3D Coordinate Systems 3D Transformation o 3D computer graphics involves the additional dimension of o The translation, scaling and rotation transformations used for 2D depth, allowing more realistic representations of 3D objects can be extended to three dimensions in the real world o In 3D, each transformation is represented by a 4x4 matrix o There are two possible ways of “attaching” the Z-axis, which o Using homogeneous coordinates it is possible to represent each gives rise to a left-handed or a right-handed system type of transformation in a matrix form and integrate transformations into one matrix

o To apply transformations, simply multiply matrices, also easier in hardware and software implementation

o Homogeneous coordinates can represent directions

o Homogeneous coordinates also allow for non-affine transformations, e.g., perspective projection

Homogeneous Coordinates Basic 2D Transformations

o In 2D, use three numbers to represent a point o Translation:  1 0 b x    0 1 b y o (x,y) = (wx,wy,w) for any constant w≠0   0 0 1  o To go backwards, divide by w, (x,y) becomes (x,y,1) sx 0 0 o Scaling:  0 s 0  y  o Transformation can now be done with matrix multiplication  0 0 1 x′ a a b x    xx xy x   cosθ − sinθ 0      y′ = a a b y sinθ cosθ 0    yx yy y   o Rotation:    0 0 1  1   0 0 1 1  

1 Translation and Scalling Matrice Translation o The translation and scaling transformations may be represented in 3D as follows:

1 0 0 trX Translation 0 1 0 trY original matrix 0 0 1 trZ 0 0 0 1

S 0 0 0 Scaling X matrix 0 SY 00 0 0 SZ 0 0 0 0 1 V=(ai + bj +ck) == (a, b, c) translation along y, or V = (0, k, 0)

Scaling 3D Shearing

Shearing:

The change in each coordinate is a linear combination of all three

Transforms a cube into a general parallelepiped Original scale all axes scale Y axis offset from origin x′ 1 a b 0x x′ a 0 0 0x y′ c 1 d 0y y′ 0 b 0 0y   =      =    z′ e f 1 0z z′ 0 0 c 0z            1  0 0 0 11  1  0 0 0 11

2 Rotation Major Axis Rotation Matrices

o In 2D, rotation is about a point 1 0 0 0 0 cosθ − sinθ 0 o about X axis R =   o In 3D, rotation is about a vector, which can be done through x 0 sinθ cosθ 0 rotations about x, y or z axes   0 0 0 1  cosθ 0 sinθ 0 Rotations are o Positive rotations are anti-clockwise, negative rotations are o about Y axis orthogonal matrices,  0 1 0 0 clockwise, when looking down a positive axis towards the R =   preserving distances and y − θ θ origin  sin 0 cos 0 angles.    0 0 0 1 y y y o about Z axis cosθ − sinθ 0 0  sinθ cosθ 0 0 x x =   x R z  0 0 1 0   0 0 0 1 z z z  

Rotation Rotation Axis

o In general rotation vector does not pass through origin

rotation of 45o about the Z axis

offset from origin translation then rotate about Z axis

3 Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis

o Rotation about an Arbitrary Axis o Step 1. Translation

y BasicBasic IdeaIdea y

1. Translate (x1, y1, z1) to the origin TT 1. Translate (x1, y1, z1) to the origin (x2,y2,z2) (x2,y2,z2) 1 0 0 − x  2.2. Rotate Rotate (x’ (x’2, ,y’ y’2, ,z’ z’2) )on on to to the the z z axis axis 1 R 2 2 2   R 3. Rotate the object around the z-axis 0 1 0 − y 3. Rotate the object around the z-axis T =  1  TR 4. Rotate the axis to the original 1 1 1 − (x1,y1,z1) 4. Rotate the axis to the original (x ,y ,z ) 0 0 1 z1  orientation   -1 orientation 0 0 0 1 x RR-1 x   5.5. Translate Translate the the rotation rotation axis axis to to the the

z -1 originaloriginal position position z TT-1

−1 −1 −1 [T ] = [T ] [T (α )] [T ()φ ] [T ()θ ][T ()φ ][T (α )][T ] Rarb TR Rx Ry Rz Ry Rx TR

Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis

o Step 2. Establish o Step 3. Rotate about y axis by φ [T α ] Rx ( ) y b b y a d sin α = = sin φ = , cosφ = b2 + c2 d l l l 2 = a2 + b2 + c2 = a 2 + d 2 (0,b,c) α = c = c (a,b,c) cos (a,b,c) b2 + c2 d d = b2 + c2

Projected α Projected l φ − φ − Point α 1 0 0 0 1 0 0 0 Point cos 0 sin 0 d / l 0 a / l 0 0 cosα − sinα 0 0 c / d − b / d 0 d  0 1 0 0  0 1 0 0 x []=   =   φ x []=   =   TR (α ) T ()φ x 0 sinα cosα 0 0 b / d c / d 0 Ry φ φ (a,0,d) sin 0 cos 0 a / l 0 d / l 0 z         0 0 0 1 0 0 0 1  0 0 0 1  0 0 0 1 Rotated Rotated Point z Point

4 Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis o Step 4. Rotate about z axis by the desired angle θ o Step 5. Apply the reverse transformation to place the axis back in its initial position y y 1 0 0 x1 1 0 0 0   α α  −1 −1 −1 0 1 0 y1 0 cos sin 0 [T ] [T ()α ] [T ()φ ] =    TR Rx Ry   − α α  cosθ − sinθ 0 0 0 0 1 z1 0 sin cos 0 l    sinθ cosθ 0 0 0 0 0 1 0 0 0 1 []T ()θ =   φ φ Rz  cos 0 sin 0  0 0 1 0   l 0 1 0 0 x   l   θ  0 0 0 1 x − sinφ 0 cosφ 0    0 0 0 1

z z −1 −1 −1 [T ] = [T ] [T (α )] [T ()φ ] [T ()θ ][T ()φ ][T (α )][T ] Rarb TR Rx Ry Rz Ry Rx TR

Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis

o Step1. Translate point A (2,1,0) to the origin Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1).

y 1 0 0 − 2   B’(1,2,1) 0 1 0 −1 [T ] =   TR 0 0 1 0  A’(0,0,0)   x 0 0 0 1  z

5 Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis o Step 2. Rotate axis A’B’ about the x axis by and angle α, until it lies o Step 3. Rotate axis A’B’’ about the y axis by and angle φ, on the xz plane. until it coincides with the z axis.

α = 2 = 2 = 2 5 sin φ = 1 = 6 22 +12 5 5 sin y 6 6 1 5 cos α = = 5 30 y 5 5 cos φ = = 6 6 Projected point (0,2,1) B’(1,2,1) = 2 + 2 + 2 = l 1 2 1 6   l 30 − 6 α l φ x  0 0 1 0 0 0 (0,0,√6)  6 6  x  5 2 5  √ []= 0 1 0 0 − B”(1,0, 5) TR ()φ   0 0 y  6 30  z √ []= 5 5 0 0 B”(1,0, 5) TR ()α     x 2 5 5 z 6 6 0 0  0 0 0 1  5 5    0 0 0 1

Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis ° o Step 4. Rotate the cube 90 about the z axis 1 0 0 0   30 6  − 1 0 0 2 5 2 5  0 00 1 0 0 0 0 6 6 − 0 1 0 1  1 0 0 0 0 1 0 0 [T ] =   5 5  0 1 0 0  Rarb 0 0 1 0 2 5 5 6 30 0 0 1 0   0 − 0− 0 0 1 0 0 0        0 0 0 1 5 5 6 6 0 0 0 1 []T ()° = 0 0 0 1 0 0 0 1 Rz 90 0 0 1 0     1 0 0 0 30 − 6 −    0 0 5 2 5 1 0 0 2 0 0 0 1 6 6 0 − 0     0 1 0 −1  0 1 0 0 5 5   6 30 2 5 5 0 0 1 0   0 00 0 Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,     6 6 5 5 0 0 0 1   0 0 0 10 0 0 1 − − − 1 1 1 0.166 − 0.075 0.983 1.742 [T ] = [T ] [T ()α ] [T ()φ ] [T ][T ()φ ][T ()α ][T ]   Rarb TR Rx Ry R ()90o Ry Rx TR z  0.742 0.667 0.075 −1.151 =   − 0.650 0.741 0.167 0.560     0 0 0 1 

6 Rotation about an Arbitrary Axis Rotation about an Arbitrary Axis o Multiplying [TR]AB by the point matrix of the original cube o Reflection Relative to the xy Plane

 x' 1 0 0 0 x  * y y [P ]=[T ]⋅[]P  y' 0 1 0 0 y Rarb   =    z  z' 0 0 − 1 0 z       1 0 0 0 1 1  0.166 − 0.075 0.983 1.742 0 0 1 1 0 0 1 1 z x x       0.742 0.667 0.075 −1.1511 1 1 1 0 0 0 0 []P* =    o Z-axis Shear − 0.650 0.741 0.167 0.560 1 0 0 1 1 0 0 1     0 0 0 1 1 1 1 1 1 1 1 1  x' 1 0 a 0 x   2.650 1.667 1.834 2.816 2.725 1.742 1.909 2.891   y' 0 1 b 0 y   =    − 0.558 − 0.484 0.258 0.184 −1.225 −1.151 − 0.409 − 0.483 =    z' 0 0 1 0 z   1.467 1.301 0.650 0.817 0.726 0.560 − 0.091 0.076          1  0 0 0 11   1 1 1 1 1 1 1 1 

Q1 - Translate by <1, 1, 1> Q2- Rotate by 45 degrees about x axis

o A translation by an offset (tx, ty, tz) is achieved o So to rotate by 45 degrees about the x-axis, we using the following matrix: use the following matrix: 1 0 0 0 1 0 0 tx  1 0 0 0  1 − 1  0 1 0 t  o So to translate by a vector   0 0 =  y  0 cos 45 − sin 45 0  2 2  M T (tx ,ty ,tz ) (1, 1, 1), the matrix is simply: R (45) =   =   0 0 1 t  x 1 1 z 0 sin 45 cos 45 0 0 0     0 0 0 1  1 0 0 1 0 0 0 1  2 2    0 0 0 1 0 1 0 1   M (1,1,1) =   T 0 0 1 1   0 0 0 1

7 Q3 - Rotate by 45 about axis <1, 1, 1> Q3 - Arbitrary Axis Rotation o So a rotation by 45 degrees about <1, 1, 1> can o The composite transformation can then be be achieved by a few succesive rotations about obtained as follows: the major axes. Which can be represented as a M (1,1,1) = R −1 (α) • R −1 (β ) • R (θ )• R (β )• R (α) single composite transformation R y x z x y

− − − = 2 + 2 = =  cos( 45) 0 sin( 45) 0 1 0 0 0 cos(45) sin(45) 0 0 d n n 2 1.414  0 1 0 0 0 cos(−35.2) −sin(−35.2) 0 sin(45) cos(45) 0 0 x z       n = 1 = −sin(−45) 0 cos(−45) 0 0 sin(−35.2) cos(−35.2) 0  0 0 1 0 x n       β = −1 y = −1 1 =  0 0 0 1 0 0 0 1  0 0 0 1 n = 1 SO tan tan 35.264 y d 2 1 0 0 0 cos( 45) 0 sin(45) 0 0 cos( 35.2) − sin(35.2) 0  −1 n −1 1 0 1 0 0 x nz = 1 α = x = =    ?? tan tan 45 0 sin(35.2) cos( 35.2) 0− sin(45) 0 cos( 45) 0 y n z 1    0 0 0 1 0 0 0 1 z

Directions vs. Points Homogeneous Directions o We have looked at transforming points o Translation does not affect directions! o Directions are also important in graphics o Homogeneous coordinates give us a clear way of handling o Viewing directions this, e.g., direction (x,y) becomes homogeneous direction o Normal vectors (x,y,0), and remains the same after translation: o Ray directions (1,1) (-2,-1) 1 0 bx x x 0 1 b y = y o Directions are represented by vectors, like points, and can  y     be transformed, but not like points 0 0 1 0 0 o Say we define a direction as the difference of two points:      d=a–b. This represents the direction of the line between two o (x, y, 0) is a vector, (x,y,1) is a point. points o The same applies to rotation and scaling, e.g., scaling o Now we translate the points by the same amount: changes the length of vector, but not direction a’=a+t, b’=b+t o Normal vectors are slightly different though (can’t always o Have we transformed d? use the matrix for points to transform the normal vector)

8 Alternative Rotations OpenGL Transformations o Specify the rotation axis and the angle (OpenGL method) o OpenGL internally stores two matrices that control viewing o Euler angles: Specify how much to rotate about X, then how of the scene much about Y, then how much about o The GL_MODELVIEW matrix for modelling and world to view o These are hard to think about, and hard to compose transformations oQuaternions o The GL_PROJECTION matrix captures the view to canonical o 4-vector related to axis and angle, unit magnitude, e.g., conversion rotation about axis (nx,ny,nz) by angle θ: o Mapping from canonical view volume into window space is through a glViewport function call (n cos()θ / 2 ,n cos ()θ / 2 ,n cos ()()θ / 2 ,sin θ / 2 ) x y z o Matrix calls, such as glRotate, glTranslate, glScale right multiply the transformation matrix M with the current matrix o Only normalized quaternions represent rotations, but you can C (e.g., identity matrix initially), resulting in CM - the last one normalize them just like vectors, so it isn’t a problem is the first applied o But we don’t want to learn all the maths about quaternions in this module, because we have to learn how to create a basic application before trying to make rotation faster

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