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Plane curves
Adrian Down
September 19, 2006
1 Review of multivariable calculus
1.1 Motivation
We will now study the properties of curves in a plane. We will be able to investigate more of the global properties of the curve. Green’s theorem will be an important tool that will allow us to relate the shape of the image of a curve to the geometry of the region bounded by that curve.
1.2 Notation: integration along a curve
Let α be a curve that maps from (a, b) ∈ R to an image, call it C, in the xy plane. The curve can be written in component form as α = α(x(t), y(t)). Let f = f (x, y) be a vector valued function. Path integrals of f along C are defined,
∫
C
f dx =
∫ (^) b
a
f (x(t), y(t))
dx dt
(t)dt ∫
C
f dy =
∫ (^) b
a
f (x(t), y(t))
dy dt
(t)dt
Note. It can be shown that these integrals are independent of parameteriza- tion.
The integral with respect to arc length is, ∫
C
f ds =
∫ (^) b
a
f (x(t), y(t))
dx dt
dy dt
) 2 )^12
ds The integral of the differential of a function along C is defined as, ∫
C
df ≡
C
df dx
dx +
dy dy
dy
The fundamental theorem of line integrals states that, ∫
C
df = f (b) − f (a)
where b and a are the endpoints of C.
Note. The fundamental theorem of line integrals may be more familiar in the notation, ∫ ∇f · dr = f (b) − f (a)
1.3 Green’s theorem
Theorem. Given a curve C bounding a region R and two functions f and g defined on R, ∫
C
f dx + gdy =
R
∂g ∂x
∂f ∂y
dxdy
Note. • We omit the proof of Green’s theorem, which is presented in many standard calculus texts. The proof essentially involves breaking of the region R into many infinitesimally small area elements and sum- ming the integral over each element, using cancellation along interior edges between adjacent area elements.
- Green’s theorem is the two-dimensional form of Stokes theorem.
- Green’s theorem is of the form of all other standard fundamental the- orems of calculus, which state that the integral over some quantity, in this case the integral over the area R, is equal to the value at the boundaries, in this case the integral over C.
Note. Lower case letters are used to denote the Frenet-Serret basis vectors in the plane.
In the plane, the choice of the normal is easier than in space. Instead of orientating a normal in three dimensional space, we now need only a single line perpendicular to the plane in which α lies. There are two candidates for this normal. The normal vector n(s) is defined to be the unit vector such that { t(s), n(s) } is a right-handed basis.
Note. In two dimensions, a right-handed coordinate system { a, b } is one such that,
det
a^1 b^1 a^2 b^2
With these definitions of t and n, we can define a curvature,
k(s) = 〈t′(s), n(s)〉
Note. A lowercase k is used to refer to the planar curvature, whereas the greek κ is reserved for the three-dimensional curvature.
k is called the signed planar curvature. It is different from the three dimensional curvature κ in that k can be negative.
2.3 Frenet-Serret theorem
We can show that an analogy to the Frenet-Serret apparatus in three dimen- sions holds in the plane as well.
Theorem. Given a curve α : (a, b) → R^2 with tangent t, normal n, and signed planar curvature k,
( t′ n′
0 k −k 0
t n
Proof. Using the same technique used in the proof of the Frenet-Serret appa- ratus in three dimensions, expand the derivative in terms of the orthonormal basis,
t′^ = 〈t′, t〉 t + 〈t′, n〉 n
The second inner product is equal to the curvature. The first inner product is identically equal to 0 because t is a unit vector. Thus t′^ = kn. To obtain the second line of the desired formula, expand the derivative of the other basis vector in terms of the orthonormal basis,
n′^ = 〈n′, t〉 t + 〈n′, n〉 n
As usual, the second inner product is equal to 0 because the magnitude of n is constant. To obtain the first inner product, differentiate the inner product of n and t, using the fact that these vectors are orthonormal,
〈n, t〉 ≡ 0
⇒
d ds
〈n, t〉 ≡ 0
⇒ 〈n′, t〉 + 〈n, t′〉 = 0 ⇒ 〈n′, t〉 = − 〈n, t′〉
Thus n′^ = − 〈n, t′〉 t = −kt.
Note. We can also regard α as a curve in three dimensions from which we could create the usual Frenet-Serret apparatus. The correspondence between the planar and three-dimensional quantities is,
T = t N = ∓n k = |κ|
2.4 Computing t and n
Since α is a plane curve, we can obtain simple forms forms for t and n by differentiating the curve componentwise,
α(s) = (x(s), y(s)) t(s) = (x′(s), y′(s))
We can try a guess for n, motivated by what we would obtain by expressing α in terms of a sine and a cosine term,
n(s) = (−y′(s), x′(s))
This choice of n must be a unit vector, be orthonormal with t, and must form a right-handed coordinate system. With this choice of n,
|n| =
(x′)^2 + (y′)^2
= |t| = 1
3.1 Closed curves
Definition (Closed curve). A regular curve β : R 7 → R^2 is closed if β is periodic, i.e. ∃ a > 0 such that β(t + a) = β(t) ∀ t. The smallest such number a is called the period.
Definition (Simple curve). A curve is simple if either β is one-to-one or β is a closed curve of period a such that
β(t 1 ) = β(t 2 ) ⇔ t 1 − t 2 = n for some n ∈ Z
Note. The first case in this definition corresponds to a curve that maps to a line. The second case implies that a closed curve is not simple if it intersects itself.
3.2 Rotation index
We would like examine the angle that t makes with the horizontal along the curve. There is a problem with continuity in the definition of this angle, since this angle is only defined modulo two pi. If we restrict the angle to only be between 2π, we will not always be able to define the angle continuously. The following definition will ensure that θ will be continuous,
θ(s) = θ(0) +
∫ (^) s
0
k(r)dr
Definition (Rotation index). The rotation index of the curve to be,
iα =
θ(L) − θ(0) 2 π By the way in which θ is defined, the rotation index can be written,
iα =
2 π
C
kds =
2 π
∫ L
0
k(s)ds
Note. This definition is foreshadowing an important definition later in the course. Everything we do for curves, we would like to find an analog for surfaces. We will see that the integral of the Gaussian curvature for a surface will be integer-valued.
The most important theorem relating to rotation indices states that the rotation index of a simple closed curve is ±1.
Theorem. Suppose α is a simple closed curve, then iα = ± 1.
We will see that this result is difficult to prove. As is usually the case, proofs of global theorems are intricate, although they provide useful informa- tion. We will define a unit vector pointing from one location on the curve to another location on the curve. We will use the fact that the curve is simple. This method of proof would not be possible if the curve crossed itself.
