A Compendium of Curve Formulae
In the following r(t) = x(t) , y(t) , z(t) is a parametrization of a curve. The vectors
T̂(t), N̂(t), and B̂(t) = T̂(t) × N̂(t) are the unit tangent, normal and binormal vectors,
respectively, at r(t). The tangent vector points in the direction of travel (i.e. direction of
increasing t) and the normal vector points toward the centre of curvature. The arc length
from time 0 to time t is denoted s(t). Then
◦ the velocity
v(t) =
dr
dt (t)
◦ the acceleration
a(t) =
d2 r
(t)
dt2
◦ the speed
ds
dt (t)
◦ the arc length
=
=
ds
dt (t) T̂(t)
d2 s
(t) T̂(t)
dt2
+ κ(t)
2
ds
(t)
N̂(t)
dt
= |v(t)| = dr
dt (t)
R1p
Rt
(τ
)
dτ
=
x′ (τ )2 + y ′ (τ )2 + z ′ (τ )2 dτ
s(t) = 0 ds
dt
0
◦ the curvature
κ(t) =
◦ the radius of curvature
ρ(t) =
|v(t) × a(t)|
3
ds
(t)
dt
1
κ(t)
◦ the centre of curvature is r(t) + ρ(t)N̂(t)
◦ the torsion
(t)
v(t) × a(t) · da
dt
τ (t) =
2
|v(t) × a(t)|
◦ the binormal
B̂(t) = T̂(t) × N̂(t) =
v(t) × a(t)
|v(t) × a(t)|
Under arclength parametrization (i.e. if t = s) we have T̂(s) =
formulae
dT̂
(s) = κ(s) N̂(s)
ds
dN̂
ds (s)
=
dB̂
ds (s)
= −τ (s) N̂(s)
dr
(s)
ds
and the Frenet-Serret
τ (s) B̂(s) − κ(s) T̂(s)
When the curve lies in the x–y plane
dx d2 y
(t) 2 (t) − dy (t) d2 2x (t)
dt
dt
dt
κ(t) = hdt
2
2 i3/2
dy
dx
(t) + dt (t)
dt
When the curve lies in the x–y plane and the parameter is x (so that y is given as a
function y(x) of x)
d2 y
2 (x)
dx
κ(x) = h
2 i3/2
dy
(x)
1 + dx
January 10, 2016
A Compendium of Curve Formulae