Arclength Calculation
r(t) = Position at a time.
 To calculate distance, multiply Speed and Time
 Speed is the magnitude of velocity
 Distance (S), also known as arclength is equal to the integral of the magnitude of V, times dt (a tiny chunk at that time) from time 1, to time 2.
FrenetSerret Frame (Moving Triad)
 This is a new coordinate system
 Made up of Unit Tangent (T), Unit Normal (N), and a perpendicular vector known as the Binormal (B)
 The Binormal is found by taking the cross product of T and N

Right hand rule applies here!
 Fingers along T
 Curl fingers towards N.
 Thumb points in direction of B

Another nice trick, is that N always points to the center of curvature.
 If T is going in a direction where the curve is increasing, N is facing up
 If T is going in a direction where the curve is decreasing, N is going down

Another unique thing about this coordinate system, is that T, N and B are all functions of time

T(t), N(t), B(t)

This coordinate system can warp with time
Black = B Blue = T Red = N
Acceleration
Whatβs cool about Acceleration along a curved path like such, is acceleration has two components
at = Transintestinal Acceleration an = Normal Acceleration
Acceleration
$a=a_{t}β+a_{n}β$at = Transintestinal Acceleration
$a_{t}=aβT^$ $a_{t}=aββ£β£vβ£β£vβ$an = Normal Acceleration
$a_{n}=aβN^$ $a_{n}=β£β£vβ£β£β£β£aΓvβ£β£β$Other helpful FrenetSerret Stuff
$v=β£β£vβ£β£T^$V = Magnitude of Velocity (Speed) T = Direction of Unit Tangent (Direction)
$B^=β£β£vΓaβ£β£vΓaβ$ This is quite a rigorous proof
 Note, in the bottom, he does switch
a X v
tov X a
 Normally, this would switch the direction of the perpendicular product, but since weβre taking the magnitude, we do not care.
 A rearrangement of the previous formula