It is obvious that the shape of a surface influences the curvature of curves on the surface. For example, a curve on a plane or a cylinder can have zero curvature everywhere, but this is not possible for curves on a sphere since no segment of a straight line can lie on a sphere. Thus, another natural way to investigate how much a surface curves is to look at the curvature of curves on the surface.
We shall see that this leads, once again, to the second fundamental form of the surface.
N×g
g .
N .
k
Á g..
g
kn
kg
Ifγ is a unit-speed curve on an oriented surfaceS, then ˙γ is a unit vector and is, by definition, a tangent vector to S. Hence, ˙γ is perpendicular to the unit normal N of S, so ˙γ, N and N×γ˙ are mutually perpendicular unit vectors. Again since γ is unit-speed, ¨γ is perpendicular to ˙γ, and hence is a linear combination ofN andN×γ˙:
γ¨ =κnN+κgN×γ˙. (7.7)
Definition 7.3.1
The scalars κn and κg in Eq. 7.7 are called the normal curvature and the geodesic curvatureofγ, respectively.
Note thatκn and κs both change sign whenN is replaced by−N, so on a general (not necessarily orientable) surface only the magnitudes ofκn andκs
are well defined.
Proposition 7.3.2
With the above notation, we have
κn = ăγãN, κg = ăγã(Nìγ˙), (7.8)
κ2=κ2n+κ2g, (7.9)
κn=κcosψ, κg =±κsinψ, (7.10)
whereκis the curvature ofγ andψis the angle betweenN and the principal normalnofγ.
Proof
Equations7.8 and7.9follow from Eq.7.7and the fact that NandN×γ˙ are perpendicular unit vectors. The first equation in (7.10) follows from ¨γ =κn, and the second then follows from Eq.7.9.
If γ is regular, but not necessarily unit-speed, we define the geodesic and normal curvatures of γ to be those of a unit-speed reparametrization of γ (see Exercise 7.3.1). When a unit-speed parametertis changed to another such parameter±t+c, wherecis a constant, it is clear thatκn→κnandκg → ±κg, soκn is well defined for any regular curve, whileκg is well defined up to sign.
Equations7.9and7.10continue to hold ifγ is any regular curve.
The following proposition is the most important single fact about the normal curvature, and reveals its relation to the second fundamental form, .
Proposition 7.3.3
Ifγis a unit-speed curve on an oriented surfaceS, its normal curvature is given by
κn =γ˙,γ˙ .
Ifσis a surface patch ofS and γ(t) =σ(u(t), v(t)) is a curve inσ, κn=Lu˙2+ 2Mu˙˙v+Nv˙2
in the notation of Section7.1.
This result means that two curves whichtoucheach other at a pointpof a surface (i.e., which intersect atpand have parallel tangent vectors atp) have the same normal curvature atp.
Proof
Since ˙γ is a tangent vector toS,N.γ˙ = 0. Hence,Nãγă=−N˙ ãγ˙ so κn=Nãγă =−N˙ ãγ˙ =W( ˙γ),γ˙ =γ˙,γ˙ ,
since
N˙ = d
dtG(γ(t)) =−W( ˙γ).
The second part follows from the first and Proposition7.2.2.
It turns out that, while the normal curvature depends on the second funda- mental form of the surface, the geodesic curvatureκg depends only on itsfirst fundamental form (see Exercise 7.3.4). But we leave further discussion ofκg to Chapter 9.
Here is a classical application of Proposition7.3.3. It takes almost as long to state as to prove.
Proposition 7.3.4 (Meusnier’s Theorem)
Letpbe a point of a surfaceS and letv be a unit tangent vector toS atp.
Let Πθ be the plane containing the line throughp parallel to v and making an angleθ with the tangent planeTpS, and assume that Πθ is not parallel to TpS. Suppose that ΠθintersectsS in a curve with curvatureκθ. Then,κθsinθ is independent ofθ.
°à Πà
P
à
S v
Proof
Assume thatγθ is a unit-speed parametrization of the curve of intersection of Πθ andS. Then, atp, ˙γθ=±v, so ¨γθis perpendicular tovand is parallel to Πθ. Thus, in the notation of Proposition7.3.2,ψ=π/2−θand so Eq.7.10gives
κθsinθ=κn. Butκn depends only onpandv, and not onθ.
An important special case is that in whichγis anormal sectionof the sur- face, i.e.,γis the intersection of the surface with a plane Π that is perpendicular to the tangent plane of the surface at every point ofγ.
Corollary 7.3.5
The curvatureκ, normal curvatureκn and geodesic curvatureκg of a normal section of a surface are related by
κn=±κ, κg= 0.
Proof
As in the proof of Proposition7.3.3,κn=κsinθ, whereθ=±π/2 for a normal section. This gives the first equation; the second follows from it and Eq.7.9.
EXERCISES
7.3.1 Letγ be a regular, but not necessarily unit-speed, curve on a sur- face. Prove that (with the usual notation) the normal and geodesic curvatures ofγare
κn= γ˙,γ˙
γ˙,γ˙ and κg=γăã(Nìγ˙) γ˙,γ˙ 3/2 .
7.3.2 Show that the normal curvature of any curve on a sphere of radius ris±1/r.
7.3.3 Compute the geodesic curvature of any circle on a sphere (not nec- essarily a great circle).
7.3.4 Show that, ifγ(t) =σ(u(t), v(t)) is a unit-speed curve on a surface patch σ with first fundamental form Edu2+ 2F dudv+Gdv2, the geodesic curvature ofγ is
κg= (¨vu˙−v˙u)¨
EG−F2+Au˙3+Bu˙2v˙+Cu˙v˙2+Dv˙3, where A, B, C and D can be expressed in terms of E, F, G and their derivatives. FindA, B, C, D explicitly whenF = 0.
7.3.5 Suppose that a unit-speed curveγ with curvatureκ >0 and princi- pal normalnis a parametrization of the intersection of two oriented surfacesS1 and S2 with unit normalsN1 and N2. Show that, ifκ1 andκ2are the normal curvatures ofγ when viewed as a curve inS1 andS2, respectively, then
κ1N2−κ2N1=κ(N1×N2)×n.
Deduce that, ifαis the angle between the two surfaces, κ2sin2α=κ21+κ22−2κ1κ2cosα.
7.3.6 A curveγon a surfaceSis calledasymptoticif its normal curvature is everywhere zero. Show that any straight line on a surface is an asymptotic curve. Show also that a curveγ with positive curvature is asymptotic if and only if its binormal b is parallel to the unit normal ofS at all points ofγ.
7.3.7 Prove that the asymptotic curves on the surface σ(u, v) = (ucosv, usinv,lnu) are given by
lnu=±(v+c), wherec is an arbitrary constant.