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**| = √(*r*² + *b*²) *s* = *t* √(*r*² + *b*²)

*u* = |**u**| = √(*q*² + *c*²) *w* = *x* √(*q*² + *c*²)

Pitch and slope

pitch length, P = 2π*b* slope, P/S = *b*/*r*

pitch time, M = 2π*c* 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* = √(*x*_{1}² + *x*_{2}² + *x*_{3}²)

**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* = √(*t*_{1}² + *t*_{2}² + *t*_{3}²)

**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

d**x**(*t*)/d*t* = **v**(*t*) = − *ωr* sin(*ωt*) **i** + *ωr* cos(*ωt*) **j** + *b***k**

d**v**(*t*)/d*t* = **a**(*t*) = −* ω*²*r* cos(*ωt*) **i** + *ω*²*r* sin(*ωt*) **j**

d**t**(*x*)/d*x* = **u**(*x*) = − *ψq* sin(*ψx*) **i** + *ψq* cos(*ψx*) **j** + *c***k**

d**u**(*x*)/d*x* = **b**(*x*) = −* ψ*²*q* cos(*ψx*) **i** + *ψ*²*q* sin(*ψx*) **j**