e******r
发帖数: 220 | 1 Every rotation in space has an axis , and is given by an orthogonal matrix(
Why?). The first fact is the one that is needed to prove that in three
dimensional space, rotating around one axis and then rotating around another
axis has the same result as rotating around some third axis. This is a far
from obvious geometric fact(and isn't true in four dimensional space...Why?).
In which dimensions is it true?
Thanks | c*******v
发帖数: 2599 | 2 他这个结论都不对，i,j,k三个坐标轴都是独立的，
例如绕i,j旋转了，转轴就是x=a i+b j
,怎么可能把x用k表示出来? 难道他说的是x是第三个轴？
那这不是数学问题，是个运动的分解问题，就跟从A走到B等价于
从A->O,然后从O->B一样道理，这是实验决定的。
. | b*****e
发帖数: 1 | 3 By definition, rotations preserve length and origin, which means
for all points x and y in the space. |T(x)-T(y)|=|x-y|, where T is
the rotation, and T(0)=0. Easy to see such a rotation is a
linear operation.
Let ' donate transpose of vectors and matrices.
● is a dot product.
Since |T(x)-T(y)|=(T(x)-T(y●(T(x)-T(y))=(x-y) ●x-y), let y=0,
we can get T(x) ●T(x)=x●x. Also, since T is linear, we can
write T(x) as Ax, where A is a n by n matrix. Then we have:
T(x) ●T(x)= T(x)'T(x) =(Ax)'(Ax)=x'A'Ax=
【在 e******r 的大作中提到】
: Every rotation in space has an axis , and is given by an orthogonal matrix(
: Why?). The first fact is the one that is needed to prove that in three
: dimensional space, rotating around one axis and then rotating around another
: axis has the same result as rotating around some third axis. This is a far
: from obvious geometric fact(and isn't true in four dimensional space...Why?).
: In which dimensions is it true?
: Thanks
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