Background Material
We need to discuss direction cosines. Consider the following:
A vector along the axis of rotation is:
vec(v) = [a b c] = [(x'-x) (y'-y) (z'-z)]
If this vector is normalized, then it becomes the vector of direction cosines.
Properties of direction cosines:
cos(alpha) is the angle between the axis of rotation and the x-axis.
cos(alpha) = a / sqrt((x'-x)^2+(y'-y)^2+(z'-z)^2)
cos(beta) = b / sqrt((x'-x)^2+(y'-y)^2+(z'-z)^2)
cos(gamma) = c / sqrt((x'-x)^2+(y'-y)^2+(z'-z)^2)
cos(alpha)^2 + cos(beta)^2 + cos(gamma)^2 = 1
This may seem a little connfusing, so we start with realizations in 2D as follows:
If the rotation axis passes through the origin (0,0) and a point (a,b) as illustrated above:
1) What is the length of the vector along the rotation axis through (a,b) starting at the origin?
2) What are the direction cosines of the rotation axis knowing that cos(theta) = adjacent / hypotenuse?
3) If we add a point (a',b') along the axis as follows:
4) What is a vector along the axis of rotation, given points (a,b) and (a',b')?
5) What is the length of this vector?
6) What are the direction cosines?
Problem1: Assume an axis of rotation defined by the two points (3,3) and (5,5). Find the direction cosines.
Problem2: Assume an axis of rotation defined by the two points (1,2) and (4,5). Find the direction cosines.
Are the direction cosines for Problem 1 the same as Problem 2 or different? What is the explanation for your result?