Introduction to the Local Theory of
Curves and Surfaces
Notes of a course for the ICTP-CUI Master of Science in Mathematics
Marco Abate, Fabrizio Bianchi (with thanks to Francesca Tovena)
(Much more on this subject can be found in [1])
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Introduction to the Local Theory of
Curves and Surfaces
Notes of a course for the ICTP-CUI Master of Science in Mathematics
Marco Abate, Fabrizio Bianchi (with thanks to Francesca Tovena)
(Much more on this subject can be found in [ 1 ])
2 1. LOCAL THEORY OF CURVES
that is a graph can always be described as a vanishing locus too. Moreover, it also is the image of the map σ : I → R^2 given by σ(t) =
t, f (t)
Remark 1.1. To be pedantic, the graph defined in last example is a graph with respect to the first coordinate. A graph with respect to the second coordinate is a set of the form
f (t), t
∣ (^) t ∈ I
, and has the same right to be considered a curve. Since we obtain one kind of graph from the other just by permuting the coordinates (an operation which geometrically amounts to reflecting with respect to a line), both kinds of graphs are equally suitable, and in what follows dealing with graphs we shall often omit to specify the coordinate we are considering.
Example 1.3. A circle (or circumference) with center (x 0 , y 0 ) ∈ R^2 and ra- dius r > 0 is the curve having equation
(x − x 0 )^2 + (y − y 0 )^2 = r^2.
Note that it is not a graph with respect to either coordinate. However, it can be represented as the image of the map σ : R → R^2 given by
σ(t) = (x 0 + r cos t, y 0 + r sin t). Example 1.4. Open sets in the plane, closed disks and, more generally, subsets of the plane with non-empty interior do not correspond to the intuitive idea of curve, so they are to be excluded. The set [0, 1] × [0, 1] \ Q^2 , in spite of having an empty interior, does not look like a curve either.
Let us see which clues we can gather from these examples. Confining ourselves to graphs for defining curves is too restrictive, since it would exclude circles, which we certainly want to consider as curves (however, note that circles locally are graphs; we shall come back to this fact later). The approach via vanishing loci of functions looks more promising. Indeed, all the examples we have seen (lines, graphs, circles) can be described in this way; on the other hand, an open set in the plane or the set [0, 1] × [0, 1] \ Q^2 cannot be the vanishing locus of a continuous function (why?). So we are led to consider sets of the form C = {(x, y) ∈ Ω | f (x, y) = 0} ⊂ R^2
for suitable (at least) continuous functions f : Ω → R, where Ω ⊆ R^2 is open. We must however be careful. Sets of this kind are closed in the open set Ω, and this is just fine. But the other implication hold as well:
Proposition 1.1. Let Ω ⊆ Rn^ be an open set. Then a subset C ⊆ Ω is closed in Ω if and only if there exists a continuous function f : Ω → R such that C = {x ∈ Ω | f (x) = 0} = f −^1 (0).
Proof. It is enough to define f : Ω → R by setting f (x) = d(x, C) = inf{‖x − y‖ | y ∈ C} ,
where ‖ · ‖ is the usual Euclidean norm in Rn. Indeed, f is obviously continuous, and x ∈ C if and only if f (x) = 0 (why?).
So, using continuous functions we get sets that clearly cannot be considered curves. However, the problem could be caused by the fact that continuous functions are too many and not regular enough; we might have to confine ourselves to smooth functions.
1.1. HOW TO DEFINE A CURVE 3
(Un)fortunately this precaution is not enough. Indeed, it is possible to prove the following
Theorem 1.1 (Whitney). Let Ω ⊆ Rn^ be an open set. Then a subset C ⊆ Ω is closed in Ω if and only if there exists a function f : Ω → R of class C∞^ such that C = f −^1 (0).
In other words, any closed subsets is the vanishing locus of a C∞^ function, not just of a continuous function, and the idea of defining the curves as vanishing loci of arbitrary smooth functions has no chance of working. Let’s take a step back and examine again Examples 1.1, 1.2, and 1.3. In all those cases, it is possible to describe the set as the image of a mapping. This corresponds, in a sense, to a dynamic vision of a curve, thought of as a locus described by a continuously (or differentiably) moving point in a plane or in space or, more in general, in Rn. With some provisos we shall give shortly, this idea turns out to be the right one, and leads to the following definition.
Definition 1.1. Given k ∈ N ∪ {∞} and n ≥ 2, a parametrized curve of class Ck^ in Rn^ is a map σ : I → Rn^ of class Ck, where I ⊆ R is an interval. The image σ(I) is often called support (or trace) of the curve; the variable t ∈ I is the parameter of the curve. If I = [a, b] and σ(a) = σ(b), we shall say that the curve is closed.
Remark 1.2. If I is not an open interval, and k ≥ 1, saying that σ is of class Ck in I means that σ can be extended to a Ck^ function defined in an open interval properly containing I. Moreover, if σ is closed of class Ck, unless stated otherwise we shall always assume that
σ′(a) = σ′(b), σ′′(a) = σ′′(b),... , σ(k)(a) = σ(k)(b).
In particular, a closed curve of class Ck^ can always be extended to a periodic map ˆσ : R → Rn^ of class Ck.
Example 1.5. The graph of a map f : I → Rn−^1 of class Ck^ is the image of the parametrized curve σ : I → Rn^ given by σ(t) =
t, f (t)
Example 1.6. For v 0 , v 1 ∈ Rn^ with v 1 6 = O, the parametrized curve σ : R → Rn given by σ(t) = v 0 + tv 1 has as its image the straight line through v 0 in the direction v 1.
Example 1.7. The two parametrized curves σ 1 , σ 2 : R → R^2 given by σ 1 (t) = (x 0 + r cos t, y 0 + r sin t) and σ 2 (t) = (x 0 + r cos 2t, y 0 + r sin 2t)
both have as their image the circle having center (x 0 , y 0 ) ∈ R^2 and radius r > 0.
Example 1.8. The parametrized curve σ : R → R^3 given by σ(t) = (r cos t, r sin t, at) ,
with r > 0 and a ∈ R∗, has as its image the circular helix having radius r e pitch a; see Fig 1.(a). The image of the circular helix is contained in the right circular cylinder having equation x^2 + y^2 = r^2. Moreover, for each t ∈ R the points σ(t) and σ(t + 2π) belong to the same line parallel to the cylinder’s axis, and have distance 2π|a|.
1.1. HOW TO DEFINE A CURVE 5
Proposition 1.2. Let Ω ⊆ R^2 be an open set, and f : Ω → R a function of class Ck, with k ∈ N∗^ ∪ {∞}. Choose p 0 ∈ Ω such that f (p 0 ) = 0 but ∇f (p 0 ) 6 = O. Then there exists a neighborhood U of p 0 such that U ∩ {p ∈ Ω | f (p) = 0} is the graph of a function of class Ck.
Proof. Since the gradient of f in p 0 = (x 0 , y 0 ) is not zero, one of the partial derivatives of f is different from zero in p; up to permuting the coordinates we can assume that ∂f /∂y(p 0 ) 6 = 0. Then the implicit function Theorem 1.2 tells us that there exist a neighborhood U of p 0 , an open interval I ⊆ R including x 0 , and a function g : I → R of class Ck^ such that U ∩ {f = 0} is exactly the graph of g.
Remark 1.3. If ∂f /∂x(p) 6 = 0 then in a neighborhood of p the vanishing locus of f is a graph with respect to the second coordinate.
In other words, the vanishing locus of a function f of class C^1 , being locally a graph, is locally the support of a parametrized curve near the points where the gradient of f is nonzero.
Example 1.12. The gradient of the function f (x, y) = (x−x 0 )^2 +(y−y 0 )^2 −r^2 is zero only in (x 0 , y 0 ), which does not belong to the vanishing locus of f. Accordingly, each point of the circle with center (x 0 , y 0 ) and radius r > 0 has a neighborhood which is a graph with respect to one of the coordinates.
Remark 1.4. Actually, it can be proved that a subset of R^2 which is locally a graph always is the support of a parametrized curve.
However, the definition of a parametrized curve is not yet completely satisfying. The problem is that it may well happen that two parametrized curves that are different as maps describe what seems to be the same geometric set. An example is given by the two parametrized curves given in Example 1.7, both having as their image a circle; the only difference between them is the speed with which they describe the circle. Another, even clearer example (one you have undoubtedly stumbled upon in previous courses) is the straight line: as recalled in Example 1.1, the same line can be described as the image of infinitely many distinct parametrized curves, just differing in speed and starting point. On the other hand, considering just the image of a parametrized curve is not correct either. Two different parametrized curves might well describe the same support in geometrically different ways: for instance, one could be injective whereas the other comes back more than once on sections already described before going on. Or, more simply, two different parametrized curves might describe the same image a different number of times, as is the case when restricting the curves in Example 1.7 to intervals of the form [0, 2 kπ]. These considerations suggest to introduce an equivalence relation on the class of parametrized curves, such that two equivalent parametrized curves really describe the same geometric object. The idea of only allowing changes in speed and starting point, but not changes in direction or retracing our steps, is formalized using the notion of diffeomorphism.
Definition 1.2. A diffeomorphism of class Ck^ (with k ∈ N∗^ ∪ {∞}) between two open sets Ω, Ω 1 ⊆ Rn^ is a homeomorphism h : Ω → Ω 1 such that both h and its inverse h−^1 are of class Ck.
6 1. LOCAL THEORY OF CURVES
More generally, a diffeomorphism of class Ck^ between two sets A, A 1 ⊆ Rn is the restriction of a diffeomorphism of class Ck^ of a neighborhood of A with a neighborhood of A 1 and sending A onto A 1.
Example 1.13. For instance, h(x) = 2x is a diffeomorphism of class C∞^ of R with itself, whereas g(x) = x^3 , even though it is a homeomorphism of R with itself, is not a diffeomorphism, not even of class C^1 , since the inverse function g−^1 (x) = x^1 /^3 is not of class C^1.
Definition 1.3. Two parametrized curves σ : I → Rn^ and ˜σ : I˜ → Rn^ of class Ck^ are equivalent if there exists a diffeomorphism h : I˜ → I of class Ck^ such that ˜σ = σ ◦ h; we shall also say that ˜σ is a reparametrization of σ, and that h is a parameter change.
In other words, two equivalent curves only differ in the speed they are traced, while they have the same image, they curve (as we shall see) in the same way, and more generally they have the same geometric properties. So we have finally reached the official definition of what a curve is:
Definition 1.4. A curve of class Ck^ in Rn^ is an equivalence class of pa- rametrized curves of class Ck^ in Rn. Each element of the equivalence class is a parametrization of the curve. The support of a curve is the support of any parametrization of the curve. A plane curve is a curve in R^2.
Remark 1.5. We shall almost always use the phrase “let σ : I → Rn^ be a curve” to say that σ is a particular parametrization of the curve under consideration.
Some curves have a parametrization keeping an especially strong connection with its image, and so they deserve a special name.
Definition 1.5. A Jordan (or simple) arc of class Ck^ in Rn^ is a curve admitting a parametrization σ : I → Rn^ that is a homeomorphism with its image, where I ⊆ R is an interval. In this case, σ is said to be a global parametrization of C. If I is an open (closed) interval, we shall sometimes say that C is an open (closed ) Jordan arc.
Definition 1.6. A Jordan curve of class Ck^ in Rn^ is a closed curve C admitting a parametrization σ : [a, b] → Rn^ of class Ck, injective both on [a, b) and on (a, b]. In particular, the image of C is homeomorphic to a circle (why?). The periodic extension ˆσ of σ mentioned in Remark 1.2 is a periodic parametrization of C. Jordan curves are also called simple curves (mostly when n > 2).
Example 1.14. Graphs (Example 1.2), lines (Example 1.6) and circular he- lices (Example 1.8) are Jordan arcs; the circle (Example 1.3) is a Jordan curve.
Example 1.15. The ellipse E ⊂ R^2 with semiaxes a, b > 0 is the vanishing locus of the function f : R^2 → R given by f (x, y) = (x/a)^2 + (y/b)^2 − 1, that is,
E =
(x, y) ∈ R^2
x^2 a^2
y^2 b^2
A periodic parametrization of E of class C∞^ is the map σ : R → R^2 given by σ(t) = (a cos t, b sin t).
8 1. LOCAL THEORY OF CURVES
Figure 2. A non-regular curve
tangent line to the curve at the point σ(t 0 ). Finally, if σ′(t) 6 = O for all t ∈ I we shall say that σ is regular.
Remark 1.6. The notion of a tangent vector depends on the parametrization we have chosen, while the affine tangent line (if any) and the fact of being regular are properties of the curve. Indeed, let σ : I → Rn^ and ˜σ : I˜ → Rn^ be two equivalent parametrized curves of class C^1 , and h : I˜ → I the parameter change. Then, by computing ˜σ = σ ◦ h, we find
(1) σ˜′(t) = h′(t) σ′
h(t)
Since h′^ is never zero, we see that the length of the tangent vector depends on our particular parametrization, but its direction does not; so the affine tangent line in ˜σ(t) = σ
h(t)
determined by ˜σ is the same as that determined by σ. Moreover, ˜σ′^ is never zero if and only if σ′^ is never zero; so, being regular is a property of the curve, rather than of a particular representative.
Example 1.18. Graphs, lines, circles, circular helices, and the curves in Ex- amples 1.10 and 1.11 are regular curves.
Example 1.19. The curve σ : R → R^2 given by σ(t) = (t^2 , t^3 ) is a non-regular curve whose image cannot be the image of a regular curve; see Fig 2 and Exer- cises 1.4 and 1.10.
As anticipated in the previous section, what makes the theory of curves espe- cially simple to deal with is that every regular curve has a canonical parametrization (unique up to its starting point; see Theorem 1.4), strongly related to the geometri- cal properties common to all parametrizations of the curve. In particular, to study the geometry of a regular curve, we often may confine ourselves to working with the canonical parametrization. This canonical parametrization basically consists in using as our parameter the length of the curve. So let us start by defining what we mean by length of a curve.
Definition 1.9. Let I = [a, b] be an interval. A partition P of I is a (k + 1)- tuple (t 0 ,... , tk) ∈ [a, b]k+1^ with a = t 0 < t 1 < · · · < tk = b. If P is partition of I, we set
‖P‖ = max 1 ≤j≤k
|tj − tj− 1 |.
1.2. ARC LENGTH 9
Definition 1.10. Given a parametrized curve σ : [a, b] → Rn^ and a partition P of [a, b], denote by
L(σ, P) =
∑^ k
j=
‖σ(tj ) − σ(tj− 1 )‖
the length of the polygonal closed curve having vertices σ(t 0 ),... , σ(tk). We shall say that σ is rectifiable if the limit
L(σ) = lim ‖P‖→ 0
L(σ, P)
exists and is finite. This limit is the length of σ.
Theorem 1.3. Every parametrized curve σ : [a, b] → Rn^ of class C^1 is rectifi- able, and we have
L(σ) =
∫ (^) b
a
‖σ′(t)‖ dt.
Proof. Since σ is of class C^1 , the integral is finite. So we have to prove that, for each ε > 0 there exists a δ > 0 such that if P is a partition of [a, b] with ‖P‖ < δ then
∫ (^) b
a
‖σ′(t)‖ dt − L(σ, P)
< ε.
We begin by remarking that, for each partition P = (t 0 ,... , tk) of [a, b] and for each j = 1,... , k, we have
‖σ(tj ) − σ(tj− 1 )‖ =
∫ (^) tj
tj− 1
σ′(t) dt
∫ (^) tj
tj− 1
‖σ′(t)‖ dt ;
so summing over j we find
(3) L(σ, P) ≤
∫ (^) b
a
‖σ′(t)‖ dt ,
independently of the partition P. Now, fix ε > 0; then the uniform continuity of σ′^ over the compact interval [a, b] provides us with a δ > 0 such that
(4) |t − s| < δ =⇒ ‖σ′(t) − σ′(s)‖ < ε b − a
for all s, t ∈ [a, b]. Let P = (t 0 ,... , tk) be a partition of [a, b] with ‖P‖ < δ. For all j = 1,... , k and s ∈ [tj− 1 , tj ] we have
σ(tj ) − σ(tj− 1 ) =
∫ (^) tj
tj− 1
σ′(s) dt +
∫ (^) tj
tj− 1
σ′(t) − σ′(s)
dt
= (tj − tj− 1 )σ′(s) +
∫ (^) tj
tj− 1
σ′(t) − σ′(s)
dt.
1.2. ARC LENGTH 11
Theorem 1.4. Every regular oriented curve admits a unique (up to a transla- tion in the parameter) parametrization by arc length. More precisely, let σ : I → Rn be a regular parametrized curve of class Ck. Having fixed t 0 ∈ I, denote by s : I → R the arc length of σ measured starting from t 0. Then σ˜ = σ ◦ s−^1 is (up to a trans- lation in the parameter) the unique regular Ck^ curve parametrized by arc length equivalent to σ and having the same orientation.
Proof. First of all, s′^ = ‖σ′‖ is positive everywhere, so s : I → s(I) is a monotonically increasing function of class Ck^ having inverse of class Ck^ between the intervals I and I˜ = s(I). So ˜σ = σ ◦ s−^1 : I˜ → Rn^ is a parametrized curve equivalent to σ and having the same orientation. Furthermore,
σ˜′(t) =
σ′
s−^1 (t)
∥σ′(s− (^1) (t))∥∥ ,
so ‖˜σ′‖ ≡ 1, as required. To prove uniqueness, let σ 1 be another parametrized curve satisfying the hy- potheses. Being equivalent to σ (and so to ˜σ) with the same orientation, there exists a parameter change h with positive derivative everywhere such that σ 1 = ˜σ ◦ h. As both ˜σ and σ 1 are parametrized by arc length, (1) implies |h′| ≡ 1; but h′^ > 0 everywhere, so necessarily h′^ ≡ 1. This means that h(t) = t + c for some c ∈ R, and thus σ 1 is obtained from ˜σ by translating the parameter.
So, every regular curve admits an essentially unique parametrization by arc length. In some textbooks this parametrization is called the natural parametriza- tion.
Remark 1.8. In what follows, we shall always use the letter s to denote the arc-length parameter, and the letter t to denote an arbitrary parameter. Moreover, the derivatives with respect to the arc-length parameter will be denoted by a dot (˙), while the derivatives with respect to an arbitrary parameter by a prime (′). For instance, we shall write ˙σ for dσ/ds, and σ′^ for dσ/dt. The relation between ˙σ and σ′^ easily follows from the chain rule:
(5) σ′(t) =
dσ dt
(t) =
dσ ds
s(t)
) (^) ds dt
(t) = ‖σ′(t)‖ σ˙
s(t)
Analogously we have
σ˙(s) =
∥σ′
s−^1 (s)
∥ σ
′(s− (^1) (s))^ ,
where in last formula the letter s denotes both the parameter and the arc length function. As you will see, using the same letter to represent both concepts will not cause, once you get used to it, any confusion.
Example 1.20. Let σ : R → Rn^ be a line parametrized as in Example 1.6. Then the arc length of σ starting from 0 is s(t) = ‖v 1 ‖t, and thus s−^1 (s) = s/‖v 1 ‖. In particular, a parametrization of the line by arc length is ˜σ(s) = v 0 + sv 1 /‖v 1 ‖.
Example 1.21. Let σ : [0, 2 π] → R^2 be the parametrization of the circle with center (x 0 , y 0 ) ∈ R^2 and radius r > 0 given by σ(t) = (x 0 + r cos t, y 0 + r sin t). Then the arc length of σ starting from 0 is s(t) = rt, so s−^1 (s) = s/r. In par- ticular, a parametrization ˜σ : [0, 2 πr] → R^2 by arc length of the circle is given by ˜σ(s) =
x 0 + r cos(s/r), y 0 + r sin(s/r)
12 1. LOCAL THEORY OF CURVES
Example 1.22. The circular helix σ : R → R^3 with radius r > 0 and pitch a ∈ R∗^ described in Example 1.8 has ‖σ′‖ ≡
r^2 + a^2. So an arc length param- etrization is
σ˜(s) =
r cos
s √ r^2 + a^2
, r sin
s √ r^2 + a^2
as √ r^2 + a^2
Example 1.23. The catenary is the graph of the hyperbolic cosine function; so a parametrization is the curve σ : R → R^2 given by σ(t) = (t, cosh t). It is one of the few curves for which we can explicitly compute the arc length parametrization using elementary functions. Indeed, σ′(t) = (1, sinh t); so
s(t) =
∫ (^) t
0
1 + sinh^2 τ dτ =
∫ (^) t
0
cosh τ dτ = sinh t
and s−^1 (s) = arc sinh s = log
s +
1 + s^2
Now, cosh
log
s +
1 + s^2
1 + s^2 , and thus the parametrization of the cate- nary by arc length is
˜σ(s) =
log
s +
1 + s^2
1 + s^2
Example 1.24. Let E be an ellipse having semiaxes a, b > 0, parametrized as in Example 1.15, and assume b > a. Then
s(t) =
∫ (^) t
0
a^2 sin^2 τ + b^2 cos^2 τ dτ = b
∫ (^) t
0
a^2 b^2
sin^2 τ dτ
is an elliptic integral of the second kind, whose inverse is expressed using Jacobi elliptic functions. So, to compute the arc-length parametrization of the ellipse we have to resort to non-elementary functions.
Remark 1.9. Theorem 1.4 says that every regular curve can be parametrized by arc length, at least in principle. In practice, finding the parametrization by arc length of a particular curve might well be impossible: as we have seen in the previous examples, in order to do so it is necessary to compute the inverse of a function given by an integral. For this reason, from now on we shall use the parametrization by arc length to introduce the geometric quantities (like curvature, for instance) we are interested in, but we shall always explain how to compute those quantities starting from an arbitrary parametrization too.
1.3. Curvature and torsion In a sense, a straight line is a curve that never changes direction. More precisely, the image of a regular curve is contained in a line if and only if the direction of its tangent vector σ′^ is constant (see Exercise 1.22). As a result, it is reasonable to suppose that the variation of the direction of the tangent vector could tell us how far a curve is from being a straight line. To get an effective way of measuring this variation (and so the curve’s curvature), we shall use the tangent versor.
Definition 1.12. Let σ : I → Rn^ be a regular curve of class Ck. The tangent versor (also called unit tangent vector ) to σ is the map ~t : I → Rn^ of class Ck−^1 given by
~t = σ
′ ‖σ′‖
14 1. LOCAL THEORY OF CURVES
Example 1.28. Let σ : R → R^2 be the catenary, parametrized by arc length as in Example 1.23. Then
~t(s) =
1 + s^2
s √ 1 + s^2
and ~t˙(s) =
s (1 + s^2 )^3 /^2
(1 + s^2 )^3 /^2
so the catenary has curvature
κ(s) =
1 + s^2
Now, it stands to reason that the direction of the vector ~t˙ should also contain significant geometric information about the curve, since it gives the direction the curve is following. Moreover, the vector ~t˙ cannot be just any vector. Indeed, since ~t is a versor, we have
〈~t, ~t〉 ≡ 1 , where 〈· , ·〉 is the canonical scalar product in Rn; hence, after taking the derivative, we get 〈 ~t, ~˙t〉 ≡ 0.
In other words, ~t˙ is orthogonal to ~t everywhere.
Definition 1.14. Let σ : I → Rn^ be a biregular curve of class Ck^ (with k ≥ 2) parametrized by arc length. The normal versor (also called unit normal vector ) to σ is the map ~n : I → Rn^ of class Ck−^2 given by
~n =
~t˙
‖ ~t˙‖
~t˙ κ
The plane through σ(s) and parallel to Span
~t(s), ~n(s)
is the osculating plane to the curve at σ(s). The affine normal line of σ at the point σ(s) is the line through σ(s) parallel to the normal versor ~n(s).
Before going on, we must show how to compute the curvature and the normal versor without resorting to the arc-length, fulfilling the promise we made in Remark 1.9:
Proposition 1.3. Let σ : I → Rn^ be any regular parametrized curve. Then the curvature κ : I → R+^ of σ is given by
(6) κ =
‖σ′‖^2 ‖σ′′‖^2 − |〈σ′′, σ′〉|^2 ‖σ′‖^3
In particular, σ is biregular if and only if σ′^ and σ′′^ are linearly independent every- where; in this case,
(7) ~n =
‖σ′′‖^2 − |〈σ
′′,σ′〉| 2 ‖σ′‖^2
σ′′^ −
〈σ′′, σ′〉 ‖σ′‖^2 σ′
1.3. CURVATURE AND TORSION 15
Proof. Let s : I → R be the arc length of σ measured starting from an arbi- trary point. Equation (5) gives
~t
s(t)
σ′(t) ‖σ′(t)‖
since d dt
~t
s(t)
d~t ds
s(t)
) (^) ds dt
(t) = ‖σ′(t)‖~ t˙
s(t)
we find
~t˙(s(t))^ = 1 ‖σ′(t)‖
d dt
σ′(t) ‖σ′(t)‖
‖σ′(t)‖^2
σ′′(t) −
〈σ′′(t), σ′(t)〉 ‖σ′(t)‖^2
σ′(t)
note that ~t˙
s(t)
is a multiple of the component of σ′′(t) orthogonal to σ′(t). Finally,
κ(t) =
∥~ (^) t˙
s(t)
‖σ′(t)‖^2
‖σ′′(t)‖^2 − |〈σ′′(t), σ′(t)〉|^2 ‖σ′(t)‖^2
and the proof is complete, as the last claim follows from the Cauchy-Schwarz in- equality, and (7) follows from (8).
Let us see how to apply this result in several examples.
Example 1.29. Let σ : R → R^2 be the ellipse having semiaxes a, b > 0, with the parametrization described in Example 1.15. Then σ′(t) = (−a sin t, b cos t), and hence σ′′(t) = (−a cos t, −b sin t). Therefore
~t(t) = σ
′(t) ‖σ′(t)‖
a^2 sin^2 t + b^2 cos^2 t
(−a sin t, b cos t)
and the curvature of the ellipse is given by
κ(t) =
ab (a^2 sin^2 t + b^2 cos^2 t)^3 /^2
Example 1.30. The normal versor of a circle with radius r > 0 is ~n(s) =
− cos(s/r), − sin(s/r)
that of a circular helix with radius r > 0 and pitch a ∈ R∗^ is
~n(s) =
− cos s √ r^2 + a^2
, − sin s √ r^2 + a^2
that of the catenary is
~n(s) =
s √ 1 + s^2
1 + s^2
and that of the ellipse with semiaxes a, b > 0 is
~n(t) =
a^2 sin^2 t + b^2 cos^2 t
(−b cos t, −a sin t).
1.3. CURVATURE AND TORSION 17
equal to 1/r 6 = 0. Then the coordinates of σ satisfy the linear system of ordinary differential equations { ¨σ 1 = − (^1) r σ˙ 2 , ¨σ 2 = (^1) r σ˙ 1.
Keeping in mind that ˙σ^21 + ˙σ 22 ≡ 1, we find that there exists a s 0 ∈ R such that
σ˙(s) =
− sin
s + s 0 r
, cos
s + s 0 r
so the support of σ is contained (why?) in a circle with radius |r|. In other words, circles are characterized by having a constant nonzero oriented curvature.
As we shall shortly see (and as the previous example suggests), the oriented curvature completely determines a plane curve in a very precise sense: two plane curves parametrized by arc length having the same oriented curvature only differ by a rigid plane motion (Theorem 1.6 and Exercise 1.48). Space curves, on the other hand, are not completely determined by their cur- vature. This is to be expected: in space, a curve may bend and also twist, that is leave any given plane. And, if n > 3, a curve in Rn^ may hypertwist in even more dimensions. For the sake of clarity, in the rest of this section we shall (almost) uniquely consider curves in the space R^3. If the support of a regular curve is contained in a plane, it is clear (why? see the proof of Proposition 1.4) that the osculating plane of the curve is constant. This suggests that it is possible to measure how far a space curve is from being plane by studying the variation of its osculating plane. Since a plane (through the origin of R^3 ) is completely determined by the direction orthogonal to it, we are led to the following
Definition 1.16. Let σ : I → R^3 be a biregular curve of class Ck. The binormal versor (also called unit binormal vector ) to the curve is the map ~b : I → R^3 of class Ck−^2 given by ~b = ~t ∧ ~n, where ∧ denotes the vector product in R^3. The affine binormal line of σ at the point σ(s) is the line through σ(s) parallel to the binormal versor ~b(s). Finally, the triple {~t, ~n,~b} of R^3 -valued functions is the Frenet frame of the curve. Sometimes, the maps ~t, ~n, ~b : I → R^3 are also called spherical indicatrices because their image is contained in the unit sphere of R^3.
So we have associated to each point σ(s) of a biregular space curve σ an or- thonormal basis {~t(s), ~n(s),~b(s)} of R^3 having the same orientation as the canonical basis, and varying along the curve (see Fig. 3).
Remark 1.15. The Frenet frame depends on the orientation of the curve. Indeed, if we denote by {~t−, ~n−,~b−} the Frenet frame associated with the curve σ−(s) = σ(−s) equivalent to σ having opposite orientation, we have
~t−(s) = −~t(−s) , ~n−(s) = ~n(−s) , ~b−(s) = −~b(−s).
On the other hand, since it was defined using a parametrization by arc length, the Frenet frame only depends on the oriented curve, and not on the specific parametrization chosen to compute it.
18 1. LOCAL THEORY OF CURVES
t n b
t
t
n
n
b
b
Figure 3. The Frenet frame
Example 1.33. Let σ : R → R^3 be the circular helix with radius r > 0 and pitch a ∈ R∗, parametrized by arc length as in Example 1.22. Then
~b(s) =
a √ r^2 + a^2
sin
s √ r^2 + a^2
a √ r^2 + a^2
cos
s √ r^2 + a^2
r √ r^2 + a^2
Example 1.34. If σ : I → R^3 is the graph of a map f = (f 1 , f 2 ) : I → R^2 such that f ′′^ is nowhere zero, then
~b = √^1 ‖f ′′‖^2 + | det(f ′, f ′′)|^2
det(f ′, f ′′), −f 2 ′′ , f 1 ′′
Example 1.35. If we identify R^2 with the plane {z = 0} in R^3 , we may consider every plane curve as a space curve. With this convention, it is straightforward (why?) to see that the binormal versor of a biregular curve σ : I → R^2 is everywhere equal to (0, 0 , 1) if the oriented curvature of σ is positive, and everywhere equal to (0, 0 , −1) if the oriented curvature of σ is negative.
Remark 1.16. Keeping in mind Proposition 1.3, we immediately find that the binormal versor of an arbitrary biregular parametrized curve σ : I → R^3 is given by
(12) ~b = σ′^ ∧ σ′′ ‖σ′^ ∧ σ′′‖
In particular, we obtain another formula for the computation of the normal versor of curves in R^3 :
~n = ~b ∧ ~t = (σ′^ ∧ σ′′) ∧ σ′ ‖σ′^ ∧ σ′′‖ ‖σ′‖
Moreover, formula (6) for the computation of the curvature becomes
(13) κ =
‖σ′^ ∧ σ′′‖ ‖σ′‖^3
The next proposition confirms the correctness of our idea that the variation of the binormal versor measures how far a curve is from being plane:
Proposition 1.4. Let σ : I → R^3 be a biregular curve of class Ck^ (with k ≥ 2 ). Then the image of σ is contained in a plane if and only if the binormal versor is constant.