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lecture05-3

# lecture05-3 - Introduction to Computer Graphics CS 445 645...

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Unformatted text preview: Introduction to Computer Graphics CS 445 / 645 Lecture 5 Transformations M.C. Escher – Smaller and Smaller (1956) Modeling Transformations Specify transformations for objects Specify • Allows definitions of objects in own coordinate systems • Allows use of object definition multiple times in a scene – Remember how OpenGL provides a transformation Remember stack because they are so frequently reused stack Chapter 5 from Hearn and Baker H&B Figure 109 Overview 2D Transformations • Basic 2D transformations • Matrix representation • Matrix composition 3D Transformations • Basic 3D transformations • Same as 2D 2D Modeling Transformations Modeling Coordinates Scale Translate y x Scale Rotate Translate World Coordinates 2D Modeling Transformations Modeling Coordinates y x Let’s look at this in detail… World Coordinates 2D Modeling Transformations Modeling Coordinates y x Initial location at (0, 0) with x- and y-axes aligned 2D Modeling Transformations Modeling Coordinates y x Scale .3, .3 Rotate -90 Translate 5, 3 2D Modeling Transformations Modeling Coordinates y x Scale .3, .3 Rotate -90 Translate 5, 3 2D Modeling Transformations Modeling Coordinates y x Scale .3, .3 Rotate -90 Translate 5, 3 World Coordinates Scaling Scaling a coordinate means multiplying each of its components by a scalar components Uniform scaling means this scalar is the same for all components: components: ×2 Scaling Non-uniform scaling: different scalars per component: Non-uniform X × 2, Y × 0.5 How can we represent this in matrix form? Scaling Scaling operation: Or, in matrix form: x' ax y ' = by x ' a y ' = 0 0 x y b scaling matrix 2-D Rotation (x’, y’) (x, y) θ x’ = x cos(θ) - y sin(θ) y’ = x sin(θ) + y cos(θ) 2-D Rotation (x’, y’) (x, y) θ φ x = r cos (φ) y = r sin (φ) x’ = r cos (φ + θ) y’ = r sin (φ + θ) Trig Identity… x’ = r cos(φ) cos(θ) – r sin(φ) sin(θ) y’ = r sin(φ) sin(θ) + r cos(φ) cos(θ) Substitute… x’ = x cos(θ) - y sin(θ) y’ = x sin(θ) + y cos(θ) 2-D Rotation This is easy to capture in matrix form: x ' cos(θ ) y ' = sin (θ ) − sin (θ ) x cos(θ ) y Even though sin(θ) and cos(θ) are nonlinear functions are of θ, • x’ is a linear combination of x and y • y’ is a linear combination of x and y 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*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ Transformations can be combined (with simple algebra) 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*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ Basic 2D Transformations Translation: • x’ = x + tx • y’ = y + ty (x,y) Scale: • x ’ = x * sx x’ (x’,y’) • y ’ = y * sy y’ Shear: • x’ = x + hx*y • y’ = y + hy*x Rotation: • x’ = x*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ x’ = x*sx y’ = y*sy 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*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ (x’,y’) x’ = (x*sx)*cosΘ - (y*sy)*sinΘ y’ = (x*sx)*sinΘ + (y*sy)*cosΘ Basic 2D Transformations Translation: • x’ = x + t x x’ • y’ = y + t y y’ Scale: (x’,y’) • x’ = x * sx • y’ = y * sy Shear: • x’ = x + hx*y • y’ = y + hy*x Rotation: • x’ = x*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ x’ = ((x*sx)*cosΘ - (y*sy)*sinΘ) + tx y’ = ((x*sx)*sinΘ + (y*sy)*cosΘ) + ty 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*cosΘ - y*sinΘ • y’ = x*sinΘ + y*cosΘ x’ = ((x*sx)*cosΘ - (y*sy)*sinΘ) + tx y’ = ((x*sx)*sinΘ + (y*sy)*cosΘ) + ty Overview 2D Transformations • Basic 2D transformations • Matrix representation • Matrix composition 3D Transformations • Basic 3D transformations • Same as 2D Matrix Representation Represent 2D transformation by a matrix a c b d Multiply matrix by column vector ⇔ apply transformation to point x' = a y ' c b x d y x' = ax + by y ' = cx + dy Matrix Representation Transformations combined by multiplication x' = a y ' c b e d g f i h k j x l y Matrices are a convenient and efficient way to represent a sequence of transformations! 2x2 Matrices What types of transformations can be What represented with a 2x2 matrix? represented 2D Identity? x' = x y' = y x' = 1 y ' 0 0 x 1 y 2D Scale around (0,0)? x' = s x * x y' = s y * y x ' s x y ' = 0 0 x s y y 2x2 Matrices What types of transformations can be What represented with a 2x2 matrix? represented 2D Rotate around (0,0)? x ' = cos Θ* x − sin Θ* y y ' = sin Θ* x + cos Θ* y x ' cos Θ − sin Θ x y ' = sin Θ cos Θ y 2D Shear? x ' = x + shx * y y ' = shy * x + y x ' 1 y ' = sh y shx x 1 y 2x2 Matrices What types of transformations can be What represented with a 2x2 matrix? represented 2D Mirror about Y axis? x ' = −x y' = y x' = −1 0 x y ' 0 1 y 2D Mirror over (0,0)? x ' = −x y' = −y x' = −1 0 x y ' 0 −1 y 2x2 Matrices What types of transformations can be What represented with a 2x2 matrix? represented 2D Translation? x' = x + t x y' = y + t y NO! Only linear 2D transformations can be represented with a 2x2 matrix Linear Transformations Linear transformations are combinations of … • Scale, • Rotation, • Shear, and x' a y ' = c b x d y • Mirror Properties of linear transformations: • Satisfies: T ( s1p1 + s2p 2 ) = s1T (p1 ) + s2T (p 2 ) • Origin maps to origin • Lines map to lines • Parallel lines remain parallel • Ratios are preserved • Closed under composition Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix? x' = x + t x y' = y + t y Homogeneous Coordinates Homogeneous coordinates • represent coordinates in 2 represent dimensions with a 3-vector dimensions x x homogeneous coords y → y 1 Homogeneous coordinates seem unintuitive, but they Homogeneous make graphics operations much easier much Homogeneous Coordinates Q: How can we represent translation as a 3x3 matrix? x' = x + t x y' = y + t y A: Using the rightmost column: 1 Translation = 0 0 0 1 0 tx ty 1 Translation Example of translation Homogeneous Coordinates x ' 1 0 t x x x + t x y ' = 0 1 t y = y + t y y 1 0 0 1 1 1 α tx = 2 ty = 1 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 y 2 (2,1,1) or (4,2,2) or (6,3,3) 1 Convenient coordinate system to represent many useful transformations 1 2 x Basic 2D Transformations Basic 2D transformations as 3x3 matrices x ' 1 y ' = 0 1 0 0 1 0 x ' s x y ' = 0 1 0 t x x t y y 1 1 Translate x' cos Θ y ' = sin Θ 1 0 − sin Θ cos Θ Rotate 0 0 x 0 y 11 0 sy 0 Scale 0 x 0 y 11 x ' 1 y ' = sh y 1 0 shx 1 0 Shear 0 x 0 y 11 Affine Transformations Affine transformations are combinations of … • Linear transformations, and • Translations x ' a y ' = d w 0 Properties of affine transformations: • Origin does not necessarily map to origin • Lines map to lines • Parallel lines remain parallel • Ratios are preserved • Closed under composition b e 0 c x f y 1 w Projective Transformations Projective transformations … • Affine transformations, and • Projective warps x' a y ' = d w' g Properties of projective transformations: • Origin does not necessarily map to origin • Lines map to lines • Parallel lines do not necessarily remain parallel • Ratios are not preserved • Closed under composition b e h c x f y i w Overview 2D Transformations • Basic 2D transformations • Matrix representation • Matrix composition 3D Transformations • Basic 3D transformations • Same as 2D Matrix Composition Transformations can be combined by Transformations matrix multiplication matrix x ' 1 y ' = 0 w' 0 p’ = 0 1 0 T(tx,ty) tx cos Θ − sin Θ 0sx ty sin Θ cos Θ 0 0 0 1 0 1 0 R(Θ) 0 sy 0 S(sx,sy) 0 x y 0 1 w p Matrix Composition Matrices are a convenient and efficient way to Matrices represent a sequence of transformations represent • General purpose representation • Hardware matrix multiply p’ = (T * (R * (S*p) ) ) p’ = (T*R*S) * p Matrix Composition Be aware: order of transformations matters – Matrix multiplication is not commutative p’ = T * R * S * p “Global” “Local” Matrix Composition What if we want to rotate and translate? What and • Ex: Rotate line segment by 45 degrees about endpoint a Ex: and lengthen a a Multiplication Order – Wrong Way Our line is defined by two endpoints • Applying a rotation of 45 degrees, R(45), affects both points • We could try to translate both endpoints to return endpoint a to its We original position, but by how much? original a a Wrong R(45) a Correct T(-3) R(45) T(3) Multiplication Order - Correct Isolate endpoint a from rotation effects Isolate • First translate line so a is at origin: T (-3) First • Then rotate line 45 degrees: R(45) a a a • Then translate back so a is where it was: T(3) Then a Matrix Composition Will this sequence of operations work? Will 1 0 − 3 cos(45) − sin( 45) 0 1 0 3 a x a ' x 0 1 0 sin( 45) cos(45) 0 0 1 0 a = a' y y 0 0 1 0 0 1 0 0 1 1 1 Matrix Composition After correctly ordering the matrices Multiply matrices together What results is one matrix – store it (on stack)! What store Multiply this matrix by the vector of each vertex All vertices easily transformed with one matrix All multiply multiply Overview 2D Transformations • Basic 2D transformations • Matrix representation • Matrix composition 3D Transformations • Basic 3D transformations • Same as 2D 3D Transformations Same idea as 2D transformations • Homogeneous coordinates: (x,y,z,w) Homogeneous • 4x4 transformation matrices x' a y' e z' = i w' m b f j n c g k o d x h y l z p w Basic 3D Transformations x' 1 y ' 0 z ' = 0 w 0 0 1 0 0 0 0 1 0 0 x 0 y 0 z 1w Identity x ' 1 y ' 0 = z ' 0 w 0 0 1 0 0 0 0 1 0 t x x t y y t z z 1 w Translation x ' s x y ' 0 = z' 0 w 0 x' −1 y ' 0 z' = 0 w 0 0 sy 0 0 0 0 sz 0 0 1 0 0 0 0 1 0 Scale 0 x 0 y 0 z 1w 0 x 0 y 0 z 1 w Mirror about Y/Z plane Basic 3D Transformations Rotate around Z axis: x' cos Θ − sin Θ y ' sin Θ cos Θ z ' = 0 0 w 0 0 x ' cos Θ y ' 0 Rotate around Y axis: = z ' − sin Θ w 0 Rotate around X axis: x' 1 y ' 0 z ' = 0 w 0 0 1 0 0 0 0 1 0 0 x 0 y 0 z 1w sin Θ 0 x 0 0 y cos Θ 0 z 0 1w 0 0 cos Θ − sin Θ sin Θ cos Θ 0 0 0 x 0 y 0 z 1w Reverse Rotations Q: How do you undo a rotation of θ , R(θ )? Q: R( A: Apply the inverse of the rotation… R-1(θ ) = R(-θ ) How to construct R-1(θ ) = R(-θ ) How • Inside the rotation matrix: cos(θ) = cos(-θ) – The cosine elements of the inverse rotation matrix are unchanged • The sign of the sine elements will flip Therefore… R-1(θ ) = R(-θ ) = RT(θ ) Summary Coordinate systems • World vs. modeling coordinates 2-D and 3-D transformations • Trigonometry and geometry • Matrix representations • Linear vs. affine transformations Matrix operations • Matrix composition ...
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