It almost seems to me that the shear is combination of translation and rotation, while maintaining the linear map properties without using homogenous coordinates. Do you agree with this thought ? Moreover, what types of motion is shear used for in graphics ?
graphicstar11
shear is not linear, but affine. For addition and scalar transformations, shear does not seem to follow the rules of linearity
keenan
@dranzer A shear is not a combination of rotation and translation; it's a combination of two rotations and a scaling---see in particular the discussion of singular value decomposition on this slide.
keenan
@graphicstar11 Shear is definitely linear---it says it right there on the slide! ;-). In general, any transformation that can be represented as $\mathbf{x} \mapsto A\mathbf{x}$ for some fixed matrix $A$ is linear, since $A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y}$ and $A(a\mathbf{x}) = aA\mathbf{x}$.
It almost seems to me that the shear is combination of translation and rotation, while maintaining the linear map properties without using homogenous coordinates. Do you agree with this thought ? Moreover, what types of motion is shear used for in graphics ?
shear is not linear, but affine. For addition and scalar transformations, shear does not seem to follow the rules of linearity
@dranzer A shear is not a combination of rotation and translation; it's a combination of two rotations and a scaling---see in particular the discussion of singular value decomposition on this slide.
@graphicstar11 Shear is definitely linear---it says it right there on the slide! ;-). In general, any transformation that can be represented as $\mathbf{x} \mapsto A\mathbf{x}$ for some fixed matrix $A$ is linear, since $A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y}$ and $A(a\mathbf{x}) = aA\mathbf{x}$.