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| We use the [http://blake.bcm.edu/eman2/doxygen_html/classEMAN_1_1Transform.html Transform] class for storing/managing Euler angles,translations, scales and x mirroring. At any time a Transform ({{{$$X$$}}}) object defines a group of 4 transformations of a rigid body that are applied in a specific order, namely {{{$$X \equiv M T S R $$}}} Where {{{$$M }}} is a mirroring operation about the x-axis, {{{$$T} }}} is a translation, {{{$$S$$}}} is a uniform, positive, non zero scaling operation and {{{$$R$$}}} is a rotation. The Transformobject stores these transformations internally in a 4x4 matrix, as is commonly the case in computer graphics applications that use homogeneous coordinate systems (i.e. OpenGL). In these approaches the 4x4 transformation matrix {{{$$X$$}}} is constructed in this way {{{$$ X = [[sMR,M\mathbf{t}],[\mathbf{0}^T,1]]$$}}} Where {{{$$s$$}}} is the constant scaling factor, {{{$$M$$}}} is the option x-mirroring operation which identity, except in the case of x mirroring where the (0,0) entry is -1, {{{$$R$$}}} is a {{{$$3x3$$}}} rotation matrix and {{{$$\mathbf{t}=(dx,dy,dz)^T$$}}} is a post translation. In this approach a 3D point {{{$$\mathbf{p}=(x,y,z)^T$$}}} as represented in homogeneous coordinates as a 4D vector {{{$$\mathbf{p}_{hc}=(x,y,z,1)^T$$}}} and is multiplied by the matrix {{{$$M$$}}} to produce the result of applying the transformation {{{$$ X \mathbf{p}_{hc} = ( (sMR\mathbf{p} + M\mathbf{t})^T, 1 )^T $$}}} In this way the result of applying a Transform is a rotation followed by scaling operation, followed by a translation and then finally the x mirroring operation is (optionally) applied. |
What is a Transform?
We use the [http://blake.bcm.edu/eman2/doxygen_html/classEMAN_1_1Transform.html Transform] class for storing/managing Euler angles,translations, scales and x mirroring. At any time a Transform ($$X$$) object defines a group of 4 transformations of a rigid body that are applied in a specific order, namely
$$X \equiv M T S R $$
Where $$M is a mirroring operation about the x-axis, $$T} is a translation, $$S$$ is a uniform, positive, non zero scaling operation and $$R$$ is a rotation. The Transformobject stores these transformations internally in a 4x4 matrix, as is commonly the case in computer graphics applications that use homogeneous coordinate systems (i.e. OpenGL). In these approaches the 4x4 transformation matrix $$X$$ is constructed in this way
$$ X = [[sMR,M\mathbf{t}],[\mathbf{0}^T,1]]$$
Where $$s$$ is the constant scaling factor, $$M$$ is the option x-mirroring operation which identity, except in the case of x mirroring where the (0,0) entry is -1, $$R$$ is a $$3x3$$ rotation matrix and $$\mathbf{t}=(dx,dy,dz)^T$$ is a post translation. In this approach a 3D point $$\mathbf{p}=(x,y,z)^T$$ as represented in homogeneous coordinates as a 4D vector $$\mathbf{p}_{hc}=(x,y,z,1)^T$$ and is multiplied by the matrix $$M$$ to produce the result of applying the transformation
$$ X \mathbf{p}_{hc} = ( (sMR\mathbf{p} + M\mathbf{t})^T, 1 )^T $$
In this way the result of applying a Transform is a rotation followed by scaling operation, followed by a translation and then finally the x mirroring operation is (optionally) applied.