We will define an arbitrary line by a point the line goes through and a direction vector. If the axisof rotation is given by two pointsP1= (a,b,c) andP2= (d,e,f), then a direction vector can beobtained by?u,v,w?=?d-a,e-b,f-c?. We can now write a transformation for the rotation ofa point about this line.
Assuming that?u,v,w?is a unit vector so thatL= 1, we obtain a more practical result forTP1-1Txz-1Tz-1Rz(θ)TzTxzTP1.
繞任意軸旋轉(zhuǎn)
If we multiply this times?x,y,z?we can obtain a function of of ten variables that yields theresult of rotating the point (x,y,z) about the line through (a,b,c) with direction vector?u,v,w?(whereu2+v2+w2= 1) by the angleθ.
旋轉(zhuǎn)后的目標(biāo)點(diǎn)
