Pros and cons of transformation representations

doc/source/user_guide/transformations.rst

@@ -112,6 +112,22 @@ point :math:`{_A}\boldsymbol{p}` from frame :math:`A` to frame :math:`B`:
 You can use :func:`~pytransform3d.transformations.transform` to apply a
 transformation matrix to a homogeneous vector.
 
+**Pros**
+
+* It is easy to apply transformations on vectors in homogeneous coordinates by
+  matrix-vector multiplication.
+* Concatenation of transformations is trivial through matrix multiplication.
+* You can directly read the basis vectors and translation from the columns.
+* No singularities.
+
+**Cons**
+
+* We use 16 values for 6 degrees of freedom.
+* Not every 4x4 matrix is a valid transformation matrix, which means for
+  example that we cannot simply apply an optimization algorithm to
+  transformation matrices or interpolate between them. Renormalization is
+  computationally expensive.
+
 -----------------------
 Position and Quaternion
 -----------------------
@@ -131,6 +147,17 @@ a 2D array.
 pytransform3d uses a numpy array of shape (7,) to represent position and
 quaternion and typically we use the variable name pq.
 
+**Pros**
+
+* More compact than the matrix representation and less susceptible to
+  round-off errors.
+* Compact representation.
+
+**Cons**
+
+* Separation of translation and rotation component. Both have to be handled
+  individually.
+
 ----------------
 Screw Parameters
 ----------------
@@ -205,6 +232,20 @@ coordinates of transformation and typically we use the variable name Stheta.
     [4]_. They use a different order of the 3D vector components and they do
     not separate :math:`\theta` from the screw axis in their notation.
 
+**Pros**
+
+* Minimal representation.
+* Can also represent velocity and acceleration when we replace
+  :math:`\theta` by :math:`\dot{\theta}` or :math:`\ddot{\theta}` respectively,
+  which makes numerical integration and differentiation easy.
+
+**Cons**
+
+* There might be discontinuities and ambiguities. This has to
+  be considered. Normalization is recommended.
+* Concatenation and transformation of vectors requires conversion to
+  transformation matrix or dual quaternion.
+
 ---------------------------
 Logarithm of Transformation
 ---------------------------
@@ -302,6 +343,17 @@ quaternion encodes the translation component as
 :math:`\boldsymbol{t}` is a quaternion with the translation in the vector
 component and the scalar 0, and rotation quaternions have the same ambiguity.
 
+**Pros**
+
+* No singularities.
+* Efficient and compact form for representing transformations [7]_.
+
+**Cons**
+
+* The representation is not straightforward to interpret.
+* There are always two unit dual quaternions that represent exactly the same
+  transformation.
+
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 References
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