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Modeling Transformation
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Overview 2D Transformation 3D Transformation
Basic 2D transformations Matrix representation Matrix Composition 3D Transformation Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ray casting
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Modeling Transformation
Specify transformations for objects Allow definitions of objects in own coordinate systems Allow use of object definition multiple times in a scene Ref] Hearn & Baker
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2D Modeling Transformations
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2D Modeling Transformations
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Basic 2D Transformations
Translation x’ = x + tx y’ = y + ty Scale x’ = x * sx y’ = y * sy Shear x’ = x + hx * y y’ = y + hy * x Rotation x’ = x * cosq - y * sinq y’ = x * sinq + y * cosq
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Matrix Representation
Represent 2D transformation by a matrix Multiply matrix by column vector Apply transformation to point
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Matrix Representation
Transformations combined by multiplication Matrices are a convenient and efficient way to represent a sequence of transformations
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2 x 2 Matrices What types of transformations can be represented with a 2 x 2 matrix? 2D Identity 2D Scale around (0, 0)
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2 x 2 Matrices What types of transformations can be represented with a 2x 2 matrix? 2D Rotate around (0, 0) 2D Shear shx = 2
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2 x 2 Matrices What types of transformations can be represented with a 2x 2 matrix? 2D Mirror (=Reflection) over Y axis 2D Mirror over (0, 0)
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Only linear 2D transformation can be presented with a 2 x 2 matrix
2 x 2 Matrices What types of transformations can be represented with a 2x 2 matrix? 2D Translation ? NO! Only linear 2D transformation can be presented with a 2 x 2 matrix
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Linear Transformations
Linear transformations are combinations of … Scale Rotation Shear Mirror (=Reflection) Properties of linear transformations Satisfies: Origin maps to origin Lines map to lines Parallel line remain parallel Ratios are preserved Closed under composition
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2D Translation 2D Translation represented by a 3 x 3 matrix
Point represented with homogeneous coordinate
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Homogeneous Coordinates
Add a 3rd coordinate to every 2D point (x, y, w) represents a point at location (x/w, y/w) (x, y, 0) represents a point at infinity (0, 0, 0) is not allowed
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Basic 2D Transformations
Basic 2D Transformations as 3 x 3 matrices
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Affine Transformations
Affine Transformations are combinations of … Linear transformation Translation Properties of affine transformations Origin does not necessarily map to origin Lines map to lines Parallel line remain parallel Ratios are preserved Closed under composition
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Transformation : Rigid-Body / Euclidean Transforms
Rigid Body Transformation (=rigid motion) A transform made up of only translation and rotation Preserve the shape of the objects that they act on Preserves distances Preserves angles Rigid / Euclidean Identity Translation Rotation
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Transformation : Similitudes / Similarity Transforms
Similarity Transformations Preserves angles Translation, Rotation, Isotropic scaling Similitudes Rigid / Euclidean Identity Translation Isotropic Scaling Rotation
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Transformation : Linear Transformations
Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear Translation is not linear
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Transformation : Affine Transformations
preserves parallel lines Affine Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear
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Transformation : Projective Transformations
preserves lines Projective Affine Similitudes Linear Rigid / Euclidean Scaling Identity Translation Isotropic Scaling Reflection Rotation Shear Perspective
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Projective Transformations
Affine transformation Projective warps Properties of projective transformations Origin does not necessarily map to origin Lines map to lines Parallel line do not necessarily remain parallel Ratios are not preserved (but “cross-ratios” are) Closed under composition
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Cross-ratio Given any four points A, B, C, and D, taken in order, define their cross ratio as Cross Ratio (ABCD) = cross ratio (ABCD) = cross ratio (A'B'C'D')
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Matrix Composition Transformations can be combined by matrix multiplication
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Matrix Composition Matrices are a convenient and efficient way to represent a sequence of transformations General purpose representations Hardware matrix multiply Efficiency with premultiplication
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Matrix Composition Be aware: order of transformations matter
Matrix multiplication is not commutative “Global” “Local”
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Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p (5,3) Scale(2,2) (2,2) Translate(3,1) (1,1) (3,1) (0,0) (0,0) Translate then Scale: p' = S ( T p ) = ST p (8,4) Translate(3,1) (4,2) Scale(2,2) (1,1) (6,2) (3,1) (0,0)
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Matrix Composition Rotate by q around arbitrary point (a, b)
M = T(a, b)* R(q) * T(-a, -b) Translate (a, b) to the origin do the rotation about origin translate back Scale by sx, sy around arbitrary point (a, b) M = T(a, b)* S(sx, sy) * T(-a, -b) do the scale about origin
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Transformation Game
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Overview 2D Transformation 3D Transformation
Basic 2D transformations Matrix representation Matrix Composition 3D Transformation Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ray casting
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3D Transformations Same idea as 2D transformations
Homogeneous coordinates: (x, y, z, w) 4 x 4 transformation matrices
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Basic 3D transformations
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Basic 3D transformations
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Non-commutative Composition : Order of Rotations
Order of Rotation Affects Final Position X-axis Z-axis Z-axis X-axis
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Matrix Composition : Rotations in space
Rotation about an Arbitrary Axis Rotate about some arbitrary a through angle Rotate about the z axis through an angle The axis a lies in the yz plane Rotate about the x axis through an angle The axis a coincident with the z axis Rotate about the a axis through Same as a rotation about the now coincident z axis Reverse the second rotation about the x axis Reverse the first rotation about the z axis Returning the axis a to its original position z a y x
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Transformation Hierarchies
Scene may have hierarchy of coordinate systems Each level stores matrix representing transformation from parent’s coordinate system
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Transformation Example 1
Well-suited for humanoid characters
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Transformation Example 1
Ref] Princeton University
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Transformation Example 2
An object may appear in a scene multiple times
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Transformation Example 2
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Summary Coordinate systems
World coordinates Modeling coordinates Representations of 3D modeling transformations 4 x 4 Matrices Scale, rotate, translate, shear, projections, etc. Not arbitrary warps Composition of 3D transformations Matrix multiplication (order matters) Transformation hierarchies
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