Representation of a Smooth Isometric Deformation of a Planar Material Region into a Curved Surface
Abstract
We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset \({\mathcal{D}}\) of twodimensional Euclidean point space \(\mathbb{E}^{2}\) into a surface \({\mathcal{S}}\) in threedimensional Euclidean point space \(\mathbb{E}^{3}\). To be isometric, such a deformation must preserve the length of every possible arc of material points on \({\mathcal{D}}\). Characterizing the curves of zero principal curvature of \({\mathcal{S}}\) is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in \(\mathbb{E}^{2}\), based upon an à priori chosen preimage form of the curves of zero principal curvature in \({\mathcal{D}}\), and use that coordinate system to construct the most general isometric deformation of \({\mathcal{D}}\) to a smooth surface \({\mathcal{S}}\). A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of \({\mathcal{S}}\) are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).
Keywords
Isometry Unstretchable Inextensional Ruled Developable EmbeddingMathematics Subject Classification
53A05 74K15 74K35 57R40 53A451 Introduction
Recently, we [1] established an explicit necessary and sufficient representation for a threetimes continuously differentiable, isometric deformation of a planar material region identified with an open, connected region \({\mathcal{D}}\) in twodimensional Euclidean point space \(\mathbb{E}^{2}\) into a surface \({\mathcal{S}}\) in threedimensional Euclidean point space \(\mathbb {E}^{3}\). Each such deformation is determined by a sufficiently smooth space curve \({\mathcal{C}}\), the directrix, and a family of straight lines, the generators. A condition necessary, but not sufficient, for the deformation to be isometric is that the generator at each point of \({\mathcal{C}}\) lies in the plane orthogonal to \({\mathcal{C}}\) at that point, with its precise orientation within that plane being determined by the cumulative torsion of \({\mathcal{C}}\). Additionally, however, each ordered combination \((u,v)\) of arclength \(u\) along the directrix and distance \(v\) along the generators must correspond isometrically to a unique material point \(\boldsymbol{x}\) in \({\mathcal{D}}\). That correspondence takes the form of an implicit relation, involving convoluted dependence on the curvature and torsion of \({\mathcal{C}}\), and admits a closedform solution only in very simple examples, encountered for instance in the construction of the isometric deformation that bends a half disk into a conical surface.
In the present paper, we describe an alternative strategy designed to mitigate the aforementioned difficulties. This strategy produces a different, but equivalent, necessary and sufficient representation for the class of isometric deformations of planar material regions and it corrects a fundamental misunderstanding concerning an interpretation of the coordinate representation that has circulated in the mainstream literature on the subject. We consider only kinematical issues, leaving questions surrounding the variational characterization of equilibrium configurations for future consideration.
Expressions for the curvature tensor and potentially nonvanishing principal curvature of a general smooth surface \({\mathcal{S}}\) determined by an isometric deformation of a rectangular material strip are derived on the basis of our representation in Sect. 6. In Sect. 7, we specialize our results to obtain the most general smooth isometric deformation of a planar material region to a cylindrical form and provide two elementary examples involving isometric deformations of rectangular material strips. A summary of our strategy for determining any smooth isometric deformation of a planar material region is provided in Sect. 8. This strategy is then used, in Sect. 9, to determine the isometric deformation of a rectangular material strip to a conical form. Next, in Sect. 10, we show the equivalence of the representation given in our [1] previous work and that obtained here. In particular, that equivalence rests on working with orthogonal curvilinear coordinates \((\zeta ^{1},\zeta^{2})\). Finally, in Sect. 11, we briefly review the conceptual position we have taken in this work regarding the isometric mappings of planar material regions. We contrast our position with a few other notable works that do not regard the surfaces as material entities and, rather, apply the concept of isometry as it is defined in differential geometry.
2 Notion of an Isometric Deformation
It is important to distinguish between our notion of an isometric deformation and an alternative notion that is encountered in differential geometry—a notion that has been applied naively when dealing with deformations of twodimensional bodies which cannot withstand stretching. Such bodies are referred to as “inextensional” in the classical theories of plates (see, for example, Simmonds and Libai [11, 12]) and shells (see, for example, Libai and Simmonds [13, 14]) but are often referred to as “inextensible” in recent works on ribbonlike forms.
In differential geometry, it is commonly understood that a mapping of a part \({\mathcal{D}}\) of a surface \({\mathcal{A}}\subset\mathbb{E}^{3}\) onto a part \({\mathcal{S}}\) of a surface \({\mathcal{B}}\subset \mathbb {E}^{3}\) is isometric, or lengthpreserving, if the length of any arc on \({\mathcal{S}}\) is the same as the length of the inverse image of the arc on \({\mathcal{D}}\). If such a mapping exists, then the surfaces \({\mathcal{D}}\) and \({\mathcal{S}}\) are said to be isometric. In the differential geometric concept of isometry, the surfaces \({\mathcal {A}}\) and ℬ are considered as given and the objective is to determine conditions which ensure that a lengthpreserving mapping exists between the corresponding parts \({\mathcal{D}}\subset{\mathcal {A}}\) and \({\mathcal{S}}\subset{\mathcal{B}}\). Statements to the effect that “isometric surfaces must have the same Gaussian curvature at corresponding points of such a mapping” and “if the Gaussian curvatures of \({\mathcal{D}}\) and \({\mathcal{S}}\) are constant and equal to one another then the surfaces are isometric” are commonplace.^{1} So also is the statement “corresponding curves on isometric surfaces have the same geodesic curvature at corresponding points”. Furthermore, it is wellknown that if \({\mathcal{D}}\) and \({\mathcal{S}}\) are developable then the Gaussian curvatures of both are zero and thus, in particular, \({\mathcal{D}}\) and \({\mathcal{S}}\) are isometric to one another from the differential geometric point of view.
In the kinematics of continuous twodimensional material bodies, as is the concern of this paper, it is commonly understood that a mapping considered as a deformation of a given material surface \({\mathcal{D}}\subset\mathbb{E}^{3}\) into a surface \({\mathcal {S}}\subset\mathbb{E}^{3}\) is isometric (i.e., lengthpreserving, unstretchable, or inextensional), if the length of any arc of material points on \({\mathcal{D}}\) is the same as the length of the image of this material arc on the surface \({\mathcal{S}}\subset\mathbb{E}^{3}\)under the deformation. Any such mapping is considered to be a deformation of the surface \({\mathcal{D}}\subset\mathbb{E}^{3}\), which is identified as a given reference configuration, and the objective is to determine conditions on the deformation necessary and sufficient to ensure that it is lengthpreserving. If, for example, the material surface \({\mathcal{D}}\) is planar and its mapping, considered as a deformation of \({\mathcal{D}}\mapsto{\mathcal{S}}\), produces a developable image \({\mathcal{S}}\), then the Gaussian curvatures of both \({\mathcal{D}}\) and \({\mathcal{S}}\) are zero but the mapping is not necessarily an isometric deformation. To illustrate, \({\mathcal {D}}\subset\mathbb{E}^{2}\) could be an undistorted, rectangular material ribbon which is mapped to \({\mathcal{S}}\subset{\mathcal {T}}_{c}\), where \({\mathcal{T}}_{c}\subset\mathbb{E}^{3}\) is a circular cylindrical surface. In this case, both \({\mathcal{D}}\) and \({\mathcal{S}}\) have zero Gaussian curvature, but the mapping need not be an isometric deformation because stretching of material filaments may have taken place. To be an isometric deformation, the developability of the reference surface \({\mathcal{D}}\) and its target image \({\mathcal{S}}\) is not sufficient, as we show later in Sect. 4 of this paper.
The notions of isometry that arise in differential geometry and in the kinematics of continuous twodimensional material bodies are fundamentally different. Importantly, however, only the second of these notions is relevant when studying the deformation of a twodimensional body that cannot withstand stretching.
In the setting of differential geometry, the surfaces \({\mathcal{A}}\) and ℬ in \(\mathbb{E}^{3}\) are preconceived and given a priori without regard for how one is obtained from the other, and the central question concerns whether lengths measured on a part \({\mathcal{D}}\subset{\mathcal{A}}\) can be made to correspond to (i.e., be equal to) lengths measured on a part \({\mathcal{S}}\subset {\mathcal{B}}\) for any mapping in the collection of all mappings of \({\mathcal{D}}\mapsto{\mathcal{S}}\). When such a mapping exists then the surfaces \({\mathcal{D}}\) and corresponding \({\mathcal{S}}\) are said to be geometrically isometric. From this standpoint, no surface is considered to be a twodimensional continuous material region and no mapping is considered to be the deformation of such a body. Suppose, for example, that \({\mathcal{A}}\) and ℬ are planar surfaces in \(\mathbb{E}^{3}\). Then, it is clear that a square part \({\mathcal{D}}\subset{\mathcal{A}}\) and a square part of equal size \({\mathcal{S}}\subset{\mathcal{B}}\) are isometric in the differential geometric sense. However, from the standpoint of the kinematics of twodimensional continuous material regions, if \({\mathcal{D}}\subset {\mathcal{A}}\) is considered to be an undistorted material reference configuration and if \({\mathcal{S}}\subset{\mathcal{B}}\) is a distorted (i.e., stretched) image of \({\mathcal{D}}\), then no square parts of equal size in \({\mathcal{D}}\) and \({\mathcal{S}}\), respectively, are related by an isometric deformation.
If \({\mathcal{A}}\) and ℬ are developable surfaces in \(\mathbb{E}^{3}\), then in the differential geometric sense there generally exists an isometric image \({\mathcal{S}}\subset{\mathcal {B}}\) of a part \({\mathcal{D}}\) of \({\mathcal{A}}\). But, in the kinematics of twodimensional continuous material regions there need not exist an isometric deformation which maps the same part \({\mathcal {D}}\subset{\mathcal{A}}\) to \({\mathcal{S}}\subset{\mathcal{B}}\). The differential geometric isometric image of \({\mathcal{D}}\) does not necessarily represent a deformation of the relevant points of \({\mathcal {A}}\) into the part \({\mathcal{S}}\subset{\mathcal{B}}\). In general, it is simply an image or overlay that defines a region \({\mathcal{S}}\) on ℬ in which lengths can be measured in the same way that they were measured in \({\mathcal{D}}\subset{\mathcal{A}}\). From the standpoint of the kinematics of twodimensional continuous material region, if \({\mathcal{A}}\) is developable then an isometric deformation of a part \({\mathcal{D}}\subset{\mathcal{A}}\) in \(\mathbb{E}^{3}\) will produce a developable surface \({\mathcal{S}}\) which has the additional property that the length between material points on \({\mathcal{D}}\) and the length between corresponding material points on \({\mathcal{S}}\) under the deformation mapping are equal. From this point of view, the requirement that a deformation maps a developable surface to a developable surface is necessary for the underlying mapping to be an isometric deformation but it does not suffice to ensure that material lengths are preserved. Thus, when considering the characterization of the deformation (i.e., bending and twisting) of a rectangular material ribbon under the constraint that material lengths cannot be changed—which, for example, is the common hypothesis in deforming a rectangular strip of paper into a Möbius band—deformations are the only physically relevant class of mappings.
3 General Analysis and Setup: Isometric Deformation of \({\mathcal{D}}\subset\mathbb{E}^{2}\) to \({\mathcal{S}}\subset \mathbb{E}^{3}\)

Strips of \({\mathcal{S}}\) each of which contains a single oneparameter family of straight lines that do not intersect in \({\mathcal{S}}\) but run through \({\mathcal{S}}\). These families describe the bent regions of \({\mathcal{S}}\) in which only one principal curvature of \({\mathcal{S}}\) vanishes.

Planar strips of \({\mathcal{S}}\) each of which are bounded by straight lines of zero principal curvature of \({\mathcal{S}}\) that do not intersect in \({\mathcal{S}}\). Clearly, all continuous curves in such regions are curves of zero principal curvature of \({\mathcal{S}}\).
Even for the class of surfaces containing only regular points, as noted in the second of the above bullet items there may be continuous, nondifferentiable curves of zero principle curvature of \({\mathcal {S}}\) that lie on \({\mathcal{S}}\). However, in this case such curves must again lie in a common planar part of \({\mathcal{S}}\) and the closure of the interior of its convex hull must also be in \({\mathcal {S}}\). Of course, on planar parts of \({\mathcal{S}}\)all curves are curves of zero principal curvature of \({\mathcal{S}}\).
4 Coordinate Representation of an Isometric Deformation: Necessary and Sufficient Condition
5 Alternative Representative Forms of an Isometric Deformation
6 Curvature Tensor of \({\mathcal{S}}\)
7 General Cylindrical Bending
7.1 Example 1: Circular Cylindrical Bending. Helical Forms
7.2 Example 2: Spiral Cylindrical Bending. Helical Forms
For \(\theta_{0} < \pi/2\), \(\hat{\boldsymbol{y}}_{0}\) defined by (7.34) parametrizes a helical spiral on the spiral cylindrical surface \({\mathcal{T}}_{s}\) with generators parallel to \(\boldsymbol{\imath }_{3}\). For \(\theta_{0} = \pi/2\), we see from (7.26) that \(a = 0\) in which case (7.34) corresponds to a logarithmic spiral lying in the plane spanned by \(\boldsymbol{l}_{1}\) and \(\boldsymbol{l}_{2}\) and, of course, \(\hat{\boldsymbol {y}}_{0}(0)=\boldsymbol{0}\).
In words, according to (7.36) the rectangle \({\mathcal{D}}\) is deformed isometrically to the form \({\mathcal{S}}\) which lies on \({\mathcal{T}}_{s}\) such that the parallel straight lines originally through \({\mathcal{D}}\) at angle \(\theta_{0}\) with the base vector \(\boldsymbol{\imath}_{1}\) are coincident with the generators of \({\mathcal{T}}_{s}\), which are, of course, parallel to \(\boldsymbol{\imath}_{3}\). The helical spiral on \({\mathcal{T}}_{s}\) given in (7.38) is the image of the \(x_{2} = 0\) coordinate centerline of \({\mathcal{D}}\) and runs through the point \(\boldsymbol {y}_{0}=\hat{\boldsymbol{y}}_{0}(0)\) on \({\mathcal{T}}_{s}\). The ‘left upper corner’ of the rectangle \({\mathcal{D}}\) at \(\boldsymbol{x}\to (w/2)\boldsymbol {\imath}_{2}\) is mapped to the point (7.42) on the boundary of \({\mathcal{S}}\subset{\mathcal{T}}_{s}\subset\mathbb {E}^{3}\), which is on the axis, \(\boldsymbol{\imath}_{3}\), of the spiral cylindrical surface \({\mathcal{T}}_{s}\). This corresponds to a limiting corner ‘tip’ of the domain \({\mathcal{S}}\) into which \({\mathcal{D}}\) is mapped. The remainder of the rectangle \({\mathcal {D}}\) is wrapped onto the spiral cylindrical surface \({\mathcal {T}}_{s}\) into a righthanded helical spiral form \({\mathcal{S}}\) according to the direction of \(\boldsymbol{\imath}_{3}\).
In the case \(\theta_{0}<\pi/2\), we know that \(a:= w\cot\theta _{0}/2\neq0\) and, to simplify, we may set \(b=0\) in all of the equations developed in this subsection. For example, \(\boldsymbol {y}_{0}\) in (7.33) becomes \(\boldsymbol{y}_{0}=(a/\sqrt {2})\sin\theta_{0}\boldsymbol{l}_{1}\). In the case \(\theta _{0}=\pi/2\), we have \(a = 0\) and the rectangle \({\mathcal{D}}\subset \mathbb{E}^{2}\) is deformed isometrically into a logarithmic spiral cylindrical form \({\mathcal{S}}\subset{\mathcal{T}}_{s}\subset \mathbb {E}^{3}\). In this case, the constant \(b\) may be chosen as any number such that \(b>0\).
8 Summary: Strategy for Determining an Isometric Deformation of \({\mathcal{D}}\subset\mathbb{E}^{2}\) to \({\mathcal{S}}\subset \mathbb{E}^{3}\)
 1.Recall from (4.14) that the fundamental proper orthogonal linear transformation \(\boldsymbol{Q}\), which is at the basis for constructing any isometric deformation, must satisfywhere \(\lambda\) is scalarvalued and where the unit vectorvalued field \(\boldsymbol{a}_{2}\) defines the direction of the straight lines of zero principal curvature on the deformed surface \({\mathcal{S}}\).$$ \mbox{ax}\bigl(\dot{\boldsymbol{Q}}\boldsymbol{Q}^{{\top}}\bigr)= \lambda\boldsymbol{a}_{2}, $$
 2.Choose \(\lambda\) and \(\boldsymbol{a}_{2}\), and define \(\boldsymbol{w}\) byIn addition, let \(\boldsymbol{W}\) be the skew linear transformation whose axial vector is \(\mbox{ax}(\boldsymbol{W}):=\boldsymbol{w}\) and note from Step 1 that \(\boldsymbol{Q}\) must satisfy$$ \boldsymbol{w}\bigl(\eta^{1}\bigr):=\lambda\bigl(\eta^{1}\bigr) \boldsymbol{a}_{2}\bigl(\eta^{1}\bigr). $$in which the field \(\boldsymbol{W}\) is now considered to be known. Clearly, if \(\boldsymbol{A}\) is the skew linear transformation whose axial vector is \(\mbox{ax}(\boldsymbol{A}):=\boldsymbol{a}_{2}\), then we may make the following replacement above: \(\boldsymbol{W}=\lambda \boldsymbol{A}\). Now, the initial condition \(\boldsymbol {Q}(0)=\boldsymbol{Q}_{0}\in\mathrm{Orth}^{+}\) must be chosen and the nowformulated tensor initial value problem for \(\boldsymbol{Q}\in \mathrm{Orth}^{+}\) must be solved. Note that this problem is equivalent to (8.1).$$ \dot{\boldsymbol{Q}}\bigl(\eta^{1}\bigr)=\boldsymbol{W}\bigl( \eta^{1}\bigr)\boldsymbol{Q}\bigl(\eta^{1}\bigr), $$
 3.Determine the unit vectorvalued field \(\boldsymbol {b}_{2}\) using (4.9) according toand use (4.2b) to determine \(\theta\) with values in \((0,\pi)\) according to$$ \boldsymbol{b}_{2}\bigl(\eta^{1}\bigr) = \boldsymbol{Q}^{{\top}}\bigl(\eta^{1}\bigr)\boldsymbol{a}_{2} \bigl(\eta^{1}\bigr) $$$$ \boldsymbol{b}_{2}=\cos\theta\boldsymbol{ \imath}_{1}+\sin\theta\boldsymbol{\imath}_{2}. $$
 4.Interpret the two parameters \(\eta^{1}\) and \(\eta ^{2}\) such that all points \(\boldsymbol{x}\in{\mathcal{D}}\) are located according to (4.3), that isNote, in particular, that this provides a definitive interpretation of the parameter \(\eta^{1}\) as the \(\eta^{1}\) (not \(x_{1}\)) coordinate of any point \(\boldsymbol{x}\in{\mathcal{D}}\).$$ \boldsymbol{x}=\hat{\boldsymbol{x}}\bigl(\eta^{1},\eta^{2}\bigr)= \eta^{1}\boldsymbol{\imath}_{1}+\eta^{2} \boldsymbol{b}_{2}\bigl(\eta^{1}\bigr). $$
 5.Determine \(\hat{\boldsymbol{y}}_{0}\) according to (4.11) by integratingHere, \(\eta^{1}=0\) is the origin of the midline of the region \({\mathcal{D}}\) in \(\mathbb{E}^{2}\) and \(\boldsymbol{y}_{0}=\hat {\boldsymbol{y}} _{0}(0)\) is specified as the limit point in \(\mathbb{E}^{3}\) where \(\boldsymbol{x}\to\hat{\boldsymbol{x}}(0,0)=\boldsymbol{0}\) is to be transformed under the isometric deformation \(\tilde{\boldsymbol{y}}\) from \({\mathcal{D}}\) to \({\mathcal{S}}\).^{8}$$ \dot{\hat{\boldsymbol{y}}}_{0}\bigl(\eta^{1}\bigr)=\boldsymbol {Q}\bigl( \eta^{1}\bigr)\boldsymbol{\imath}_{1} \quad\mbox{subject to} \quad\hat{\boldsymbol{y}}_{0}(0)=\boldsymbol{y}_{0}\in \mathbb{E}^{3}. $$
 6.Determine the component \(\hat{\boldsymbol{y}}\) of the parametric representation of the isometric deformation \(\tilde{\boldsymbol{y}}\) defined implicitly through (4.3) and (4.6), in accord with$$\begin{aligned} \hat{\boldsymbol{y}}\bigl(\eta^{1},\eta^{2}\bigr) &=\hat {\boldsymbol{y}}_{0} \bigl(\eta^{1}\bigr)+\eta^{2}{\boldsymbol{Q}}\bigl( \eta^{1}\bigr)\boldsymbol{b}_{2}\bigl(\eta^{1} \bigr) \\ &=\hat{\boldsymbol{y}}_{0}\bigl(\eta^{1}\bigr)+\eta^{2} \boldsymbol{a}_{2}\bigl(\eta^{1}\bigr). \end{aligned}$$
Finally, after determining the form of the isometric deformation \(\tilde{\boldsymbol{y}}\) by replacing \((\eta^{1},\eta^{2})\) in Step 6 with \((x_{1},x_{2})\) using Step 4, observe that the curvature tensor for \({\mathcal{S}}\) is given by (6.12).
9 Isometric Deformation of a Rectangular Material Strip \({\mathcal{D}}\subset\mathbb{E}^{2}\) to Portion \({\mathcal{S}}\) of a Conical Surface \({\mathcal{K}}\subset\mathbb{E}^{3}\)
9.1 General Case
9.2 Particular Example
Now, to interpret \(\hat{\boldsymbol{y}}_{0}\) in (9.64): Clearly, at \(\eta^{1} = 0\), \(\theta(0) = \pi/4\), and \(\hat{\boldsymbol{y}}_{0}(0) = R\boldsymbol{\imath }_{1}\); at \(\eta^{1} = \sqrt{2} R\), \(\theta(\sqrt{2} R) = \pi/2\) and \(\hat{\boldsymbol{y}}_{0}(\sqrt{2} R) = (R/\sqrt{2})\boldsymbol {\imath}_{2} + \sqrt {3} R(1  1/\sqrt{2})\boldsymbol{\imath}_{3}\); at \(\eta^{1} =4R/\sqrt {2}\), \(\theta(4R/\sqrt{2}) = 3\pi/4\) and \(\hat{\boldsymbol {y}}_{0}(4R/\sqrt {2})=R\boldsymbol{\imath}_{1}\). In words, the curve parametrized by \(\hat{\boldsymbol{y}}_{0}\) smoothly raises with increasing \(\eta ^{1}\) from the base of the conical surface \({\mathcal{K}}\) at \(R\boldsymbol{\imath}_{1}\) and crosses the plane spanned by \(\boldsymbol{\imath}_{2}\) and \(\boldsymbol {\imath }_{3}\) on \({\mathcal{K}}\) at the point \((R/\sqrt{2})\boldsymbol {\imath }_{2} + \sqrt{3} R(1  1/\sqrt{2})\boldsymbol{\imath}_{3}\). It then lowers on \({\mathcal{K}}\) with increasing \(\eta^{1}\) and intersects the base of \({\mathcal{K}}\) at the point \(R\boldsymbol{\imath}_{1}\). It continues to lower on \({\mathcal{K}}\) with increasing \(\eta^{1}\) and as \(\eta^{1}\longrightarrow+\infty\), \(\theta(\eta^{1})\to\pi \) and, asymptotically, \(\hat{\boldsymbol{y}}_{0}(\eta^{1})\to\sqrt {2} R\boldsymbol {\imath}_{1} \infty\boldsymbol{\imath}_{2}  \infty\boldsymbol {\imath}_{3}\), never again crossing the \(\boldsymbol{\imath }_{2}\)\(\boldsymbol{\imath}_{3}\) plane. The curve determined by \(\hat{\boldsymbol{y}}_{0}\) on \((\infty,+\infty)\) is symmetric with respect to the plane spanned by \(\boldsymbol{\imath}_{2}\) and \(\boldsymbol{\imath}_{3}\).
10 Orthogonal Coordinate Representation of an Isometric Deformation: Necessary and Sufficient Form
10.1 The Isometric Deformation \(\tilde{\boldsymbol{y}}\) in Terms of the Curve \({\mathcal{C}}\in{\mathcal{S}}\) and Its Coordinate Preimage Curve \({\mathcal{C}}_{0}\in{\mathcal{D}}\)
10.2 Recovery of the Component \(\bar{\boldsymbol{y}}\), or Its Equivalent \(\hat{\boldsymbol{y}}\), of the Parametric Representation of an Isometric Deformation \(\tilde{\boldsymbol{y}}\)
10.3 Curvature Tensor of \({\mathcal{S}}\)
11 Discussion
It is common to find in the literature works in which twodimensional continuous material regions in threedimensional Euclidean point space \(\mathbb{E}^{3}\) are considered within the general class of surfaces that are related by being developable, namely the class of surfaces that are isometrically related in the differential geometric sense described in Sect. 3. For examples of this approach, see Hangan [3], Sabitov [4], Starostin and van der Heijden [2], Kurono and Umehara [5], Chubelaschwili and Pinkall [6], Naokawa [7], Kirby and Fried [8], and Shen et al. [9]. A somewhat different but related approach is taken by Dias and Audoly [16], who consider smooth mappings between flat reference surfaces and ruled target surfaces. It can be shown that a reference surface and a ruled target surface are related by an isometric deformation if arclength is measured identically along the directrix of the reference surface and the directrix of the target surface and the ruled target surface is assumed to be developable. Dias and Audoly [16] concentrate on developability but do not, however, provide a strategy for constructing such deformations. Nor do they explain how to identify material points in the reference surface and their coordinates, which measure distance along the directrices and the generators. We postpone further discussion of these issues and their bearing on the variational strategy proposed by Dias and Audoly [16] for future work. To ensure that the arc length of every curve of material points on an unstretchable material surface is preserved under a deformation, we have found in the present work that the generators, the curvature, and the rotation field of the deformed surface must satisfy a specific nontrivial relationship. Incompleteness in characterizing the isometric deformation is a common oversight in the literature concerned with the forms of ribbons in \(\mathbb{E}^{3}\).
Throughout this paper we have considered the kinematics related to the isomeric deformation of an unstretchable planar material region identified with an open, connected subset \({\mathcal{D}}\) of \(\mathbb {E}^{2}\) to a surface \({\mathcal{S}}\) in \(\mathbb{E}^{3}\). Such a deformation is a mapping which is restricted so that all material fibers of \({\mathcal{D}}\) remain unchanged in length. From this vantage point, a surface is a twodimensional material object that is embedded in \(\mathbb{E}^{3}\) and the corresponding notion of isometry is fundamentally different from the differential geometric counterpart in which a surface is considered purely as twodimensional nonEuclidean manifold that may be embedded in \(\mathbb{E}^{3}\) without regard to the identification of material points.
In differential geometry, surfaces are parametrically represented by coordinate systems and are endowed with first and second fundamental forms which describe their metric and curvature properties, respectively. The coordinate systems for different surfaces may, but are not required to, be taken as the ‘same’ in the sense that the values of the coordinates of an image point are the same as those of the corresponding inverse image point. According to Kreyszig [15, p. 161], given two surfaces, a portion of one is isometrically mapped onto a portion of the other if and only if at corresponding points of the two surfaces—when referred to the same coordinate systems on the two surfaces—the coefficients of the first fundamental forms for the two surfaces are the same. This has the consequence that for the isometric mapping of a portion of one surface onto a portion of another surface, the length of any arc on one surface must be the same as that of its inverse image. It is wellknown in differential geometry that two surfaces which have the same constant Gaussian curvature may be isometrically mapped, one onto the other. It is also wellknown in the kinematics of deformable twodimensional continuous material regions that the isometric deformation of a planar, undistorted reference configuration must produce a surface with the same, vanishing, Gaussian curvature. However, it does not generally follow that a reference and a target surface that have the same constant Gaussian curvature from the differential geometric point of view represent the isometric deformation of a material reference configuration into its deformed image.
The first fundamental form of a deformed twodimensional continuous material region characterizes its metric properties and is generated by the deformation itself. The deformation from a given reference configuration of the body is said to be isometric if the length of any arc of material fibers in the reference configuration and the length of the image of that arc in the deformed configurations is identical.
In differential geometry, an isometric mapping of a portion of one surface onto a portion of another surface requires that the length of any arc on one must be the same as that of its inverse image on the other. Since lengths on a surface are determined by the first fundamental form on that surface, and this depends on its parametrization, then there exists an isometric mapping if and only if when the same coordinates are used to parameterize the surfaces the coefficients of their first fundamental forms are equal. A test for this draws upon the important theorem that isometric surfaces necessarily have the same Gaussian curvature at corresponding points. This test is also sufficient if the surfaces have the same constant Gaussian curvature. Since developable surfaces are ruled surfaces of zero Gaussian curvature that characterize the class of all surfaces that are geometrically isometric to a plane, and since an isometric deformation of planar, undistorted material surfaces produces a developable image, the two distinct concepts of isometry from the differential geometric and the deformation points of view have become fuzzy and are sometimes mistaken to be equivalent. The characterization of an isometric deformation of a planar material region involves more than the geometric restriction of developability. All fibers of the material region must remain unchanged in length and this yields an additional restriction on the changing rotation field that is responsible for the bending that takes place around the generators of the ruled and developable deformed material surface.
In this paper, we have established several equivalent necessary and sufficient representations of a smooth, isometric deformation of a planar material region into a curved surface and we have emphasized the essential nature of the nontrivial tensorial ordinarydifferential initialvalueproblem that must be solved to properly relate the generators, the curvature, and the rotation field of the isometrically deformed material surface. For illustrative purposes, we have also provided examples involving isometric deformations of rectangular material strips into cylindrical and conical surfaces.
Footnotes
 1.
See, for example, Kreyszig [15, p. 164].
 2.
Throughout this work, Roman indicies range over \(\{1,2\}\), with summation over twice repeated indices being implicit, and a subscripted comma denotes partial differentiation.
 3.
The linear transformation \(\mbox{grad}_{s}\boldsymbol{n}\) is also called the “Weingarten mapping” or the “shape operator” and its symmetry property is a fundamental theorem of differential geometry. The principal values of \(\mbox{grad}_{s}\boldsymbol{n}\) are the principal curvatures of \({\mathcal {S}}\) at \(\boldsymbol{y}\).
 4.
 5.
In this regard, note that \(\boldsymbol{a}^{2}(\eta^{1})\) is the orientation of the line \({\mathcal{L}}(\eta^{1})\) of zero curvature in \({\mathcal{S}}\) which corresponds to the line \({\mathcal{L}}_{0}(\eta ^{1})\) that passes through the \(x_{1}\)axis at \((\eta_{1}, 0)\) with orientation \(\boldsymbol{b}_{2}(\eta^{1})\) in \({\mathcal{D}}\).
 6.For example, Starostin and van der Heijden [2] consider a parametrizationof a strip of length \(L\) and width \(2w\), where \(\boldsymbol{r}\) is the centerline of the strip, \(s\) denotes the arc length along the centerline and \(\boldsymbol{t}\), \(\boldsymbol{b}\), \(\kappa\), \(\tau\) and \(\eta:= \tau/\kappa\) are as above in this section. They take this to be a representation that preserves all intrinsic distance. While this parametrization is similar to (5.12)_{4}, Starostin and van der Heijden [2] do not explain how to correlate \(s\) and \(t\) with the corresponding material points of the flat reference rectangle. Since \(L\) and \(2w\) are designated as the length and width of the rectangular strip and the domains for \(s\) and \(t\) are given as \([0,L]\) and \([w,w]\), respectively. it is natural to take \(s\) as \(x_{1}\) and \(t\) as \(x_{2}\). However, this would be incorrect, and this is what defines the ‘confusion’ noted in the text of the paragraph containing this footnote. Such a correlation is obtained in the present work through the use of (4.1) in \((5.1)_{4}\). Moreover, Starostin and van der Heijden [2] make no mention of any condition such as (4.13) which would ensure that their parametric representation is indeed isometric in the sense that it preserves distances between all pairs of material points. Without further restriction the parametrization does not represent an isometric deformation. See, also, the discussion of Chen and Fried [10].$$ \boldsymbol{y}(s,t)=\boldsymbol{r}(s)+t \bigl(\boldsymbol {b}(s)+\eta(s) \boldsymbol{t}(s) \bigr),\quad s\in[0, L],\ t\in[w, w], $$(∗)
 7.
 8.
 9.
We calculate the nonzero principal curvature of \({\mathcal{S}}\) in (9.58) and show how the inequality \(w < 2L\sin\theta_{0}\) arises from the condition that the curvature of \({\mathcal{S}}\) is bounded. Due to this inequality, the nonvanishing principal curvature is, in fact, everywhere negative on \({\mathcal{S}}\).
 10.
See also the interpretation of (9.58), below.
 11.
 12.
 13.Of course, the initial value for (10.15)_{2} is \(\bar{\boldsymbol{Q}}_{0} := \bar {\boldsymbol{Q}}(0)\), which, according to (10.42) and (10.44), can be shown to depend upon the initial values of \({\mathcal{C}}\) through \(\bar {\boldsymbol{t}}(0)\), \(\bar{\boldsymbol{t}}^{\prime}(0)\), and \(\bar{\varphi}(0)\), and the initial value of \({\mathcal{C}}_{0}\) through \(\bar{\theta}_{0}\). Note, as an alternative, that \(\bar{\boldsymbol{R}}:= \bar{\boldsymbol{Q}}\bar {\boldsymbol{Q}}_{0}^{{\top} }\in\mathrm{Orth}^{+}\) satisfies(10.49)
Notes
Acknowledgements
The work of Eliot Fried was supported by the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan. Yichao Chen thanks the Okinawa Institute of Science and Technology for hospitality and generous support during a sabbatical and several subsequent visits.
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