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) + aθt +
1
2
θ
2
an + o(θ
3
).
Since the arclength s = aθ, 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/κ(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.