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.

The Transforms object in Python

Constructing a Transform

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

Setting/getting rotations

   1 t = Transform()
   2 t.set_rotation({"type":"spider","phi":32,"theta":12,"psi":-100})
   3 spider_rot = t.get_rotation("spider")
   4 eman_rot = t.get_rotation("eman")
   5 s = Transform(eman_rot) # works fine

Setting/getting scale

   1 t = Transform()
   2 t.set_scale(2.0)
   3 scale = t.get_scale()
   4 s = Transform({"scale":scale}) # set scale as part of construction

Setting/getting mirror

   1 t = Transform()
   2 t.set_mirror(2.0)
   3 mirror = t.get_mirror()
   4 s = Transform({"mirror":mirror}) # set mirror as part of construction

Setting/getting translation

   1 t = Transform()
   2 t.set_trans(1,2,3) # method 1
   3 t.set_trans(Vec3f(1,2,3)) # method 2
   4 t.set_trans([1,2,3]) # method 3 - the tuple is converted to a Vec3f automatically
   5 v = t.get_trans()
   6 s = Transform("dx":v[0],"dy":v[1],"dz":v[2]) # set translation as part of construction

Setting/getting parameters

You can tell a Transform deduce any of its parameters from a dictionary. Similarly you can get the parameters of a Transform as a dictionary

   1 t = Transform()
   2 t.set_params({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"dx":3.4})
   3 d = t.get_params("eman") # must specify the euler convention
   4 s = Transform(d) # s is the same as t
   5 d = t.get_params("spider")
   6 s = Transform(d) # s is the same as t
   7 d = t.get_params("matrix")
   8 s = Transform(d) # s is the same as t