1. Review: Transformations
Need to transform points and vectors
Transform object data from object space to world space
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2. Transformations Modeling transformations
build complex models by positioning (transforming) simple components relative to each other
Viewing transformations
placing virtual camera in the world
transformation from world coordinates to camera coordinates
Perspective projection of 3D coordinates to 2D
Animation
vary transformations over time to create motion
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3. Transformations - Modeling
Object parts defined in their own object coordinate spaces
Transform them relative to each other to assemble object
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4. Transformations - Viewing
CAMERA
OBJECT
Object defined in object space is transformed into world space
Camera with its own local coordinate system is transformed into world space
Need object relative to camera coordinate system
WORLD
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5. Modeling Transformations
Transform objects/points
Transform coordinate system
Can think of transformation either as:
Transforming object relative to fixed coordinate system
transforming coordinate system that object is in (e.g. moving camera)
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6. Affine Transformations
Transform P = (x, y, z) to Q = (x’, y’, z’).
Affine transformation:
x’ = m11 x + m12 y + m13 z + m14
y’ = m21 x + m22 y + m23 z + m24
z’ = m31 x + m32 y + m33 z + m34
Math name of what we’re interested in: Affine transformations
Means: Linear equations.
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7. Translation Translation by (tx, ty, tz): x’ = x + tx; y’ = y + ty;
z’ = z + tz.
(tx, ty, tz)
Translation is an affine transformation.
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8. Scaling Scaling by (sx, sy, sz): x’ = sx x; y’ = sy y; z’ = sz z.
Scaling is an affine transformation.
Note: Location of lower left corner of house is changed
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9. Rotation Rotation counter-clockwise by angle  around the z-axis:
x’ = x cos() – y sin() ;
y’ = x sin() + y cos() ;
z’ = z. x z y   r
Proof:
x = r cos();
y = r sin();
x’ = r cos( + ) = r cos() cos() – r sin() sin() = x cos() – y sin();
y’ = r sin( + ) = r cos() sin() + r sin() cos() = x sin() + y cos();
USE RIGHT HAND RULE
Rotation is an affine transformation.
(x,y,z) =( r*cos(alpha), r*sin(alpha) , z)
Another viewpoint: Where do the points (1, 0, 0) and (0, 1, 0) go?
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10. Rotation around x-axis
Rotation counter-clockwise by angle  around the x-axis:
x’ = x;
y’ = y cos() – z sin() ;
z’ = y sin() + z cos() . z  x y
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11. Rotation around y-axis
Rotation counter-clockwise by angle  around the y-axis:
y’ = y;
z’ = z cos() – x sin() ;
x’ = z sin() + x cos(). Or
x’ = x cos() + z sin();
z’ = – x sin() + z cos(). x
Note: Here x plays role of y in rotation around z-axis.  y z
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12. Matrix Multiplication
Scaling by (sx, sy, sz):
x’ = sx x;
y’ = sy y;
z’ = sz z.
Rotation counter-clockwise by angle  around the z-axis:
x’ = x cos() – y sin() ;
y’ = x sin() + y cos() ;
z’ = z.
Translation by (tx, ty, tz):
x’ = x + tx;
y’ = y + ty;
z’ = z + tz.
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13. Homogeneous Coordinates
Represent P by (x, y, z, 1) and Q by (x’, y’, z’, 1).
(Homogeneous coordinates.)
Translation by (tx, ty, tz):
x’ = x + tx;
y’ = y + ty;
z’ = z + tz.
Scaling by (sx, sy, sz):
x’ = sx x;
y’ = sy y;
z’ = sz z.
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14. Rotation Matrices
Rotation counter-clockwise by angle  around the z-axis:
x’ = x cos() – y sin() ;
y’ = x sin() + y cos() ;
z’ = z.
Rotation counter-clockwise by angle  around the x-axis:
x’ = x;
y’ = y cos() – z sin() ;
z’ = y sin() + z cos() .
Rotation counter-clockwise by angle  around the y-axis:
x’ = x cos() + z sin();
y’ = y;
z’ = – x sin() + z cos().
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15. Affine Transformation Matrix
x’ = m11 x + m12 y + m13 z + m14
y’ = m21 x + m22 y + m23 z + m24
z’ = m31 x + m32 y + m33 z + m34
Transformation Matrix:
Translation in right colum
Scale along diagonal (uniform or non uniform)
Rotation elsewhere
Rigid transformations (w/ uniform scale)
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16. Shearing Shear along the x-axis: x’ = x + hy; y’ = y; z’ = z. y x z
Note change in location of lower left corner of house.
Can construct shear by combination of rotations and non-uniform scaling.
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17. Reflection Reflection across the x-axis: x’ = (-1) x; y’ = y; z’ = z.
Reflection is a special case of scaling!
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18. Elementary Transformations
Translation;
Rotation;
Scaling;
Shear.
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19. Compose Transformations
Rotate by 30° counter-clockwise around the z-axis; x z y
Scale by (2, 1, 1);
Translate by (20, 5, 0);
Translate by (0, -50, 0).
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20. Compose Transformation Matrices
Rotate by 30° counter-clockwise around the z-axis;
Rotate
Scale by (2, 1, 1);
Scale x z y
Translate by (20, 5, 0);
Translate
Translate by (0, -50, 0).
Translate
Multiply matrices together.
Real advantage of matrix multiplication/homogeneous coordinates.
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21. Compose Transformation Matrices
Scale by (2, 1, 1); x z y
Translate by (20, 5, 0);
Rotate by 30° counter-clockwise around the z-axis;
Translate by (0, -50, 0).
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22. Order Matters! Transformations are not necessarily commutative!
Translate by (20, 0, 0).
Rotate by 30.
Rotate by 30.
Translate by (20, 0, 0). x z y x z y
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23. Order of Transformation Matrices
Apply transformation matrices from right to left.
Translate by (20, 0, 0).
Rotate by 30.
Rotate by 30.
Translate by (20, 0, 0).
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24. Which transformations commute?
Translation and translation?
Scaling and scaling?
Rotation and rotation?
Translation and scaling?
Translation and rotation?
Scaling and rotation?
Uniform scaling and rotation commute.
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25. Elementary Transformations
Translation;
Rotation;
Scaling;
Shear.
Theorem: Every affine transformation can be decomposed into elementary operations.
Euler’s Theorem: Every rotation around the origin can be decomposed into a rotation around the x-axis followed by a rotation around the y-axis followed by a rotation around the z-axis. (or a single rotation about an arbitrary axis).
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26. Vectors Vector V = (Vx, Vy, Vz).
Homogeneous coordinates: (Vx, Vy, Vz, 0).
Affine transformation:
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27. Vectors
Rotation:
Scaling:
Translation:
(Note: No change.)
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28. Line Line through P0 = (x, y, z, 1) in direction V = (Vx, Vy, Vz, 0):
P(u) = P0T + u* VT
{(x0, y0, z0, 1) + u* (Vx, Vy, Vz, 0): u  Reals}.
Apply affine transformation matrix M to line P(u):
M *P(u)
{ M (P0T + u VT) : u  Reals}
= { M P0T + uM VT : u  Reals}
= { Q0 + u V’ : u  Reals}.
Theorem: Affine transformations transform lines to lines.
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29. Affine Transformation of Linesy y
Which points don’t change? z x z x
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30. Affine Combinations Q R b a P b a Q’ R’ P’
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31. Affine Combinations
Affine combinations of P = (x, y, z, 1) and Q = (x’, y’, z’, 1):
{R = a (x, y, z, 1) + b (x’, y’, z’, 1): a,b  Reals and a + b = 1}
Apply affine transformation matrix M to (a PT + b QT) gives:
M (a PT + b QT) = M a PT + M b QT
= a M PT + b M QT
= a P’ + b Q’ = MR = R’
Theorem: Affine transformations preserve affine combinations.
CIS 681

32. Affine Combinations
Theorem: The affine combination of points P and Q is the line through P and Q.
Affine combinations of P = (x, y, z, 1) and Q = (x’, y’, z’, 1):
{a (x, y, z, 1) + b (x’, y’, z’, 1): a,b  Reals and a + b = 1}.
a = 1- b so
a (x, y, z, 1) + b (x’, y’, z’, 1)
= (1-b) (x, y, z, 1) + b (x’, y’, z’, 1)
= (x, y, z, 1) + b ((x’, y’, z’, 1) – (x, y, z, 1))
= (x, y, z, 1) + b(x’-x, y’-y, z’-z, 0)
= P + b*V
Where V = Q-P
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33. Properties of Affine Transformations
Affine transformations map lines to lines;
Affine transformations preserve affine combinations;
Affine transformations preserve parallelism;
Affine transformations change volume by | Det(M) |;
Any affine transformation can be decomposed into elementary transformations.
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34. Coordinate Frame y j  i k z x
Coordinate frame is given by origin  and three mutually orthogonal unit vectors, i, j, k - defined in (x,y,z) space
Mutually orthogonal (dot products): i•j = ?; i•k = ?; j•k = ?.
Unit vectors (dot products): i•i = ?; j•j = ?; k•k = ?.
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35. Orientation Right handed coordinate system:
Left handed coordinate system: x z y y
(Into page) z x
(Out of page)
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36. Orientation
Right handed coordinate system:
Left handed coordinate system: k x z y i j k  x z y j i 
How could you test if the I,j,k coordinate frame is right-handed or left-handed
Cross product: i x j = ?
Cross product: i x j = ?
How do you test whether (i,j,k) is left handed or right handed?
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37. Coordinate Transformationsx z y i j k 
Given object data points defined in the (i, j, k, ) coordinate frame,
Given the definition of (i, j, k, ) in into (x, y, z, 0) coordinates,
How do you determine the coordinates of the object data points in the (x, y, z, 0) frame?
x,y,z,0 are mapped to i,j,k,, respectively.
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38. Coordinate change (Translation)b x z y
(x, y, z) a
(0,0,0) c
Change from (a,b,c,) coordinates to (x,y,z,0) coordinates:
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39. Coordinate change (Rotation)
(x, y, z) a b y x 
z-axis rotation by 
(0,0,0) z
Change from (a,b,c,) coordinates to (x,y,z,0) coordinates:
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40. Coordinate Transformationsx z y i j k 
Given (I,j,k,p) defined in the (x,y,z,0) coordinate frame,
Transform points in (i, j, k, ) coordinate frame to (x, y, z, 0) coordinate frame.
x,y,z,0 are mapped to i,j,k,, respectively.
Affine transformation matrix:
Maps (i, j, k, ) coordinates into (x, y, z, 0) coordinates!
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41. Composition of coordinate changey y’ M1 y”  z M2 x ’ x’ z’ z” x”
M1 changes from coordinate frame (x,y,z,) to (x’,y’,z’,’).
M2 changes from coordinate frame (x’,y’,z’,’) to (x’’,y’’,z’’,’’).
Change from coordinate frame (x,y,z,) to (x’’,y’’,z’’,0): ?
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42. Object vs. coordinate transformations
Translate and then rotate object: z  x y z  x y z  x y
Translate and then rotate coordinate frame: z  x y
z’’
’’
x’’
y’’ z  x y z’ ’ x’ y’ z  x y
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43. Object vs. coordinate transformations (2)
Translate and then rotate object: z  x y z  x y z  x y
Rotate and then translate coordinate frame:
z’’
’’
x’’
y’’ z  x y z  x y y’ z  x y x’ ’ z’
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44. Order of transformations
Let Mi be the transformation matrix for transformation Ti.
Sequence of object (point) transformations, T1, T2, T3.
Transformation matrix = M3 x M2 x M1.
Sequence of coordinate system transformations, T1, T2, T3.
Transformation matrix = M1 x M2 x M3.
Note: OpenGL, OpenInventor use coordinate system transformations.
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45. Transformations of normal vectors
n is a unit normal vector to plane .
M is an affine transformation matrix.
How is n transformed under:
translation?
rotation by ?
uniform scaling by s?
shearing or non-uniform scaling?
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46. Shear transformation n n’ n n is a unit normal vector to plane .
M is the shear transformation matrix.
M nT = n! Want n’.
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47. Transformations of normal vectors
Planar equation: a x + b y + c z + d = 0.
Let N = (a, b, c, d). (Note: n = (a, b, c) for some real .)
Let P = (x, y, z, 1) be a point in plane .
Planar equation: N • P = 0.
M is an affine transformation matrix. (MT is M transpose.)
Let P’ = M PT.
Find N’ such that N’ • P’ = 0.
N • P = 0;
N PT = 0;
N (M-1 M) PT = 0;
(N M-1) (M PT) = 0;
(N M-1) P’T = 0;
So N’ = NM-1
To put in column-vector form, (N’)T = (NM-1)T = (M-1)TNT
So N’ = ((M-1)T NT)T and (M-1)T is the transformation matrix for N.
Note: If M is a rotation matrix, (M-1)T = M. n P 
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