Homogeneous Coordinates

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1 COMS W4172 3D Math 2 Steven Feiner Department of Computer Science Columbia Universit New York, NY Februar 1, Homogeneous Coordinates w X W Y X W Y 1 W w = 1 plane X X W W Point Y W Y, for nonzero W W W W 1 1 Vectors have the form Note: Vectors (points at infinit) reside in the w = plane 2

2 Homogeneous Transformations Given P, P' MP, where M is 1 1 d Translation Td (, d) 1 d 1 s Scale Ss (, s) s 1 cos sin Rotation R( ) sin cos 1 3 Composition of Transformations R(θ) rotates about the origin 1 To rotate about point P 1 = 1 Translate P 1 to the origin 1 Rotate b R(θ) Translate origin back to P 1 T( 1, 1) R( ) T( 1, 1) 1 1 cos sin sin cos cos sin 1(1 cos ) 1sin sin cos 1(1cos ) 1sin 1 Rotate about origin P 1 Original house Rotate about P 1 After translation of P 1 to origin After rotation b θ After translation back to original P 1 4

3 Eample: Transforming a Template To scale and rotate a template about point P 1 and position at point P 2 Translate P 1 to the origin Scale Rotate Translate origin to P 2 P 2 P 1 Original house Translate P 1 to origin Scale Rotate Translate to position P 2 5 Matri Multiplication Matri multiplication is Associative but not in general Commutative However, M 1 M 2 is commutative in the following 2D cases M 1 M 2 Translate Translate Scale Scale Rotate Rotate Scale, where s = s Rotate Scale Rotate, where θ = n * 18 for integral n 6

4 Rigid Bod Transformations Rigid bod transformations result from an sequence of rotations and translations Upper left 22 submatri rows a1 b1 ab are unit vectors (length = 1) ab a b a2 b2 are perpendicular (dot product = ) rotate into the and aes Upper left 22 submatri columns ab a b cos, are unit vectors (length = 1) are perpendicular (dot product = ) are vectors into which the and aes rotate a aa cosθ sinθ d' Upper left 22 matri sinθ cosθ d' is special orthogonal: 1 AA = I, det A = where θ is the angle between a and b 7 Affine Transformations Affine transformations are defined b an arbitrar sequence of rotation, scale, and translation transformations The preserve parallelism of lines but not length of lines angle between lines The matri is of the form rs11 rs12 d' rs21 rs22 d' 1 8

5 Shear Transformations 1 a a 1 a SH SH b 1 b b (,1) (1,1) (a,1) (1+a,1) Shears are affine SH (,) (1,) (,) (1,) (,1) (1,1+b) SH (,) (1,b) 9 3D Coordinate Sstems OpenGL Right-handed coordinate sstem Looking toward origin from + ais, a 9 CCW rotation takes one + ais into another, z, z Direct3D and Unit Left-handed coordinate sstem Looking toward origin from + ais, a 9 CW rotation takes one + ais into another, z, z z z 1

6 Direct3D vs. OpenGL and Unit Direct3D Points documented as row vectors Matrices written conventionall Stored in row-major order v' = v A BC LH coord ss (Direct3D) OpenGL and Unit Points documented as column vectors (Same vector as a point in Direct3D) Matrices written conventionall (Transpose of a matri in Direct3D) But, stored in column-major order! (Same arra as a matri in Direct3D) v' = C B A v RH coord ss (OpenGL) vs. LH coord ss (Unit) Wh? To maintain code compatibilit between original IRIS GL and later OpenGL! Vectors and arras are stored and processed identicall in each. Onl the documentation differs. 11 3D Points OpenGL and Unit 4 element column vector z 1 Direct3D and XNA 4 element row vector z 1 Note: Points in Unit are tpicall epressed as 3 element vectors Transformations in Unit are tpicall epressed as specific fields in the UI and functions in code Rotations are represented internall in Unit as quaternions 12

7 3D Transformations (for column vectors) 1 d 1 d Td (, d, dz) 1 dz 1 s s Ss (, s, sz) sz 1 R ( ) z R ( ) R ( ) cos sin sin cos cos sin sin cos 1 cos sin 1 sin cos D Rigid Bod Transformations Upper left 33 submatri rows are unit vectors (length = 1) are mutuall perpendicular (dot product = ) rotate into the,, and z aes Upper left 33 submatri columns are unit vectors (length = 1) are mutuall perpendicular (dot product = ) are vectors into which the,, and z aes rotate R ( ) z R ( ) R ( ) cos sin sin cos cos sin sin cos 1 cos sin 1 sin cos 1 14

8 3D Shear Transformations Representing shear relative to z,, and aes 1 sh 1 sh SH ( sh, sh ) sh 1 SHz( sh, shz ) shz sh 1 SHz( sh, shz ) shz More on Rotation Rotation matrices can accumulate error Rows/columns can grow/shrink to be other than unit length Rows/columns can skew so that the are no longer mutuall orthogonal 16

9 More on Rotation A rotation can also be represented b an ordered set of 3 rotations about a set of mutuall orthogonal aes E.g., roll, pitch, aw Can result in gimbal lock 17 Gimbal Lock 3 gimbals: 1, 2, 3 Gimbals are concentric mounting rings Each ring pivots on an ais to provide 1 degree of rotational freedom Rotation aes are orthogonal Gimbal lock occurs when 2 aes align 1 degree of rotational freedom is lost Platform containing inertial measurement unit (3 gros) 18

10 Gimbal Lock Apollo Inertial Measurement Unit (Museum of Flight) 19 Gimbal Lock Rotation about Y ais 2

11 Gimbal Lock Rotation about Z ais 21 Gimbal Lock Rotation about X ais Lost degree of rotational freedom 1 and 3 are aligned! Can t respond to roll 22

12 Gimbal Lock Eperiencing gimbal lock with a 3D tripod head 23 Gimbal Lock Eperiencing gimbal lock with a 3D tripod head Pitch Roll Adjust starting from object (camera) roll, pitch, aw around aes of parent s (tripod) coord ss Adjust starting from parent (tripod) aw, pitch, roll around aes of object s (camera) coord ss Yaw 24

13 Demo: The Perils of Roll, Pitch, & Yaw 25