iSoul In the beginning is reality.

# Spiral/helical motion

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