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We know that in spherical coordinates angle $\theta$ and $\phi$ (two angles)are enough to express three dimensional rotation of matrix. But to express rotation mathematically as a transformation matrix we require three angles. But intuitively I expect only two parameters for rotation matrix based on the knowledge of spherical coordinates. What is wrong here?

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    $\begingroup$ Two angles represent the direction of the axis (in spherical coordinates) about which the rotation is to be performed. The third angle is the magnitude of the rotation. $\endgroup$ – lemon Apr 10 '15 at 18:34
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As pointed out by lemon, two angles are enough to specify a direction in a three dimensional coordinate system, but another is needed to specify a complete coordinate transformation. You can think of a rotation transformation in three dimensions as a mapping between two different coordinate systems. Two angles are needed to specify the relative pointing between the two z axes, but another is needed to specify the relative pointing of the x axis. Without this third angle the x and y axes could lie anywhere in the plane perpendicular to the new z axis.

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  • $\begingroup$ How many parameters would be needed to fully specify a rotation transformation in R^4? $\endgroup$ – Ethan Reesor Apr 10 '15 at 19:09
  • $\begingroup$ will it be ${}^n C_2 = 1/2 n(n-1)$ for $R^n$? so 6.You can make rotations on ${}^n C_2$ planes in $n$-dimensional space, which coincides with the number of generators of SO(n). $\endgroup$ – innisfree Apr 10 '15 at 19:18
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    $\begingroup$ @FireLizzard: Rotations are orthogonal transformations, that is $R \cdot R^T = 1$. This gives $n(n+1)/2$ constrains to a $n \times n$ matrix which therefore leaves $n^2 - n(n+1)/2$ free parameters. For $n=4$ one has 6 parameters. Take for example the homogeneous lorentz group, which is nothing more than usual rotation in a (pseudo)-euclidean space with 4 dimensions (3 boost parameter + 3 space rotation parameter). $\endgroup$ – image Apr 10 '15 at 19:31
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    $\begingroup$ In other words, $N(N-1)/2$ parameters are needed to specify a rotation in $N$-dimensional space. The only case where this is equal to $N$ is N=3. As noted above, six parameters are needed to describe rotation in four dimensional space. Also noteworthy: only one parameter is needed to describe rotation in two dimensional space. $\endgroup$ – David Hammen Apr 10 '15 at 21:33

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