Transcription of Geometric transformations in 3D and coordinate frames
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Geometric transformations in 3D and coordinate frames
Geometric transformations in 3D and coordinate framesComputer GraphicsCSE 167 Lecture 3 CSE 167: Computer Graphics 3D points as vectors Geometric transformations in 3D coordinate framesCSE 167, Winter 20182 Representing 3D points using vectors 3D point as 3 vector 3D point using affine homogeneous coordinates as 4 vectorCSE 167, Winter 20183 Geometric transformations Translation Linear transformations Scale Rotation 3D rotations Affine transformation Linear transformation followed by translation Euclidean transformation Rotation followed by translation Composition of transformations Transforming normal vectorsCSE 167, Winter 201843D translationCSE 167, Winter 20185 Usinghomogeneous coordinates3D nonuniform scaleCSE 167, Winter 20186 Usinghomogeneous coordinates3D rotation about X axisCSE 167, Winter 201873D rotation about Y axisCSE 167, Winter 201883D rotation about Z axisCSE 167, Winter 20189 Rotation matrix A rotation matrix is a special orthogonal matrix Properties of special orthogonal matrices Transformation matrix using homogeneous coordinatesCSE 167, Winter 201810 The inverse of a special orthogonal matrix is also a special orthogonal matrix3D rotations A 3D rotation can be parameterized with three numbers Common 3D rotation formalisms Rotation matrix 3x3 matrix (9 parameters), with 3 degrees of freedom Euler angl
Coordinate frames • In computer graphics, we typically use at least three ... – Camera up vector u CSE 167, Winter 2018 28 World coordinates Camera coordinates u e d. Camera matrix • Construct x c, y c, z c World coordinates Camera coordinates 29 u e d CSE 167, Winter 2018. Camera matrix • Step 1: Z‐axis • Step 2: X‐axis
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