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SE 313 – Computer Graphics
Lecture 7: Mathematical Basis for 3D
Transformations
Lecturer: Gazihan Alankuş
Please look at the last three slides for assignments (marked with TODO)
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Plan for Today
• One hour lecture
• Two hour quiz and lab
2
Exam Talk
• Next week is the midterm exam
• I can ask about anything that we have done
here.
– Slides may not be enough.
3
Our Goal
• Understand the mathematical basis of 3D
transformations.
4
What we know already
• Math
– Point, vector, magnitude, dot product, cross
product, etc.
• Transformations
– Translation, rotation, scale
– They have a mathematical basis
5
What will we transform?
• Objects are made of points
• Pairs of such points define vectors
• Points
– Translate, rotate? (around origin), scale? (wrt.
origin)
• Vectors
– Translate?(no!), rotate, scale
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Translation
• How can we move an object?
– Add numbers to the coordinates of all of its points
• Explanation on the board
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Rotation
• How can we rotate an object (around the
origin)?
– Trigonometry!
• Ok but how can we make these calculations
more systematically?
– Matrix-vector multiplication!
– M x v (the column vector is on the right)
• Explanation on the board
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Scale
• How can we scale an object (wrt the origin)?
– Multiplication!
• Explanation on the board
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Combining Transformations
• Of the same kind
– Translation
• Just add the numbers that you will add for each
– Scale
• Just multiply the numbers that you will multiply for
each
– Rotate?
• Guess what, matrix multiplication from linear algebra is
actually useful
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Combining Transformations
• What if I want to do “rotate, translate, and
then rotate” like we did last week?
• How will you combine
element-wise addition,
element-wise multiplication
and matrix multiplication?
– You can’t. Unless you make
them be the same operation.
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Scaling as Matrix Multiplication
• Instead of doing element-wise multiplication,
why not get an identity matrix, multiply the
diagonals of that identity matrix with the scale
factors and then use it?
• Explanation on the board
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Scaling as Matrix Multiplication
• Now we can combine rotation and scale
however we wish!
• RxSxRxRxSxSxRxS
– The result is still a 3 by 3 matrix! Ready to be
applied to a vector or combined with whatever
else you’ve got!
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Translation as a matrix multiplication?
• Think about it for a minute.
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Translation as a matrix multiplication?
• It is possible if we add one more dimension
– The fourth column holds the translation nicely –
very clever hack
• Explanation on the board
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Homogeneous Coordinates
• Four entities in a column vector represent three
dimensions. The last entry is always 1. If not,
divide all with the last entry.
– The fourth dimension does not have any significant
meaning. It’s just a hack.
• 3x3 rotation/scale matrices -> 4x4 matrices with
the rest coming from identity
• Transformation column vectors -> 4x4 identity
matrices, with the column vector pasted to the
fourth column.
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Combining Transformations Using
Homogeneous Coordinates
• Mr x Mt x Ms x Mt x Mr x Mr x Mt
• Still a 4x4 matrix, ready to be applied to your 3
dimensional column vectors with a 1 in the
end (for points)
• For vectors, you put a 0 in the end. Remember
slide #4, you cannot translate vectors!
– The math works nicely.
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Combining Transformations and the
Order of Matrix Multiplication
• When you read the matrix multiplication from
left to right, it is the world-to-local order of
applying transformations
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Combinations of Translations and
Rotations
• Mt x Mr x Mr x Mt x Mt x Mr
• Whichever order you apply, this is equal to
Mt’ x Mr’ for some t’ and r’
– The 3x3 part of the matrix is the rotation
– The last column is the translation
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More interesting facts
• When you apply a transformation to an object
that was straight in the origin, the object’s local
coordinate frame moves with the object. The
location and the x, y, z vectors of this local frame
can be represented in global coordinates.
– The x, y, z vectors of the local coordinate frame are
the first three columns of the transformation matrix!
– The location of this local coordinate frame is the
fourth column of the transformation matrix.
• You can use this information to construct
transformation matrices.
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Quaternions
• 3x3 matrices have too much data. We can
represent it with three numbers (three
rotations). But this does not play well with
mathematics.
• Instead, we represent them with four
numbers. Quaternions happen to be a good
mathematical construct for this.
• −1, blah blah blah. DON’T CARE! We don’t
need them.
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Quaternions
• What we care about is that they represent rotations as an
angle and an axis
– http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
– They encode it in a funny way though:
– Or, the four tuple:
𝜃
𝜃
𝜃
𝜃
(𝑐𝑜𝑠 , 𝑎𝑥 𝑠𝑖𝑛 , 𝑎𝑦 𝑠𝑖𝑛 , 𝑎𝑧 𝑠𝑖𝑛 )
2
2
2
2
• We still do not care as strongly. Just know that, for you,
quaternion means rotation.
–
–
–
–
You can use it instead of 3x3 matrices
You can create it using an axis and an angle
You can extract from it an axis and an angle
With quaternion multiplication, you can combine them just like
you combine matrices
– Most 3D libraries enable you to rotate a vector with a
quaternion
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Quizzes and Lab
• For two hours, the assistant will administer
– A closed-book quiz (15 minutes, on paper)
– An open-book quiz (rest of class, on paper)
– The blender implementation of the last two
questions of the open-book quiz (graded by
demonstrating to the assistant)
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