The outline below is also available in pdf form here.
Spiral/Helical Motion
A helix is the geodesic of a cylinder; if we develop the cylinder on which the helix is traced, the helix becomes a straight line. Radius r (or a or R or A); velocity v, arc length s, arc time, w, pitch length P; pitch time, M; pitch angle α; pitch time angle β
Constants
v = |v| = √(r² + b²) s = t √(r² + b²)
u = |u| = √(q² + c²) w = x √(q² + c²)
Pitch and slope
pitch length, P = 2πb slope, P/S = b/r
pitch time, M = 2πc time slope, M/T = c/q
Pitch angle
α = atan(P/S) = atan(b/r) β = atan(M/T) = atan(c/q)
Arc length of one winding L = √(P² + S²)
Parametric equations
3D space
|x| = x = √(x1² + x2² + x3²)
x(θ) = r cos(θ) i + r sin(θ) j + (bθ/ω) k
x(t) = r cos(ωt) i + r sin(ωt) j + bt k
x(s) = r cos(s/r) i + r sin(s/r) j + (bs/r) k
3D time
|t| = t = √(t1² + t2² + t3²)
t(φ) = q cos(φ) i + q sin(φ) j + (cφ/ψ) k
t(x) = q cos(ψx) i + q sin(ψx) j + cx k
t(w) = q cos(w/q) i + q sin(w/q) j + (cw/q) k
Derivatives
dx(t)/dt = v(t) = − ωr sin(ωt) i + ωr cos(ωt) j + bk
dv(t)/dt = a(t) = − ω²r cos(ωt) i + ω²r sin(ωt) j
dt(x)/dx = u(x) = − ψq sin(ψx) i + ψq cos(ψx) j + ck
du(x)/dx = b(x) = − ψ²q cos(ψx) i + ψ²q sin(ψx) j