Injectivity of the parametrization of thin curved rods

Josip Tambača Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia, e–mail: tambaca@math.hr

1 Curved rod

Let C be a simple curve in ℝ^{3} defined by its natural parametrization Φ ∈ C^{3}([0, ℓ]; ℝ^{3}). The tangent on C at Φ(y^{1}) is defined by t(y^{1}) = Φ′(y^{1}); obviously t ∈ C^{2}([0, ℓ]; ℝ^{3}). Then, see [2],there exists a function Q ∈ C^{2}([0, ℓ]; SO(3)) such that

Q(y^{1})e_{1} = t(y^{1}), y^{1} ∈ [0, ℓ].

The local basis {t(y^{1}), ñ(y^{1}), b̃(y^{1})} on C at Φ(y^{1}) can then be defined by

t(y^{1}) = Q(y^{1})e_{1}, ñ(y^{1}) = Q(y^{1})e_{2}, b̃(y^{1}) = Q(y^{1})e_{3}, y^{1} ∈ [0, ℓ].

The smoothness of Q implies that t, ñ, b̃ ∈ C^{2}([0, ℓ]; ℝ^{3}). Note that this basis exists even though the Frenet frame does not exist globally. If Φ is a generic curve (|Φ′′(y^{1})| > 0, y^{1} ∈ [0, ℓ]), the function Q can be chosen such that ñ and b̃ are the principal normal and binormal. Let

R(y^{1}) = (\frac{d}{dy^{1}} Q^{T} (y^{1})) Q(y^{1}), y^{1} ∈ [0, ℓ].  (1.1)

Matrix R(y^{1}) is antisymmetric for all y^{1} ∈ [0, ℓ]. Moreover, the following generalization of the Frenet equations holds: \frac{d}{dy^{1}} Q^{T} = RQ^{T}. In the generic case one has

R = ( \begin{matrix}0 & κ & 0 \\ −κ & 0 & τ \\ 0 & −τ & 0 \end{matrix} ) ,

where κ and τ are the curvature and the torsion of C.

Now we define the domain in ℝ^{3} which represents the curved rod. Let ε > 0 be a small parameter and let S ⊂ ℝ^{2} be a simply connected bounded domain with the Lipschitz boundary; points in S are denoted by (z^{2}, z^{3}). We assume that coordinates z^{2} and z^{3} are chosen so that the origin (0, 0) is the center of mass of S, i.e.

∫_{S} z^{2}dz^{2}dz^{3} = ∫_{S} z^{3}dz^{2}dz^{3} = 0.  (1.2)

Let

Ω^{ε} = (0, ℓ) × εS, Γ^{ε} = (0, ℓ) × ε∂S, B_{y^{1}}^{ε} = {y^{1}} × εS, y^{1} ∈ [0, ℓ].

Γ^{ε} is the lateral boundary, ℓ is the length, B_{y^{1}}^{ε} is the cross–section at y^{1} ∈ [0, ℓ], while ε describes the thickness of the cylinder Ω^{ε}. Let P : \overline{Ω}^{ε} → ℝ^{3} be defined by

P (y) = Φ(y^{1}) + y^{2} ñ(y^{1}) + y^{3} b̃(y^{1}).  (1.3)

Since C is simple and smooth curve, there exists ε_{0} > 0 such that the function P is injective (see below), in particular one has

g(y) = (det ∇P (y))^{2} = (1 − R_{12}(y^{1})y^{2} − R_{13}(y^{1})y^{3})^{2} ≥ m_{g}^{2} > 0, y ∈ Ω^{ε_{0}}.  (1.4)

In the sequel we restrict ourselves to ε ∈ (0, ε_{0}). The domain

Ω̃^{ε} = P (Ω^{ε})

is called the curved rod; Γ̃^{ε} = P (Γ^{ε}) is its lateral boundary, B̃_{y^{1}}^{ε} = P (B_{y^{1}}^{ε}) is its cross– section at Φ(y^{1}), while B̃_{0}^{ε} and B̃_{ℓ}^{ε} denote the bases of Ω̃^{ε}. Because of (1.2) curve C passes through the centers of mass of B̃_{y^{1}}^{ε}, y^{1} ∈ [0, ℓ], thus we say that C is the middle curve of the curved rod Ω̃^{ε}.

2 The proof of the injectivity

Now we prove that for ε small enough the function P defined in (1.3) is injective on \overline{Ω}^{ε}. Let ε > 0. We define

δ_{ε} = inf_{y^{1}∈[0,ℓ]} inf_{s^{1}∈[0,ℓ]} {|Φ(y^{1}) − Φ(s^{1})| : |s^{1} − y^{1}| ≥ ε}.

Lemma 2.1 δ_{ε} > 0.

Proof. Obviously δ_{ε} ≥ 0. Let us suppose that δ_{ε} = 0 and define the function f_{ε} : [0, ℓ] → ℝ by

f_{ε}(y^{1}) = inf_{s^{1}∈[0,ℓ]} {|Φ(s^{1}) − Φ(y^{1})| : |s^{1} − y^{1}| ≥ ε} .

For given y_{1}^{1}, y_{2}^{1}, s^{1} ∈ [0, ℓ] it follows

|Φ(s^{1}) − Φ(y_{1}^{1})| ≤ |Φ(s^{1}) − Φ(s^{1} + y_{2}^{1} − y_{1}^{1})| + |Φ(s^{1} + y_{2}^{1} − y_{1}^{1}) − Φ(y_{2}^{1})| + |Φ(y_{2}^{1}) − Φ(y_{1}^{1})|, (2.5)

|Φ(s^{1}) − Φ(y_{2}^{1})| ≤ |Φ(s^{1}) − Φ(s^{1} + y_{1}^{1} − y_{2}^{1})| + |Φ(s^{1} + y_{1}^{1} − y_{2}^{1}) − Φ(y_{1}^{1})| + |Φ(y_{1}^{1}) − Φ(y_{2}^{1})|. (2.6)

Because Φ ∈ C^{1}([0, ℓ]; ℝ^{3}) there is a constant L > 0 such that

|Φ(s^{1}) − Φ(s^{1} + y_{2}^{1} − y_{1}^{1})| ≤ L|y_{2}^{1} − y_{1}^{1}| and |Φ(s^{1}) − Φ(s^{1} + y_{1}^{1} − y_{2}^{1})| ≤ L|y_{2}^{1} − y_{1}^{1}|.

Using these estimates and taking the infimum over all s^{1} ∈ [0, ℓ] such that |s^{1} − y_{1}^{1}| ≥ ε in (2.5) and all s^{1} ∈ [0, ℓ] such that |s^{1} − y_{2}^{1}| ≥ ε in (2.6) it follows

f_{ε}(y_{1}^{1}) ≤ f_{ε}(y_{2}^{1}) + 2L|y_{1}^{1} − y_{2}^{1}| and f_{ε}(y_{2}^{1}) ≤ f_{ε}(y_{1}^{1}) + 2L|y_{1}^{1} − y_{2}^{1}|.

Therefore, f_{ε} is continuous, so it attains its minima on compact set [0, ℓ]:

∃s_{0}^{1} ∈ [0, ℓ] 0 = δ_{ε} = inf_{s^{1}∈[0,ℓ]} f_{ε}(s^{1}) = f_{ε}(s_{0}^{1}).

Then the definition of f_{ε} implies the existence of a sequence (s_{n}^{1}, n ∈ ℕ), s_{n}^{1} ∈ [0, ℓ] such that

|s_{n}^{1} − s_{0}^{1}| ≥ ε and |Φ(s_{n}^{1}) − Φ(s_{0}^{1})| → 0  (2.7)

as n tends to ∞. The sequence (s_{n}^{1}, n ∈ ℕ) is bounded, so there is a subsequence (s_{n_{k}}^{1}, k ∈ ℕ) which is convergent with a limit s_{∞}^{1}. Continuity of Φ implies

Φ(s_{n_{k}}^{1}) → Φ(s_{∞}^{1})  (2.8)

as k tends to ∞. Uniqueness of the limits in (2.7) and (2.8) implies Φ(s_{∞}^{1}) = Φ(s_{0}^{1}), so injectivity of Φ gives s_{∞}^{1} = s_{0}^{1}. Therefore

s_{n_{k}}^{1} → s_{0}^{1}

when k tends to ∞ which contradicts the first estimate in (2.7). □

Theorem 2.1 Let Φ ∈ C^{2}([0, ℓ]) with |Φ(y^{1})| = 1, y^{1} ∈ [0, ℓ] be injective. Then there exists ε_{0} > 0 such that the function P defined in (1.3) is injective on \overline{Ω}^{ε_{0}}, Ω^{ε} = (0, ℓ) × εS.

Proof. If the graph of Φ is a straight line, P is obviously injective on \overline{Ω}^{ε} for all ε > 0. In what follows we assume that Φ is not a straight line; hence at least one of the functions R_{12}, R_{13} is nonzero. Then the number

ε_{1} = \frac{2}{sup_{y^{1}∈[0,ℓ]} sup_{(z^{2},z^{3})∈\overline{S}} (R_{12}(y^{1})z^{2} + R_{13}(y^{1})z^{3})}

is well defined and positive because S is an open neighborhood of the origin (0, 0). It follows for y ∈ Ω^{ε_{1}}

det ∇P (y) = 1 − R_{12}(y^{1})y^{2} − R_{13}(y^{1})y^{3}
≥ inf_{y^{1}∈[0,ℓ]} inf_{(z^{2},z^{3})∈\overline{S}} (1 − ε_{1}(R_{12}(y^{1})z^{2} + R_{13}(y^{1})z^{3}))
≥ 1 − ε_{1} sup_{y^{1}∈[0,ℓ]} sup_{(z^{2},z^{3})∈\overline{S}} (R_{12}(y^{1})z^{2} + R_{13}(y^{1})z^{3}) = \frac{1}{2} > 0.

Therefore, P is locally injective on Ω^{ε_{1}}. Hence for each y ∈ Ω^{ε_{1}} there is the maximal open ball B(y, ε_{y}) such that P is injective on that ball. Then the family

F = {B(y, ε_{y}) : y = (y^{1}, 0, 0), y^{1} ∈ [0, ℓ]}

is an open cover of [0, ℓ] in the relative topology. Compactness of [0, ℓ] implies the existence of the Lebesgue number γ > 0 of the open cover F (every set A with diam A ≤ γ is contained in some element of F). We define

ε_{0} = min \{\frac{γ}{2\sqrt{3} diam S} , \frac{1}{3 diam S} δ_{\frac{1}{\sqrt{3}} γ}\}.

We claim that P is injective on \overline{Ω}^{ε_{0}}. Let y_{1}, y_{2} ∈ \overline{Ω}^{ε_{0}}, y_{1} ≠ y_{2}. We differ two cases

|y_{1}^{1} − y_{2}^{1}| < \frac{1}{\sqrt{3}}γ or |y_{1}^{1} − y_{2}^{1}| ≥ \frac{1}{\sqrt{3}}γ.

In the first case, |y_{1}^{1} − y_{2}^{1}| < \frac{1}{\sqrt{3}}γ implies

y_{1}, y_{2} ∈ B ((\frac{y_{1}^{1} + y_{2}^{1}}{2} , 0, 0), \frac{γ}{2} ) .

Because diam B ((\frac{y_{1}^{1}+y_{2}^{1}}{2} , 0, 0), \frac{γ}{2} ) = γ there is an element of F which contains the ball. Hence P (y_{1}) ≠ P (y_{2}).

In the second case we use the estimate

|Φ(y_{1}^{1}) − Φ(y_{2}^{1})| ≤ |Φ(y_{1}^{1}) − P (y_{1})| + |P (y_{1}) − P (y_{2})| + |P (y_{2}) − Φ(y_{2}^{1})|

to obtain

|P (y_{1}) − P (y_{2})| ≥ |Φ(y_{1}^{1}) − Φ(y_{2}^{1})| − |Φ(y_{1}^{1}) − P (y_{1})| − |P (y_{2}) − Φ(y_{2}^{1})|.  (2.9)

We calculate

|Φ(y_{1}^{1}) − P (y_{1})| = \sqrt{(y_{1}^{2})^{2} + (y_{1}^{3})^{2}} ≤ ε_{0} diam S ≤ \frac{1}{3}δ_{\frac{1}{\sqrt{3}} γ},
|P (y_{2}) − Φ(y_{2}^{1})| = \sqrt{(y_{2}^{2})^{2} + (y_{2}^{3})^{2}} ≤ ε_{0} diam S ≤ \frac{1}{3}δ_{\frac{1}{\sqrt{3}} γ}.

From the definition of δ_{ε} in the case |y_{1}^{1} − y_{2}^{1}| ≥ \frac{1}{\sqrt{3}}γ it follows

|Φ(y_{1}^{1}) − Φ(y_{2}^{1})| ≥ δ_{\frac{1}{\sqrt{3}} γ}.

Inserting these estimates in (2.9) we obtain

|P (y_{1}) − P (y_{2})| ≥ δ_{\frac{1}{\sqrt{3}} γ} − \frac{1}{3}δ_{\frac{1}{\sqrt{3}} γ} − \frac{1}{3}δ_{\frac{1}{\sqrt{3}} γ} = \frac{1}{3}δ_{\frac{1}{\sqrt{3}} γ}.

Hence, P (y_{1}) ≠ P (y_{2}) in this case too. □

References

[1] M. Jurak, J. Tambača, Derivation and justification of a curved rod model, Mat. Mod. Meth. Appl. Sci. Vol. 9, No. 7 (1999) 991-1014.

[2] M. Jurak, J. Tambača, Linear curved rod model. General curve, accepted for publication in Mat. Mod. Meth. Appl. Sci..