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Affine Transformations Jim Van Verth NVIDIA Corporation (jim@essentialmath.com)jim@essentialmath.com

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Topics »Whats an affine transformation? »How to generate various forms »Things to watch out for

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What is it? »A mapping between affine spaces »Preserves lines (& planes) »Preserves parallel lines »But not angles or distances »Can represent as »(note were using column vectors)

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Affine Space »Collection of points and vectors »Can be represented using frame »Within frame, vector and point j = (0,1) O i = (1,0)

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Affine Transformation »Key idea: map from space to space by using frames Note how axes change ( A ) Note how origin changes ( y )

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Example: Translation »Axes dont change Origin moves by y y

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Example: Rotation »Axes change »Origin doesnt move But what is A ?

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Example: Rotation »Follow the axes: »New axes go in columns of matrix (cos, sin ) (-sin, cos )

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Example: Scale »Axes change »Origin doesnt move Again, what is A ?

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Example: Scale »Follow the axes: »New axes go in columns of matrix (0, 0.5) (1.5, 0)

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Example: Reflection »Axes change »Origin doesnt move But what, oh what, is A ?

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Example: Reflection »Follow the axes: »New axes go in columns of matrix (-1, 0) (0, 1)

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Example: Shear »Axes change »Origin doesnt move Hey, hey, what is A ?

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Example: Shear »Follow the axes: »New axes go in columns of matrix (0.5, 1) (1, 0)

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Transform Types »Rigid-body transformation Translation Rotation »Deformable transformation Scale Reflection Shear

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Combining Transforms »Simple function composition

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Combining Transforms »Order is important! »Scale, then rotate

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Combining Transforms »Order is important! »Rotate, then scale

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Local and World Frames »Objects built in local frame »Want to place in world frame »Local-to-world transformation »Same basic idea: determine where axes and origin end up in world frame

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World to Local Frame »Similar, but in reverse »Want transformation relative to local frame »Often easier to just take the inverse »Example: world-to-view transformation

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Inverse »Reverses the effect of a transformation »Easy to do with formula: »For matrix, just use matrix inverse

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Matrix Form »Necessary for many APIs »Is easy »Concatenate by matrix multiply »Can be faster on vector architectures »Takes more storage, though

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Object-centered Transform »Often get this case »Already have local-to-world transform »Want rotate/scale/whatever around local origin, not world origin »How?

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Object-Oriented Transform »One way »Translate to origin »Rotate there »Translate back

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Object-Oriented Transform »Using formula: »Translate to origin »Rotate there »Translate back »This works with arbitrary center in world frame!

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Alternate Format »Problem: want to Translate object in space and change rotation and scale arbitrarily Handle rotation/scale separately Cant do easily with matrix format or Ax + y form »Involves SVD, Polar decomposition »Messy, but we can do better using…

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Rigid Body Transforms »Any sequence of translation and rotation transformations »Not scale, reflection or shear »Object shape is not affected (preserves angles and lengths) »Usually include uniform scale despite this

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Alternate Format Scale – one uniform scale factor s Rotation – matrix R Translation – single vector t

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Alternate Format Want to concatenate transforms T 1, T 2 in this form, or »Do this by

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Alternate Format »Advantages Clear what each part does Easier to change individual elements »Disadvantages Eventually have to convert to 4x4 matrix anyway for renderer/video card 4x4 faster on vector architecture

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Alternate Format »Matrix conversion

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Inverting Rigid Body Xforms »We can easily invert our rigid body transforms symbolically:

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Inverting Rigid Body Xforms (2) »In fact, the result itself may be written as a rigid body transform:

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References »Rogers, F. David and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed, McGraw-Hill, 1990. »Watt, Alan, 3D Computer Graphics, Addison- Wesley, Wokingham, England, 1993.

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