Editing Rotation matrix (section)

=== Axis and angle ===
{{Main|Axis–angle representation}}
To efficiently construct a rotation matrix {{mvar|Q}} from an angle {{mvar|θ}} and a unit axis {{math|'''u'''}}, we can take advantage of symmetry and skew-symmetry within the entries. If {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are the components of the unit vector representing the axis, and

:<math>\begin{align}
c &= \cos \theta\\
s &= \sin \theta\\
C &= 1-c
\end{align}</math>

then

:<math>Q(\theta) = \begin{bmatrix}
xxC+c  & xyC-zs & xzC+ys\\
yxC+zs & yyC+c  & yzC-xs\\
zxC-ys & zyC+xs & zzC+c
\end{bmatrix}</math>

Determining an axis and angle, like determining a quaternion, is only possible up to the sign; that is, {{math|('''u''', ''θ'')}} and {{math|(−'''u''', −''θ'')}} correspond to the same rotation matrix, just like {{math|''q''}} and {{math|−''q''}}. Additionally, axis–angle extraction presents additional difficulties. The angle can be restricted to be from 0° to 180°, but angles are formally ambiguous by multiples of 360°. When the angle is zero, the axis is undefined. When the angle is 180°, the matrix becomes symmetric, which has implications in extracting the axis. Near multiples of 180°, care is needed to avoid numerical problems: in extracting the angle, a [[Atan2|two-argument arctangent]] with {{math|[[atan2]](sin ''θ'', cos ''θ'')}} equal to {{mvar|θ}} avoids the insensitivity of arccos; and in computing the axis magnitude in order to force unit magnitude, a brute-force approach can lose accuracy through underflow {{Harv|Moler|Morrison|1983}}.

A partial approach is as follows:

:<math>\begin{align}
 x &= Q_{zy} - Q_{yz}\\
 y &= Q_{xz} - Q_{zx}\\
 z &= Q_{yx} - Q_{xy}\\
 r &= \sqrt{x^2 + y^2 + z^2}\\
 t &= Q_{xx} + Q_{yy} + Q_{zz}\\
 \theta &= \operatorname{atan2}(r,t-1)\end{align}</math>

The {{mvar|x}}-, {{mvar|y}}-, and {{mvar|z}}-components of the axis would then be divided by {{mvar|r}}. A fully robust approach will use a different algorithm when {{mvar|t}}, the [[Trace (linear algebra)|trace]] of the matrix {{mvar|Q}}, is negative, as with quaternion extraction. When {{mvar|r}} is zero because the angle is zero, an axis must be provided from some source other than the matrix.