Generalized Curvatures Part 8 docx

148 15 Approximation of the Area of Surfaces

2. In particular, if |sinθ1| ≤ ε and |sinθ2| ≤ ε, then

sin(αp) ≤

√10ε

sinγ .

Proof of Lemma 4 It is a simple computation, which we do not reproduce here. The

complete proof can be found in [66].

15.4.2.2 Comparing the Length of a Geodesic and Its Chord

Proposition 10. Let S be a smooth compact surface of E3, US be a neighborhood of

S where the map pr : US → S is well defined, and p and q be two points on S such

that [p,q] ⊂ US and pr(]p,q[) ⊂ S \ ∂S. Then, the distance lpq between p and q on S

satisfies

−→pq ≤ lpq ≤

1

1−ωS(pq)

−→pq.

Proof of Proposition 10 The left inequality is trivial. On the other hand, since

pr([p,q]) is a curve on S, its length is larger than the length lpq of the minimal

geodesic on S whose ends are p and q. Therefore, using the mean value theorem,

one has

lpq ≤ l( pr([p,q])) ≤ sup

m∈]p,q[

|D pr(m)| pq.

Since Proposition 3 implies that

|D pr(m)| ≤ 1

1− |pr(m)−m||h|pr(m)

≤

1

1−ωS(pq)

,

Proposition 10 is proved.

15.4.2.3 Comparing the Normals at a Vertex

Proposition 11. Let S be a smooth surface, t be a triangle closely inscribed in S,

and p be a vertex of t. Then, the angle αp ∈ 

0, π

2



between the normals of S and t

at p satisfies

sin(αp) ≤

√

10 πS(t)

2 sinγp (1−ωS(t)),

where γp is the angle of t at p.

This proposition is a consequence of the following lemma.

Lemma 5. Let S be a smooth surface and let p,q ∈ S such that [p,q] ⊂ S. Then, the

angle θ ∈ 

0, π

2



between −→pq and the orthogonal projection of −pq onto T → pS satisfies

15.4 A Bound on the Deviation Angle 149

sinθ ≤ |hS|lpq

2 ,

where |hS| denotes the supremum over S of the norm of the second fundamental form

of S and l is the distance on S between p and q.

The proof of Lemma 5 is similar to that of Proposition 8.

Proof of Proposition 11 Denote by l1 the distance on S between p and p1, and by

l2 the distance on S between p and p2. Since T is closely inscribed in S, thanks to

Corollary 10, we obtain

l1 ≤ pp1

1−ωS(T) ≤

εt

1−ωS(t) and l2 ≤

εt

1−ωS(t)

.

Therefore, Lemma 5 implies that

sinθ1 ≤ |h| pr(t)l1

2 ≤

πS(t)

2(1−ωS(t)) and sinθ2 ≤

πS(t)

2(1−ωS(t)).

Then, Lemma 4 implies that

sin(αp) ≤

√

10

sinγp

πS(t)

2(1−ωS(t)) =

√10 πS(t)

2 sinγp (1−ωS(t)).

15.4.2.4 Comparing the Normals of a Smooth Surface

Proposition 12. Let S be a smooth compact oriented surface of E3, t be a triangle

closely inscribed in S, and p and s be two points on T . Then, the angle αsp ∈ 

0, π

2



between two normals ξpr(p) and ξpr(s) at pr(p) and pr(s) satisfies

sin(αsp) ≤

πS(t)

1−ωpr(t)(t)

.

This proposition is the consequence of the following lemma, which is a direct

application of the mean value theorem.

Lemma 6. Let S be a smooth compact oriented surface of E3 and a and b be two

points of S. Then, the angle αab ∈ 

0, π

2



between two normals ξa and ξb at a and b

satisfies

sin(αab) ≤ |hS|lab,

where lab is the distance on S between a and b.2

2 By definition, lab is the infimum of the lengths of the curves on S linking a and b.