Math
Relations
(1)/(a)±(1)/(b) = (b±a)/(ab)
Approximations
Aexp( − τ) ≈ (1 − τ)A, if τ ≪ 1.
f(x) ≃ f(x0) + (f’(x0))/(1!)(x − x0) + ... + (fn(x0))/(n!)(x − x0)n
Paraxial angles
tanθ ≈ θ
sinθ ≈ θ
cosθ ≈ 1 − (θ2)/(2) ≈ 1
Trigonometric relations
sinα + sinβ = 2sin⎛⎝(α + β)/(2)⎞⎠cos⎛⎝(α − β)/(2)⎞⎠ = 2sin(α)/(2)cos(α)/(2) + 2sin(β)/(2)cos(β)/(2)
sinα + sinβ = 2⎛⎝cos2(β)/(2) + sin2(β)/(2)⎞⎠×⎛⎝sin(α)/(2)cos(α)/(2)⎞⎠ + 2⎛⎝cos2(α)/(2) + sin2(α)/(2)⎞⎠×⎛⎝sin(β)/(2)cos(β)/(2)⎞⎠
Ellipse
(x2)/(a2) + (y2)/(b2) = 1, where a > b and the focus on x axis.
x = acosθ and y = bsinθ.
The focus are ±c = ±a×e, where the eccentricity is e = √(1 − (b2)/(a2)).
Parametric ellipse
Or Trammel of Archimedes is the definition of a ≡ p + q and b = q.
If q ≡ 1, then p = √(1 − e2) − 1.
Rotation matrix (Euler angles α, β, γ)
In python:
R = np.dot(Rz,(np.dot(R_x,R_y)))Gaussian
f(x) = ae − ((x − b)2)/(2c2) + d
∫∞ − ∞f(x) dx = ac√(2π), since ∫∞ − ∞e − x2dx = √(π).
The scale height H is P = P0e − h ⁄ H.