The moving trihedron and all that

MATH 2411 - Harrell
The moving trihedron
and all that
B
T
Copyright 2013 by Evans M. Harrell II.
Lecture 5
N
This week’s learning plan
  You will be tested on the mathematics of curves.
  You will think about the physics of moving particles
  You will start understanding functions where vectors
go in and scalars come out.
Reminders and Clarifications
  TEST TOMORROW (yikes!)
Clicker quiz
Find a normal vector to t i + t2 j at (2,4)
A
4 i – j √!
B
i+2j
C
2i-j
D
1
E
(2 i – j)/51/2
F none of the above
The moving trihedron
  The curve’s preferred coordinate system is oriented
along (T,N,B), not some Cartesian system (i,j,k) in
the sky.
The moving trihedron
A spaceship doesn’t see a big Cartesian grid in the sky.
Looked at from the inside, a better basis for vectors will
use the unit tangent T, the principal normal N., and the
binormal B.
The trihedron of unit vectors
1. T(t) = rʹ′(t)/ |rʹ′(t)| or just
.... dr/ds.
  Because of the chain rule, since the speed
|rʹ′(t)| is ds/dt.
The trihedron of unit vectors
1. T(s) = dr/ds.
2. N(s) = (dT/ds)/κ, where κ = |dT/ds|
What you need to remember:
dT/ds determines the direction of N.
Its magnitude defines the curvature.
dT/ds = (dT/dt) / (ds/dt).
The trihedron of unit vectors
1. T(s) = dr/ds.
2. N(s) = (dT/ds)/κ , where κ = |dT/ds| is our
definition of the curvature in 3D.
3. B(s) = T(s) × N(s).
Example
  What’s the formula, for example for
c(t) = (sin(πt),cos(2πt),t) at (0,1,1)?
1.  Parametric form
Example
  What’s the formula, for example for
c(t) = (sin(πt),cos(2πt),t) at (0,1,1)?
2. Single equation
Just what is curvature?
 How do you know a curve is curving?
And how much?
 The answer should depend just on the
shape of the curve, not on the speed at
which it is drawn. So it connects with
arclength s, not with a timeparameter t.
Just what is curvature?
HOW ABOUT HERE?
WHICH CURVES MORE?
Just what is curvature?
HOW ABOUT HERE?
WHICH CURVES MORE?
Just what is curvature?
HOW ABOUT HERE?
WHICH CURVES MORE?
Just what is curvature?
HOW ABOUT HERE?
WHICH CURVES MORE?
Just what is curvature?
HOW ABOUT HERE?
WHICH CURVES MORE?
How rapidly do T and N change?
T
T
N
T
T
N
N
T
N
N
T
N
T
N
Just what is curvature?
 And let’s be quantitative about it!
 2D: How about |dφ/ds|, where φ is the
direction of T with respect to the x-axis?
 To get started, notice that the direction
of T is the same as that of the tangent
line. That is,
tan φ = dy/dx = (dy/ds)/(dx/ds)
(fasten seatbelts for the next slide!)
Different expressions for κ
  κ = |dφ/ds|
  κ = |(dφ/dt)/(ds/dt)|
  κ = |xʹ′(s) yʹ′ʹ′(s) - yʹ′(s) xʹ′ʹ′(s)|
  κ = |xʹ′(t) yʹ′ʹ′(t) - yʹ′(t) xʹ′ʹ′(t)|
|(xʹ′(t))2 + (yʹ′(t))2|3/2
Huh??
Example
 Circle of radius 5.
 No calculus needed!
 If you move distance Δs along the
perimeter, the change in angle is Δs/5.
So Δφ/Δs = 1/5. The general rule for a
circle is that the curvature is the
reciprocal of the radius.
Example
 Spiral: The formula for curvature is
complicated, but the spiral is simple,
so the curvature should be simple.
 Still, we’ll be lazy and use
Mathematica:
Example
Dimensional analysis
 What units do you use to measure
curvature?
 Answer: 1/distance, for instance 1/cm.
1/κ is known as the radius of curvature.
It’s the radius of the circle that best
matches the curve at a given contact
point.
The curve equations of
Frenet and Serret
 The first of these is
 dT/ds = κ N
The curve equations of
Frenet and Serret
 The first of these is
 dT/ds = κ N
 So… what is dN/ds
?
The curve equations of
Frenet and Serret
 The first of these is
 dT/ds = κ N
 So… what is dN/ds
?
 Like all vectors, it must be of the form
_____ T + _____ N + _____ B
N• Nʹ′ = ___0_____
 Because it is ½ the derivative of
N• N = 1 (constant).
The curve equations of
Frenet and Serret
 The first of these is
0, because N • N = 1.
 dT/ds = κ N
 So… what is dN/ds
?
Well,
N •all
T =vectors,
0, so Nʹ′ •itT.+
N •be
Tʹ′ =of0.the form
 Like
must
_____ T + _____ N + _____ B
Therefore Nʹ′ • T.= - N • Tʹ′ = - N • κNʹ′ = - κ.
Nʹ′ = _- κ_ T + __0__ N + __τ__ B .
The curve equations of
Frenet and Serret
 The first of these is
 dT/ds = κ N
 Next:
 dN/ds = -κ T + τ B
 What does torsion tell us?
The curve equations of
Frenet and Serret
 The first of these is
 dT/ds = κ N
 Next:
 dN/ds = -κ T + τ B
 Finally,
 dB/ds = - τ N
The curve equations of
Frenet and Serret



T
d 
N  = (κB + τ T) × 
ds
B
�
��
��
N11 N21
a b
N11
N12 N22
b c
N21
�
� � ∂ξ ∂ξ
N11 N12
∂x
∂y
=
∂η
∂η
N21 N22
∂x
∂y

T
N 
B
�
N12
N22
�
CULTURE BREAK
Motion in 3 D
 Remember that a curve’s favorite
coordinate system is based on the
moving trihedron (T,N,B).
 What happens to a moving particle in
this moving frame?
 velocity
 acceleration
Motion in 3 D
 v = |v| T + 0 N + 0 B.
 So… what’s the acceleration in the
local frame?
Motion in 3 D
magnitude
direction
Motion in 3 D
+0B
Selected applications of vector
calculus to physics
Angular momentum,
L := r × p = r × m rʹ′.
How does this change in time? (This is called
the torque.)
Selected applications of vector
calculus to physics
Magnetic motion,
F := (q/c) v × B,
Lorentz force law.
Funny font because the magnetic field is not the
same as the binormal.
Selected applications to physics
Magnetic motion,
F = (q/c) v × B
Suppose for now that B is a constant vector.
rʹ′ʹ′ = (q/cm) rʹ′ × B
Selected applications to physics
But if rʹ′ʹ′ = (q/cm) rʹ′ × B and the initial velocity
rʹ′(t) happens to be perpendicular to B,
then rʹ′(t) and rʹ′ʹ′(t) will both remain
perpendicular to the magnetic field B. The
entire trajectory is therefore in a plane
perpendicular to B, which must be parallel to B
after all! Moreover, rʹ′ʹ′ and rʹ′ are perpendicular,
so ||rʹ′|| is constant, as we have seen. The
velocities must be of the form
rʹ′(t) = A cos(qt/cm - φ) i ± A sin(qt/cm - φ) j