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It is well known that there are only two types of orthogonal transformations of $\mathbb{R}^2$, namely rotations and reflections. Moreover, they have a nice calculus as outlined in https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions By this I mean: composition of two reflections is a rotation. Rotation before/after reflection is reflection. Reflection after reflection is rotation. All transformations are assumed to fix the origin.

My question is: What is the counterpart of this for $\mathbb{R}^3$? There are reflections and rotations. What is their "calculus"? Do we need other types? Any reference or link would be greatly appreciated!

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  • $\begingroup$ Orthogonal transformations can be represented by matrices. Reflections have determinant -1, rotations have determinant +1. These compositions you speak of can be traced back to the determinant of matrix multiplication. It works a similar way in $\mathbb{R}^3$. $\endgroup$ – Zeno Apr 23 at 23:34
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This is typically defined as the orthogonal group which is denoted $O(n)$ and can be defined as

$$ O(n) \;\; =\;\; \{A \in M_n(\mathbb{R}) \; | \; AA^T = A^TA = I\}. $$

These matrices have orthonormal columns and rows and preserve the Euclidean dot product in that if $Q \in O(n)$ then

$$ \langle Qx, Qy\rangle \;\; =\;\; x^TQ^TQy \;\; =\;\; x^Ty \;\; =\;\; \langle x,y\rangle. $$

Similarly $O(n)$ also breaks up into two connected pieces, one containing the rotations and the other containing reflections (the group of rotations typically denoted as $SO(n)$ and called the special orthogonal group). One can easily tell whether an orthogonal matrix is a reflection or a rotation by computing its determinant. You can easily show that $\det Q = \pm 1$ for all $Q \in O(n)$. Rotations are those matrices with $\det Q = 1$ and reflections are those with $\det Q = -1$.

The calculus part comes into play because both $O(n)$ and $SO(n)$ are smooth manifolds, and better yet are smooth manifolds with a compatible group structure making them both into Lie groups. Therefore it makes sense to talk about smoothly transitioning from one rotation to another (although one cannot smoothly transition between a rotation to a reflection).

In the special case of $n=3$, there are some nice properties of orthogonal matrices. One that's particularly i nice is that any orthogonal matrix can be represented by a product of fundamental rotations about each axis. In particular we can define the matrices

\begin{eqnarray*} R_z(\theta) & = & \left [ \begin{array}{ccc} \cos \theta & - \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0& 1 \\ \end{array} \right ] \\ R_y(\phi) & = & \left [ \begin{array}{ccc} \cos \phi & 0 & - \sin \phi \\ 0 & 1& 0 \\ \sin \phi & 0 & \cos \phi \\ \end{array} \right ] \\ R_x(\gamma) & = & \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \\ \end{array} \right ] \\ \end{eqnarray*}

Then any orthogonal matrix can be written as a product of the above matrix for different values of $\theta, \phi, \gamma$ which are the roll, pitch, and yaw angles.

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  • $\begingroup$ As it follows from this answer, the reflections through hyperplanes always generate the orthogonal group. Note also that reflection to a 2-codimensional subspace is actually a rotation by $180^\circ$ (in the orthogonal 2d plane). $\endgroup$ – Berci Apr 24 at 1:19
  • $\begingroup$ Thanks, as I wrote, I am interested in knowing what the basic types are, what their calculus rules are, and references. Please take a look at the wikipedia page for $n=2$ I linked to above, especially the rules for composition of rotations and reflections. That is what I am after. $\endgroup$ – max_zorn Apr 24 at 4:24
  • $\begingroup$ @max_zorn Define what you mean by "calculus." This isn't clear to me. Orthogonal transformations are an important example of differential geometry and therefore are subject to the rules of calculus (i.e. differentiation, smooth manifolds, etc.) but if by "calculus" you mean "composition of reflections and rotations", then your answer basically follows from Berci's comment above. Notice each of $R_x, R_y, R_z$ are actually inclusions of $SO(2)$. Pair each of these with the corresponding reflection and you're done. $\endgroup$ – Mnifldz Apr 24 at 4:39
  • $\begingroup$ @Mnifldz OK, I will try to be more clear. Consider $n=2$. If you consider the product of two reflections, you get a rotation. Next, a product of a rotation and a reflection is still a reflection. And finally, the product of two rotations is still a rotation. Now consider $n=3$. What is the product of two reflections geometrically? A rotation? Suppose it is. Is the rotation times a reflection a reflection? $\endgroup$ – max_zorn Apr 24 at 4:48
  • $\begingroup$ @max_zorn Oh, that's really simple. Because reflections have determinant $-1$ and rotations have determinant $+1$, and given that $\det(QS) = \det(Q)\det(S)$ we have that any rotations multiplied is another rotation, any two reflection multiplied is a rotation, and multiplying a rotation and a reflection gives another reflection. Because this is based on the determinant it holds for all dimensions $n$. $\endgroup$ – Mnifldz Apr 24 at 5:48

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