5Geometry of curves and surfaces

IA Vector Calculus



5 Geometry of curves and surfaces
Let
r
(
s
) be a curve parametrized by arclength
s
. Since
t
(
s
) =
dr
ds
is a unit vector,
t · t
= 1. Differentiating yields
t · t
0
= 0. So
t
0
is a normal to the curve if
t
0
6
= 0.
We define the following:
Definition
(Principal normal and curvature)
.
Write
t
0
=
κn
, where
n
is a unit
vector and
κ >
0. Then
n
(
s
) is called the principal normal and
κ
(
s
) is called
the curvature.
Note that we must be differentiating against
s
, not any other parametrization!
If the curve is given in another parametrization, we can either change the
parametrization or use the chain rule.
We take a curve that can Taylor expanded around s = 0. Then
r(s) = r(0) + sr
0
(0) +
1
2
s
2
r
00
(0) + O(s
3
).
We know that r
0
= t and r
00
= t
0
. So we have
r(s) = r(0) + st(0) +
1
2
κ(0)s
2
n + O(s
3
).
How can we interpret
κ
as the curvature? Suppose we want to approximate the
curve near
r
(0) by a circle. We would expect a more “curved” curve would be
approximated by a circle of smaller radius. So
κ
should be inversely proportional
to the radius of the circle. In fact, we will show that
κ
= 1
/a
, where
a
is the
radius of the best-fit circle.
Consider the vector equation for a circle passing through
r
(0) with radius
a
in the plane defined by t and n.
a
r(0)
t
n
θ
Then the equation of the circle is
r = r(0) + a(1 cos θ)n + a sin θt.
We can expand this to obtain
r = r(0) + t +
1
2
θ
2
an + o(θ
3
).
Since the arclength s = , we obtain
r = r(0) + st +
1
2
1
a
s
2
n + O(s
3
).
As promised, κ = 1/a, for a the radius of the circle of best fit.
Definition
(Radius of curvature)
.
The radius of curvature of a curve at a point
r(s) is 1(s).
Since we are in 3D, given
t
(
s
) and
n
(
s
), there is another normal to the curve.
We can add a third normal to generate an orthonormal basis.
Definition (Binormal). The binormal of a curve is b = t × n.
We can define the torsion similar to the curvature, but with the binormal
instead of the tangent.
a
Definition (Torsion). Let b
0
= τn. Then τ is the torsion.
Note that this makes sense, since
b
0
is both perpendicular to
t
and
b
, and
hence must be in the same direction as n. (b
0
= t
0
× n + t × n
0
= t × n
0
, so b
0
is perpendicular to t; and b · b = 1 b · b
0
= 0. So b
0
is perpendicular to b).
The geometry of the curve is encoded in how this basis (
t, n, b
) changes along
it. This can be specified by two scalar functions of arc length the curvature
κ
(
s
) and the torsion
τ
(
s
) (which determines what the curve looks like to third
order in its Taylor expansions and how the curve lifts out of the t, r plane).
Surfaces and intrinsic geometry*
We can study the geometry of surfaces through curves which lie on them. At a
given point
P
at a surface
S
with normal
n
, consider a plane containing
n
. The
intersection of the plane with the surface yields a curve on the surface through
P . This curve has a curvature κ at P .
If we choose different planes containing
n
, we end up with different curves of
different curvature. Then we define the following:
Definition
(Principal curvature)
.
The principal curvatures of a surface at
P
are
the minimum and maximum possible curvature of a curve through
P
, denoted
κ
min
and κ
max
respectively.
Definition
(Gaussian curvature)
.
The Gaussian curvature of a surface at a
point P is K = κ
min
κ
max
.
Theorem
(Theorema Egregium)
. K
is intrinsic to the surface
S
. It can be
expressed in terms of lengths, angles etc. which are measured entirely on the
surface. So
K
can be defined on an arbitrary surface without embedding it on a
higher dimension surface.
The is the start of intrinsic geometry: if we embed a surface in Euclidean
space, we can determine lengths, angles etc on it. But we don’t have to do so
we can “live in the surface and do geometry in it without an embedding.
For example, we can consider a geodesic triangle
D
on a surface
S
. It consists
of three geodesics: shortest curves between two points.
Let
θ
i
be the interior angles of the triangle (defined by using scalar products
of tangent vectors). Then
Theorem (Gauss-Bonnet theorem).
θ
1
+ θ
2
+ θ
3
= π +
Z
D
K dA,
integrating over the area of the triangle.
a
This was not taught in lectures, but there is a question on the example sheet about the
torsion, so I might as well include it here.