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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 object defines a group of 4 transformations of a rigid body that are applied in a specific order, namely

$$ Tr \equiv M T S R $$

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 is constructed in this way

$$ Tr = [[sMR,M\mathbf{t}],[\mathbf{0}^T,1]]$$

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

$$ Tr \mathbf{p}_{hc} = ( (sMR\mathbf{p} +  M\mathbf{t})^T, 1 )^T $$

In this way the result of applying a Transform is a rotation followed by a scaling, followed by a translation and then finally the x mirroring operation is (optionally) applied.

Constructing a Transform in Python

There a three ways to construct a Transform object in Python

   1 t = Transform() # default constructor, t is the identity
   2 t = Transform({""type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"dx":3.4}) # construction using a dictionary
   3 s = Transform(t) # copy construction - s is precisely the same as t