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CMSC427 Computer Graphics

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CMSC427 Computer Graphics CMSC427 Computer Graphics Matthias Zwicker Fall 2018 Today Transformations & matrices • Introduction • Affine transformations • Concatenating transformations 2 Introduction • Goal: Freely position rigid objects in 3D space 3 Today Transformations & matrices • Introduction • Affine transformations • Concatenating transformations 4 Affine transformations • Transformation, or mapping: function that maps each 3D point to a new 3D point „f: R3 -> R3“ • Affine transformations: class of transformations to position 3D objects in space • Affine transformations include – Rigid transformations • Rotation • Translation – Non-rigid transformations • Scaling • Shearing 5 Affine transformations • Definition: mappings that preserve colinearity and ratios of distances http://en.wikipedia.org/wiki/Affine_transformation – Straight lines are preserved – Parallel lines are preseverd • Linear transformations + translation • Nice: All desired transformations (translation, rotation) implemented using homogeneous coordinates and matrix- vector multiplication 6 Problem statement • If you know the (rigid, or affine) transformation that you want to apply, how do you construct a matrix that implements that transformation? – For example, given translation vector, rotation angles, what is the matrix that implements this? 7 Translation Point Vector 8 Matrix formulation Point Vector 9 Matrix formulation • Inverse translation • Verify that 10 Poll Given vector t and a corresponding translation matrix T(t), you want to translate by 2*t. The corresponding translation matrix is: A. T(t)+T(t) B. T(t)*T(t) C. None of the above 11 Note • What happens when you translate a vector? 12 Rotation First: rotating a vector in 2D • Convention: positive angle rotates counterclockwise • Express using rotation matrix 13 Rotating a vector in 2D 14 Rotating a vector in 2D 15 Rotating a vector in 2D 16 Rotation in 3D Rotation around z-axis • z-coordinate does not change 0 v = R z(µ)v • What is the matrix for ? 17 Other coordinate axes • Same matrix to rotate points and vectors • Points are rotated around origin 18 Poll Which of these matrices is a rotation matrix? A. Neither one B. A is, but B isn’t C. A isn’t, but B is D. Both 19 Rotation in 3D • Concatenate rotations around x,y,z axes to obtain rotation around arbitrary axes through origin Gimbal • are called Euler angles https://en.wikipedia.org/wiki/Gimbal http://en.wikipedia.org/wiki/Euler_angles • Disadvantage: result depends on order! 20 Rotation around arbitrary axis • Still: origin does not change • Counterclockwise rotation • Angle , unit axis • • Intuitive derivation see http://mathworld.wolfram.com/RotationFormula.html 21 Summary • Different ways to describe rotations mathematically – Sequence of rotations around three axes (Euler angles) – Rotation around arbitrary angles (axis-angle representation) – Other options exist (quaternions, etc.) • Rotations preserve – Angles – Lengths – Handedness of coordinate system • Rigid transforms – Rotations and translations 22 Rotation matrices • Orthonormal – Rows, columns are unit length and orthogonal • Inverse of rotation matrix? 23 Rotation matrices • Orthonormal – Rows, columns are unit length and orthogonal • Inverse of rotation matrix? – Its transpose 24 Rotations • Given a rotation matrix • How do we obtain ? 25 Rotations • Given a rotation matrix • How do we obtain ? 26 Rotations • Given a rotation matrix • How do we obtain ? • How do we obtain …? 27 Rotations • Given a rotation matrix • How do we obtain ? • How do we obtain …? 28 Scaling • Origin does not change 29 Scaling • Inverse scaling? 30 Scaling • Inverse scaling? 31 Shear • Pure shear if only one parameter is non-zero • Cartoon-like effects 32 Summary affine transformations • Linear transformations (rotation, scale, shear, reflection) + translation Vector space, Affine space http://en.wikipedia.org/wiki/Vector_space http://en.wikipedia.org/wiki/Affine_space • vectors as [xyz] • points and vectors coordinates as [xyz1], [xyz0] • represents vectors homogeneous • 3x3 matrix multiplication coordinates implements linear • distinguishes points transformations and vectors • 4x4 matrix mult. implements linear trafos. and translation 33 Summary affine transformations • Implemented using 4x4 matrices, homogeneous coordinates – Last row of 4x4 matrix is always [0 0 0 1] • Any such matrix represents an affine transformation in 3D • Factorization into scale, shear, rotation, etc. is always possible, but non-trivial – Polar decomposition http://en.wikipedia.org/wiki/Polar_decomposition 34 Today Transformations & matrices • Introduction • Affine transformations • Concatenating transformations 35 Concatenating transformations • Build “chains” of transformations • Apply followed by followed by • Overall transformation is an affine transformation 36 Concatenating transformations • Result depends on order because matrix multiplication not commutative • Thought experiment – Translation followed by rotation vs. rotation followed by translation 37 Rotating with pivot Rotation around Rotation with origin pivot 38 Rotating with pivot 1. Translation 2. Rotation 3. Translation 39 Rotating with pivot 1. Translation 2. Rotation 3. Translation 40 Concatenating transformations • Arbitrary sequence of transformations • Note: associativity T=M3.multiply(M2); Mtotal=T.multiply(M1) T=M2.multiply(M1); Mtotal=M3.multiply(T) 41 Coming up • From 3D to 2D: projections 42.
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