis as w = (w x, w y, w z, ) Step 2: pick any axis M not parallel to W Heuristic: pick the smallest term in w, set it to and renormalize to create m Step 3: create U = W M Step 3: pick an axis v perpendicular to w & u V = W U (or U W) 9/6/ 3

31 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (V) Now put those together in a rotation matrix:! # # R ω = # # # " u x u y u z v x v y v z w x w y w z & & & & & % 9/6/ 3

32 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (VI) To rotate by θ around ω: P! = ( R R R )P ω Zθ ω R ω is from the last slide R zθ is the rotation matrix about Z, by amount θ What about R -? 9/6/ 32

33 Copyright Bruce Draper & Ross Beveridge, 2 Axis Angle (V) Useful math fact: the inverse of an orthonormal matrix is its transpose P! = ( R T R R )P ω Zθ ω That s how you implement axis-angle rotation! 9/6/ 33

34 Ross Beveridge & Bruce Draper, CSU, 23 Which Brings Us To Programming Assignment #! 9/9/5 34