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| Deletions are marked like this. | Additions are marked like this. |
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| {{{$$Transform \equiv M T S R $$}}} | {{{$$ Tr \equiv M T S R $$}}} |
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| 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 | 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 |
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| {{{$$ Transform = [[sMR,M\mathbf{t}],[\mathbf{0}^T,1]]$$}}} | {{{$$ Tr = [[sMR,M\mathbf{t}],[\mathbf{0}^T,1]]$$}}} |
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| {{{$$ Transform \mathbf{p}_{hc} = ( (sMR\mathbf{p} + M\mathbf{t})^T, 1 )^T $$}}} | {{{$$ Tr \mathbf{p}_{hc} = ( (sMR\mathbf{p} + M\mathbf{t})^T, 1 )^T $$}}} |
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| 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. | 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. |
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| = Constructing a Transform = | = The Transforms object in Python = == Constructing a Transform == There a three ways to construct a Transform object in Python {{{#!python t = Transform() # default constructor, t is the identity t = Transform({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"dx":3.4}) # construction using a dictionary s = Transform(t) # copy construction - s is precisely the same as t }}} == Setting/getting rotations == {{{#!python t = Transform() t.set_rotation({"type":"spider","phi":32,"theta":12,"psi":-100}) spider_rot = t.get_rotation("spider") eman_rot = t.get_rotation("eman") s = Transform(eman_rot) # works fine }}} == Setting/getting scale == {{{#!python t = Transform() t.set_scale(2.0) scale = t.get_scale() s = Transform({"scale":scale}) # set scale as part of construction }}} == Setting/getting mirror == {{{#!python t = Transform() t.set_mirror(2.0) mirror = t.get_mirror() s = Transform({"mirror":mirror}) # set mirror as part of construction }}} == Setting/getting translation == {{{#!python t = Transform() t.set_trans(1,2,3) # method 1 t.set_trans(Vec3f(1,2,3)) # method 2 t.set_trans([1,2,3]) # method 3 - the tuple is converted to a Vec3f automatically v = t.get_trans() 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 {{{#!python t = Transform() t.set_params({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"dx":3.4}) d = t.get_params("eman") # must specify the euler convention s = Transform(d) # s is the same as t d = t.get_params("spider") s = Transform(d) # s is the same as t d = t.get_params("matrix") s = Transform(d) # s is the same as t }}} |
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
Setting/getting rotations
Setting/getting scale
Setting/getting mirror
Setting/getting translation
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