Degrees of Freedom in Affine Transformation and Homogeneous Transformation

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I understand that a 2D Affine Transformation has 6 DOF and a 2D Homogeneous Transformation has 8 DOF. However, how can I identify what those independent paramters are?



If we consider Euclidean Transformation, it has 3 DOF: rotation, translation in x and translation in y.
beginbmatrixC_theta&-S_theta&t_x\S_theta&C_theta&t_y\0&0&1endbmatrix
If we consider Similarity transform, it has 4 DOF: rotation, translation in x, translation in y and scaling.
beginbmatrixsC_theta&-sS_theta&t_x\sS_theta&sC_theta&t_y\0&0&1endbmatrix



1) Similarily, what makes up the 6 DOF of Affine matrix and 8 DOF of Homogeneous matrix?



2) Unlike the Euclidean and Similarity Transformation, is there no fixed set of DOF?



3) Can they be any six (if we take Affine as example) of rotation, translation (in x, y), scaling (in x, y), shearing, reflection etc. depending on the application?



4) If so, given an Affine matrix, can we know what the DOF are without knowledge of application?



Link1 says Affine transformation is a combination of translation, rotation, scale, aspect ratio and shear.
Link2 says it consists of 2 rotations, 2 scaling and traslations (in x, y).
Link3 indicates that it can be a combination of various different transformations.



I am a little confused about the whole idea. Thanks in advance.




geometry linear-transformations transformation projective-geometry affine-geometry




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    I understand that a 2D Affine Transformation has 6 DOF and a 2D Homogeneous Transformation has 8 DOF. However, how can I identify what those independent paramters are?



    If we consider Euclidean Transformation, it has 3 DOF: rotation, translation in x and translation in y.
    beginbmatrixC_theta&-S_theta&t_x\S_theta&C_theta&t_y\0&0&1endbmatrix
    If we consider Similarity transform, it has 4 DOF: rotation, translation in x, translation in y and scaling.
    beginbmatrixsC_theta&-sS_theta&t_x\sS_theta&sC_theta&t_y\0&0&1endbmatrix



    1) Similarily, what makes up the 6 DOF of Affine matrix and 8 DOF of Homogeneous matrix?



    2) Unlike the Euclidean and Similarity Transformation, is there no fixed set of DOF?



    3) Can they be any six (if we take Affine as example) of rotation, translation (in x, y), scaling (in x, y), shearing, reflection etc. depending on the application?



    4) If so, given an Affine matrix, can we know what the DOF are without knowledge of application?



    Link1 says Affine transformation is a combination of translation, rotation, scale, aspect ratio and shear.
    Link2 says it consists of 2 rotations, 2 scaling and traslations (in x, y).
    Link3 indicates that it can be a combination of various different transformations.



    I am a little confused about the whole idea. Thanks in advance.




    geometry linear-transformations transformation projective-geometry affine-geometry




    share|cite|improve this question





















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      I understand that a 2D Affine Transformation has 6 DOF and a 2D Homogeneous Transformation has 8 DOF. However, how can I identify what those independent paramters are?



      If we consider Euclidean Transformation, it has 3 DOF: rotation, translation in x and translation in y.
      beginbmatrixC_theta&-S_theta&t_x\S_theta&C_theta&t_y\0&0&1endbmatrix
      If we consider Similarity transform, it has 4 DOF: rotation, translation in x, translation in y and scaling.
      beginbmatrixsC_theta&-sS_theta&t_x\sS_theta&sC_theta&t_y\0&0&1endbmatrix



      1) Similarily, what makes up the 6 DOF of Affine matrix and 8 DOF of Homogeneous matrix?



      2) Unlike the Euclidean and Similarity Transformation, is there no fixed set of DOF?



      3) Can they be any six (if we take Affine as example) of rotation, translation (in x, y), scaling (in x, y), shearing, reflection etc. depending on the application?



      4) If so, given an Affine matrix, can we know what the DOF are without knowledge of application?



      Link1 says Affine transformation is a combination of translation, rotation, scale, aspect ratio and shear.
      Link2 says it consists of 2 rotations, 2 scaling and traslations (in x, y).
      Link3 indicates that it can be a combination of various different transformations.



      I am a little confused about the whole idea. Thanks in advance.




      geometry linear-transformations transformation projective-geometry affine-geometry




      share|cite|improve this question











      I understand that a 2D Affine Transformation has 6 DOF and a 2D Homogeneous Transformation has 8 DOF. However, how can I identify what those independent paramters are?



      If we consider Euclidean Transformation, it has 3 DOF: rotation, translation in x and translation in y.
      beginbmatrixC_theta&-S_theta&t_x\S_theta&C_theta&t_y\0&0&1endbmatrix
      If we consider Similarity transform, it has 4 DOF: rotation, translation in x, translation in y and scaling.
      beginbmatrixsC_theta&-sS_theta&t_x\sS_theta&sC_theta&t_y\0&0&1endbmatrix



      1) Similarily, what makes up the 6 DOF of Affine matrix and 8 DOF of Homogeneous matrix?



      2) Unlike the Euclidean and Similarity Transformation, is there no fixed set of DOF?



      3) Can they be any six (if we take Affine as example) of rotation, translation (in x, y), scaling (in x, y), shearing, reflection etc. depending on the application?



      4) If so, given an Affine matrix, can we know what the DOF are without knowledge of application?



      Link1 says Affine transformation is a combination of translation, rotation, scale, aspect ratio and shear.
      Link2 says it consists of 2 rotations, 2 scaling and traslations (in x, y).
      Link3 indicates that it can be a combination of various different transformations.



      I am a little confused about the whole idea. Thanks in advance.





      geometry linear-transformations transformation projective-geometry affine-geometry





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