1. Transformations
Review
4/18/2018
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2. Rotations & Transformations
Output from Position Trackers
Creation of Coordinate System Graphs
Animation of Objects
Forward and Inverse Kinematics
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3. Review of 2D Rotation about the Origin
From basic trigonometry we know that:
sin(+) = y2/r; cos (+) = x2/r
sin() = y1/r; cos() =x1/r
From the double angle formulas in trigonometry we also know that:
sin(+) = sincos + cossin 
cos(+) = coscos - sinsin
Substituting:
y2/r = (y1/r)cos + (x1/r)sin
y2 = x1sin + y1cos
By a similar calculation:
x2 = x1cos - y1sin X Y
(0,0)
P2 = (x2,y2)
P1 = (x1,y1)  r  y2 y1 x2 x1
4/18/2018
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4. 2D Rotation about the Origin
y2 = x1sin + y1cos; x2 = x1cos - y1sin
2D Rotations are usually expressed as a matrix multiplication.
Negative Rotation (remember that cos(-) = cos() and sin(-) = -sin):
4/18/2018
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5. 3D rotations about an x,y, or z axis are trivial extensions of 2D rotations about the origin
For example, consider a rotation of 90 degrees about the x-axis. The y and z coordinates change but the x coordinate is not affected.
A 3D rotation about one of the major axes occurs in a 2D plane defined by the point’s coordinate relative to that axis (x = 0 plane in this case).
(0,d,0) Z
(0,0,d) X
4/18/2018
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6. 3D Rotation about the x axis
By looking at the 2D rotation matrix we can write down what the 3D rotation matrix about the x axis must look like:
4/18/2018
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7. In interactive computer graphics (including VR) we need to be able to make sense of:
Notation – How we write down an operation.
Implementation – How we code the operation.
Meaning – What the operation does to an object.
4/18/2018
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8. Cartesian Coordinate System+Z -Z -Y +Y -X +X
Left Handed -Z +Z -Y +Y -X +X
Right Handed
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9. Euclidean Space Scalars Points: P = (x,y,z) Vectors: V = [x,y,z]
Magnitude or distance ||V|| = (x2+y2+z2)
Direction
No position
Position vector
Think of as magnitude and distance relative to a point, usually the origin of the coordinate system
4/18/2018
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10. Review of Common Vector Operations in 3D
Addition of vectors
V1+V2 = [x1,y1,z1] + [x2,y2,z2] = [x1+x2, y1+y2, z1+z2]
Multiply a scalar times a vector
sV = s[x,y,z] = [sx,sy,sz]
Dot Product
V1V2 = [x1,y1,z1][x2,y2,z2] = [x1x2+y1y2+z1z2]
V1V2 = ||V1|| ||v2|| cos where  is the angle between V1 and V2
Cross Product of two Vectors
V1V2 = [x1,y1,z1][x2,y2,z2] = [y1z2-y2z1, x2z1-x1z2, x1y2-x2y1] = - V2V1
Results in a vector that is orthogonal to the plane defined by V1 and V2
4/18/2018
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11. Transformations Rotations Translations Scale
About the major axes of a coordinate system
About an arbitrary vector
Translations
Relative to a coordinate system
Scale
Relative to a coordinate system and about a fixed point
4/18/2018
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12. Translations are defined relative to the x,y, and z axes
Notation: P’ = Tx(5)P
Implementation
Meaning: Move the position of point P five units in a positive direction with respect to the x axis
New point is (x+5, y, z)
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13. Scales Notation: P’ = Sx(1/2)P Implementation
Meaning: Scale the point P by ½ in the x coordinate.
New point is (1/2x, y, z)
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14. Scales can move things! Consider a line between x = 2 and x = 6
Scale by ½ on each endpoint.
New line is ½ as long but also starts at a new place.
Usually want to guarantee that you scale relative to a fixed point that makes sense.
One end
Middle
Do this by combining translations with scales
Tx(-2)Sx(1/2)Tx(2) x 1 2 3 4 5 6
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15. Shears XY sheared on Z y x z
Of course other combinations occur too, e.g. ZX sheared on Y, X sheared on Y,
etc.
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16. Rotations are usually defined relative to the x,y and z axes (Euler angles)
Axis of rotation is Direction of positive rotation is
x y to z
y z to x
z x to y x y z
How to remember
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17. Rotation about the x axis
Notation P’ = Rx(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the x-axis y z x
4/18/2018
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18. Rotation about the y axis
Notation P’ = Ry(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the y-axis z x y
4/18/2018
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19. Rotation about the z axis
Notation P’ = Rz(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the z-axis x y z
4/18/2018
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20. Describing Orientation with Euler Angles
Orientation of an object in computer graphics is often described using Euler Angles.
Rx Ry Rz
Any axis order will work and could be used.
Yaw, Pitch & Roll
BUT, order makes a difference in the result.
A related method is to use a rotation matrix
4/18/2018
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21. Rotation About An Arbitrary Axisp2 u Z θ p1 Y
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22. Rotation About An Arbitrary Axis
Translate p1 to origin p2
T(-p1) u Z p1 u' Y
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23. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z u' uz c Y a ß uy b ux
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24. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z
u'' u' uz c Y a ß uy b ux
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, Larry F. Hodges

25. Rotation About An Arbitrary Axis
After Ry(-ß), angle α and u'' lies in the y-z plane.
Next, Rotate u'' to Z axis (u''') (rotation about X axis, Rx(α))
Rx(α) ∙ Ry(-β) ∙ T(-p1)
u'' Z a c uz Y α uy ux
4/18/2018
©Babu 2009 X
, Larry F. Hodges

26. Rotation About An Arbitrary Axis
After Rx(α), u'' lies in the Z axis as u'''.
Next, rotate about Z by θ (Rz(θ))
Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1)
u''' Z
u'' Y θ
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27. Rotation About An Arbitrary Axis
Now we’ve rotated about U.
Next, apply the inverse transformations to place u''' back on u p2
R((p1,p2),θ) = T(p1) ∙ Ry(β) ∙ Rx(-α) ∙ Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1) u Z θ p1 Y
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28. Rotation About An Arbitrary Axis
4/18/2018
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29. Problem with Euler Angles
What if we want to produce a smooth animation of a rotation from point P1 to Point P2 around some axis.
The entire rotation is defined as RxRyRz but how do we get an intermediate point?
Rotations are defined as rotations about the x, y and z axes. Rotations about any other vector in space have to be broken down into equivalent rotations about the major axes.
4/18/2018
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30. Smooth Rotation y
With Euler Angles, knowing the transformations for the entire rotation does not help us with the intermediate rotations. Each is a unique set of three rotations about the x,y and z axes z x
4/18/2018
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31. Problems with Euler Angles
Rotations not uniquely defined
ex: (z, x, y) = (90, 45, 45) = (45, 0, -45) takes positive x-axis to (1, 1, 1)
Cartesian coordinates are independent of one another, but Euler angles are not
Remember, the axes stay in the same place during rotations
Gimbal Lock
Term derived from mechanical problem that arises in gimbal mechanism that supports a compass or a gyro
Second and third rotations have effect of transforming earlier rotations, we lose a degree of freedom
ex: Rot z, Rot y, Rot x
If Rot z = 90 degrees, Rot y is equivalent to Rot x
4/18/2018
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32. Quaternions – next topic!
Invented by Sir William Hamilton (1843)
Do not suffer from Gimbal Lock
Provide a natural way to interpolate intermediate steps in a rotation about an arbitrary axis
Are used in many position tracking systems and VR software support systems
4/18/2018
©Babu 2009