Computer Graphics - Transformations - Hendrik Lensch Computer - PowerPoint PPT Presentation

Computer Graphics - Transformations - Hendrik Lensch Computer Graphics WS07/08 – Transformations

Overview • Last Time – Tone Mapping • Today – Homogeneous Coordinates – Basic transformations in homogeneous coordinates – Concatenation of transformations – Projective transformations • Next Lecture – Camera transformations Computer Graphics WS07/08 – Transformations

Euclidean Vector Space • Known from Mathematics: – Elements of a 3D vector space • v = (v 1 , v 2 , v 3 ) T ∈ V 3 = R 3 – Formally: Vectors written as column vectors (n x 1 matrix)! – Vectors describe directions – not positions! – 3 linear independent vectors create a basis: • {e 1 , e 2 , e 3 } – Any vector can now uniquely be represented with coordinates • v = v 1 e 1 + v 2 e 2 + v 3 e 3 = (v 1 , v 2 , v 3 ) T – Operations e 3 e • Addition, Subtraction, Scaling, … 3 e 2 e 2 • Metric – Dot/inner product: e 1 e 1 • Used for measurements of length (|v| 2 = v · v) and angles (cos(v 1 , v 2 )= v 1 · v 2 / | v 1 ||v 2 |) – Orthonormal basis e i · e j = δ ij right-/left handed: e 1 × e 2 = +/- e 3 • |e i |= 1 Computer Graphics WS07/08 – Transformations

Affine Space • Known from Mathematics – Affine Space: A 3 • Elements are positions – no directions! – Defined via its associated vector space V 3 • a, b ∈ A 3 ⇔ v ∈ V 3 with v = b – a → : unique, ← : ambiguous – Addition of points and vectors (p + v ∈ A 3 ) • distance(a, b) = length(a - b) – Operations on A 3 • Subtraction yields a vector • No addition of affine elements a a v v b Computer Graphics WS07/08 – Transformations

Affine Basis • Affine Basis: – {o, e1, e2, e3} • Origin: o ∈ A 3 and • Basis of vector space – Position vector of point p • (p – o) ∈ V3 e 3 e 3 e 2 e 2 o e 1 e 1 Computer Graphics WS07/08 – Transformations

Affine Coordinates • Affine Combination – Linear combination of (n+1) points – Weights form a partition of unity – b 0 , … , b n ∈ A n n n n ∑ ∑ ∑ ∑ = α = + α − = + α α = b b b b b o e ( ) , mit 1 i i i i i i i 0 0 = = = i i i 0 1 1 • Affine Coordinates – Barycentric coordinates – Center of mass (R = Σ m i r i / Σ m i ) – Affine weighted sum • Weights given by the splitting ratio – p= α 1 p 1 + α 2 p 2 α 2 α 2 α 1 + α 2 = 1 p p p 1 p 1 α 1 α 1 p p 2 2 Computer Graphics WS07/08 – Transformations

Affine Mappings • Properties – Affine mapping (continuous, bijective, invertible) • T: A 3 → A 3 – Defined by two non-degenerated simplicies • 2D: Triangle, 3D: Tetrahedron, ... – Affine/Barycentric coordinates are invariant under affine transformations – Other invariants • Straight lines, parallelism, splitting ratios, surface/volume ratios – Characterization via fixed points and lines • Eigenvalues and eigenvectors of the mapping • Representation α 2 α – Linear mapping A plus a translation t 2 p p p p 1 • T p = A p + t with (n x n) matrix A 1 α 1 α – Invariance of affine coordinates 1 • Tp= T( α 1 p 1 + α 2 p 2 )= A( α 1 p 1 + α 2 p 2 ) + t = α 1 A(p 1 ) + α 2 A(p 2 ) + α 1 t + α 2 t = α 1 Tp 1 + α 2 T( p 2 ) p 2 p 2 Computer Graphics WS07/08 – Transformations

Homogeneous Coordinates for 3D • Embedding of R 3 into P(R 4 ) – For the time being ⎛ ⎞ ⎛ ⎞ x X ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ x X W / ⎜ ⎟ ⎜ ⎟ y Y ⎜ ⎟ ⎜ ⎟ ∋ → ∈ → R y Y W 3 ⎜ ⎟ 4 ⎜ ⎟ P(R ), and / ⎜ ⎟ ⎜ ⎟ z Z ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ z Z W ⎝ ⎠ ⎝ ⎠ / ⎜ ⎟ ⎜ ⎟ W ⎝ ⎠ ⎝ ⎠ 1 – Representation of transformations by 4x4 matrices – Mathematical trick • Convenient representation to express rotations and translations as matrix multiplications • Easy to find line through points, point-line/line-line intersections – Also important for projections (later) Computer Graphics WS07/08 – Transformations

Point Representation X = ( X,Y,W ) (x,y) w =1 X Y = = x y W W Computer Graphics WS07/08 – Transformations

Line Representation l= (a,b,c) ax+by+c=0 Computer Graphics WS07/08 – Transformations

Point on Line X = (X,Y,W) l = (a,b,c) ⋅ l = x 0 Computer Graphics WS07/08 – Transformations

Intersection of Lines x l’ l × = l ' l x Computer Graphics WS07/08 – Transformations

Line through 2 Points X = (X,Y,W) X’ = (X’,Y’,W’) l = (a,b,c) × = x ' x l Computer Graphics WS07/08 – Transformations

Linear Map = Matrix • Vector Matrix Product – Action of a linear map on a vector • Multiplication of matrix with column vector ⎛ ⎞ t t t t ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x x x ' ⎜ xx xy xz xw ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ t t t t y y ⎜ ⎟ y ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ' yx yy yz yw = = = = p T p T ' ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ t t t t z z z ' ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ zx zy zz zw ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ t t t t w w w ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ' ⎝ ⎠ wx wy wz ww • In some old text books: row vector times the transpose of this matrix – Composition (first T 1 , then T 2 ) • Matrix multiplication • T 2 T 1 p = T 2 (T 1 p) = (T 2 T 1 ) p = T p • Warning: In general, matrix multiplication does not commute !!! Computer Graphics WS07/08 – Transformations

Basic Transformations • Translation ⎛ ⎞ d 1 0 0 ⎜ x ⎟ d ⎜ ⎟ 0 1 0 = y T d d d ( , , ) ⎜ ⎟ x y z d 0 0 1 ⎜ ⎟ z ⎜ ⎟ ⎝ ⎠ 0 0 0 1 + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ d p p d 1 0 0 ⎜ x ⎟ ⎜ x ⎟ ⎜ x x ⎟ + d p p d ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 0 = y y = y y p T ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + d p p d 0 0 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ z z z z ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 1 1 1 Computer Graphics WS07/08 – Transformations

Translation of Vectors • So far we looked at points (affine entities) • Vectors are defined as the difference of two points • Consequently, for vectors W is always equal to zero − ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ p q p q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x x x x − p q p q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = − = y − y = y y v p q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − p q p q ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ z z z z ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 1 0 • This means that translations DO NOT act on vectors – Which is exactly what we expect to happen ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ d v v 1 0 0 ⎜ x ⎟ ⎜ x ⎟ ⎜ x ⎟ d v v ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1 0 = y y = y = v v T ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ d v v 0 0 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ z z z ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 0 1 0 0 Computer Graphics WS07/08 – Transformations

Translations • Properties – T(0,0,0)= 1 (Identity Matrix) – T(t x , t y , t z ) T(t x ’, t y ’,t z ’) = T(t x + t x ’, t y + t y ’, t z + t z ’) – T(t x , t y , t z ) T(t x ’, t y ’,t z ’) = T(t x ’, t y ’,t z ’) T(t x , t y , t z ) – T -1 (t x , t y , t z ) = T(-t x , -t y , -t z ) Computer Graphics WS07/08 – Transformations

Rotation • Rotation in 2D y = θ x r cos r sin( θ + ϕ ) = θ y r sin = θ + ϕ x r ' cos( ) r sin θ = θ + ϕ y r θ + ϕ ' sin( ) θ θ + ϕ = θ ϕ − θ ϕ cos( ) cos cos sin sin x r cos θ r cos( θ + ϕ ) θ + ϕ = θ ϕ + θ ϕ sin( ) cos sin sin cos = θ ϕ − θ ϕ = ϕ − ϕ x r r x y ' ( cos ) cos ( sin ) sin cos sin = θ ϕ + θ ϕ = ϕ + ϕ y r r x y ' ( cos ) sin ( sin ) cos sin cos Computer Graphics WS07/08 – Transformations

Basic Transformations • Rotation around major axis θ θ ⎛ ⎞ ⎛ ⎞ 1 0 0 0 cos 0 sin 0 ⎜ ⎟ ⎜ ⎟ θ − θ ⎜ ⎟ ⎜ ⎟ 0 cos sin 0 0 1 0 0 θ = θ = R R ( ) ( ) ⎜ ⎟ ⎜ ⎟ x y θ θ − θ θ 0 sin cos 0 sin 0 cos 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 0 0 0 1 0 0 0 1 θ − θ ⎛ ⎞ cos sin 0 0 ⎜ ⎟ θ θ ⎜ ⎟ sin cos 0 0 θ = R ( ) ⎜ ⎟ z 0 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0 0 0 1 – Assumes a right handed coordinate system Computer Graphics WS07/08 – Transformations

Rotation • Properties – R a (0)= 1 a ( θ ) = R a (- θ ) – R -1 – R a ( θ ) R a ( ϕ ) = R a ( θ + ϕ ) – R a ( θ ) R a ( ϕ ) = R a ( ϕ ) R a ( θ ) a ( θ ) = R a (- θ ) = R T a ( θ ) – R -1 – BUT in general: R a ( θ ) R b ( ϕ ) ≠ R b ( ϕ ) R a ( θ ) • For rotations around different axes, the order matters Computer Graphics WS07/08 – Transformations

Basic Transformations • Scaling ⎛ ⎞ s 0 0 0 ⎜ ⎟ x s ⎜ ⎟ 0 0 0 = y S s s s ( , , ) ⎜ ⎟ x y z s 0 0 0 ⎜ ⎟ z ⎜ ⎟ ⎝ ⎠ 0 0 0 1 – Uniform Scaling • s x = s y = s z Computer Graphics WS07/08 – Transformations