The outline below is also in pdf form *here*.

Angular speed (velocity) and angular pace (legerity)

- Speed,
*v*= Δ*s*/Δ*t*, Pace,*u*= Δ*t*/Δ*s*so*u*= 1/*v*and*v*= 1/*u*except if*u*or*v*are zero- Zero speed: no motion but time changes because time is independent
- Zero pace: no motion but length changes because length is independent

Circular motion in space and time

distance, *x*; distime, *t*; radius *r* (or *a* or R or A); period radius *q*; circumference S = 2π*r* = wavelength *λ*; period T = 2π*q* = wavetime *μ*; angular velocity, *v*; angular legerity, *u*; arc length, *s*; arc time, *w*

- Circle in space
- space angle
*θ*, arc length*s*, radius*r* - angle in space:
*θ*≡*s*/*r; r*=*s*/*θ*;*s*=*rθ;*1/*θ*=*r*/*s*; 1/*r*=*θ*/*s* - angular time rate:
*ω*≡*θ*/*t; t*=*θ*/*ω*;*θ*=*ωt*; 1/*ω*=*t*/*θ*; 1/*t*=*ω*/*θ*

- space angle
- Circle in time
- time angle
*φ*, arc time*w*, period radius*q* - angle in time,
*φ*≡*w*/*q*;*q*=*w*/*φ*;*w*=*qφ*; 1/*φ*=*q*/*w*; 1/*q*=*φ*/*w* - angular space rate:
*ψ*≡*φ*/*x*;*x*=*φ*/*ψ*;*φ*=*ψx*; 1/*ψ*=*x*/*φ*; 1/*x*=*ψ*/*φ*

- time angle

Angular time rates

- Independent variable is time, dependent variable is length

- Angular velocity: time rate of rotation,
*ω*≡*θ*/*t**t*=*θ*/*ω*;*θ*=*ωt* - Wave speed: wavelength per unit time,
*v = s*/*t*= S/T =*r*/*q*=*ωr*- frequency,
*f*≡ 1/T =*v*/S =*v*/*λ*=*v*/s =*vh*=*h*/*u* - wavelength,
*λ*=*v*/*f*

- frequency,
- Wave speed normalized
- revolutions: If S = 1, then
*v*= 1/T =*f* - space radians: If
*r*= 1, then*s*=*θ*=*φ*and*v*=*θ*/*t*=*φ*/*t*=*s*/*t*=*ω*= 1/*q*= 2π/T = 2π*f**ω*= 2π*f*= 2π/T =*θ*/*t*=*φ*/*t*;*q*= T/2π;*f*=*ω*/2π

- revolutions: If S = 1, then

Angular space rates

- Independent variable is length, dependent variable is time

- Angular legerity: space rate of rotation,
*ψ*≡*φ*/*x*;*x*=*φ*/*ψ*;*φ*=*ψx* - Wave pace: wavetime per unit length,
*u = w*/*s*= T/S =*q*/*r*=*ψq*- periodicity,
*h*≡ 1/S =*u*/T =*u*/*μ*= s/*v*=*uf*=*f*/*v* - wavetime,
*μ*=*u*/*h*

- periodicity,
- Wave pace normalized
- revolutions: If T = 1, then
*u*= 1/S =*h* - time radians: If
*q*= 1, then*w*=*φ*=*θ*and*u*=*φ*/*x*=*θ*/*x*=*w*/*s*=*ψ*= 1/*r*= 2π/S = 2π*h**ψ*= 2π*h*= 2π/S =*θ*/*x*=*φ*/*x*;*r*= S/2π;*h*=*ψ*/2π

- revolutions: If T = 1, then

Parametric equations

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

**x**(*θ*) = *r* cos(*θ*) **i** + *r* sin(*θ*) **j**

**x**(*t*) = *r* cos(*ωt*) **i** + *r* sin(*ωt*) **j**;* v*(*t*) = *ωr*;** a**(*t*) = −*ω*²**x**(*t*)

**x**(*s*) = *r* cos(*s*/*r*) **i** + *r* sin(*s*/*r*) **j**

**t**(*φ*) = *q* cos(*φ*) **i** + *q* sin(*φ*) **j**

**t**(*x*) = *q* cos(*ψx*) **i** + *q* sin(*ψx*) **j**; *u*(*x*) = *ψq*;** b**(*x*) = −*ω*²**t**(*x*)**
t**(

*w*) =

*q*cos(

*w*/

*q*)

**i**+

*q*sin(

*w*/

*q*)

**j**