Download given a space curve r(t)

ARC LENGTH and CURVATURE
Arc Length : given a space curve r(t) = < f (t), g(t), h(t) >;
• the arc length from a to b is
L=
b 0
a |r (t)|dt
Z
=
b
a
Z
r
f 0(t)2 + g 0(t)2 + h0(t)2dt;
• arc length function is s(t) =
Rt
a |r
0
(u)|du, so
ds
= |r0(t)|;
dt
• parameterization with respect to arc length
is r(t(s)).
Curvature : given the unit tangent T(t) =
• the curvature
κ=|
r0 (t)
|r0 (t)| ;
dT
|;
ds
• chain rule gives
|T0(t)|
;
κ(t) = 0
|r (t)|
• Curvature Theorem
|r0(t) × r00(t)|
κ(t) =
;
0
3
|r (t)|
• for planes curves y = f (x)
|f 00(x)|
κ(x) =
.
(1 + (f 0(x))2)3/2
MOTION in SPACE:
VELOCITY and ACCELERATION
Velocity and Acceleration:
assuming r(t) is position of an object in space;
• the velocity of the object is v(t) = r0(t);
• the speed of the object is v(t) = |v(t)| = |r0(t)|;
• the acceleration of the object is a(t) = v0(t) = r00(t);
• Newton’s Second Law of Motion:
F = ma = mr00(t),
for a force F acting on an object with mass m, so given
F, two integrations are needed to find r(t).
2