1. COMPUTER GRAPHICS CHAPTERS 10-11 CS 482 – Fall 2017 TRANSFORMATIONS
AFFINE TRANSFORMATIONS
QUATERNIONS

2. AFFINE TRANSFORMATIONS
TRANSFORMATION TYPES
To effectively place objects in a graphical environment, it is convenient to use certain matrix transformations to manipulate their position and orientation.
The specific types of transformations that we use are:
Scaling: multiply each dimension by a constant factor
Translation: add a constant amount in each dimension
Rotation: spin each object some angle around the origin
Carefully combining these three types of transformations will permit a graphical object to be positioned and oriented in any desired fashion.
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3. AFFINE TRANSFORMATIONS
ROTATION
2D Rotation:
(xcos - ysin, xsin + ycos) 
(x, y)
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4. AFFINE TRANSFORMATIONS
3D ROTATION
Pitch: ROTATION ABOUT X-AXIS
YAW: ROTATION ABOUT Y-AXIS
Roll: ROTATION ABOUT Z-AXIS
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5. AFFINE TRANSFORMATIONS
SCALING
2D Scaling:
(x, y)
(k1x, k2y)
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6. AFFINE TRANSFORMATIONS
TRANSLATION
2D Translation:
(x + x, y + y) y
(x, y) x
Any 2x2 matrix representing translation would require non-constant values in the matrix (requiring a separate matrix for each point being translated!)
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7. AFFINE TRANSFORMATIONS
HOMOGENEOUS COORDINATES
The translation matrix problem is remedied by the use of homogeneous coordinates, which utilize an extra coordinate (i.e., a “weight” factor) for every 2D point.
Rotation
Scaling
Translation
(Note: In most of our work, the weight factor will merely be 1.)
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8. AFFINE TRANSFORMATIONS
COMBINING TRANSFORMATIONS
By applying different transformations in a particular order, graphical objects can be manipulated to form complex variations of their original forms.
Enlarge one copy of the original image via scaling, and then translate it to the right.
Shrink another copy of the original image via scaling, rotate it slightly counterclockwise, and then translate it higher and to the right.
Original Image
Shrink a third copy of the original image via scaling, rotate it clockwise a bit less than the second image, and then translate it slightly higher and further to the right.
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9. AFFINE TRANSFORMATIONS
ORDER OF TRANSFORMATIONS
Care must be taken when applying transformations, since the resulting image varies with the order in which the transformations are applied.
R=60 Counterclockwise Rotation S=150% x-Scaling & 50% Y-Scaling T=Translation by (-3,-1) R
first S
second T
third S
first T
second R
third
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10. CHAPTERS 10-11: TRANSFORMATIONS
QUATERNIONS
SMOOTH ROTATIONS
Traditionally, a graphical object’s orientation has been defined via “Euler angle” rotations about the three coordinate axes (i.e., pitch, yaw, and roll).
However, this approach can result in “gimbal lock”, in which rotation about one axis effectively overrides rotation about another axis.
OpenGL uses rotation about a user-specified axis, but that approach can result in interpolations between orientations that do not appear smooth.
A much smoother approach involves quaternions, four-dimensional versions of complex numbers that can essentially be viewed as a 3D vector, combined with a scalar.
q=𝑤+𝑥i+𝑦j+𝑧k
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11. QUATERNIONS ROTATING IN OBJECT SPACE
To clarify, after combining several Euler angle rotations, it becomes difficult to rotate objects relative to their own space (due to Gimbal lock).
Note in the image at right that an attempt to perform yaw rotation yields a spin around the y-axis in world space, but not in object space (as demonstrated by the non-stationary green arrow).
Using quaternions eliminates this problem by performing operations in object space.
Technically, quaternions space is a four-dimensional vector space that may be viewed as an extension of the two-dimensional complex plane, with a variation of cross products used for its (non-commutative) multiplication.
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12. QUATERNIONS P  C Q SPHERICAL LINEAR INTERPOLATION
quaternions may be used to create slerps, spherical linear interpolations, which compel the interpolated orientations to be along the great circle of a sphere.
Given a sphere with center C and surface points P and Q, where the angle between the CP and CQ vectors is . P 
The SLERP equation for all of the points on the great circle of the sphere joining P and Q is: C Q
𝑃 𝑡 =𝑃 sin 1−𝑡)𝜙 sin 𝜙 +𝑄 sin 𝑡𝜙 sin 𝜙
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