ABSTRACT
This chapter presents some plane curves1 that are useful in designing nonimaging optics.
The magnitude of a vector v is given by
v = √ ____
A normalized vector with the same direction as v, but with unit magnitude can be obtained from
nrm v = v___ v
= v_____√ ____
The distance [A, B] between two points A and B is given by the magnitude of the vector B – A, that is, B – A or
[A, B] = √ ________________
The angle between two vectors u and v is given by
ang(v, u) = θ = arc cos (v ⋅ u______ v u) = arc cos ( v ⋅ u__________√
____ v ⋅ v √ u ⋅ u ) (17.4)
This angle, however, is 0 ≤ θ ≤ π. Consider then that vectors u and v are 2-D and that u = (u1, u2) and v = (v1, v2). We can defi ne the vectors U = (u1, u2, 0) and V = (v1, v2, 0) in three dimensions. The cross product of U and V is
U × V = (0, 0, u1v2 − u2v1) (17.5)
If the third component of U × V is positive, then v is in the counterclockwise direction from u. Also, if the third component of U × V is negative, then v is in the clockwise direction from u. We now defi ne the angle in the positive direction from u to v as
angp(v, u) = ang(v, u) if u1v2 − u2v1 ≥ 0
angp(v, u) = 2π − ang(v, u) if u1v2 − u2v1 < 0 (17.6)
the range 0 ≤ φ < 2π.
