Riemannian
structures
and free curve-straightening
Anders Linnér
Submitted January 15, 2005 and, in revised form, August 24, 2005.
Abstract. The choice of Riemannian structure on a space of curves affects computational speed, smoothness, and the preservation of symmetries along steepest descent. It is in general impossible to address all of these concerns at once. For instance, to preserve symmetry of reflection it is necessary to sacrifice either computational speed, or smoothness. Another peculiarity is the failure of some Riemannian structures to commute with the derivative operator as the flow projects onto the space of velocities of the curves. By choosing a structure that avoids this behavior, a flow-invariant is discovered for free curve-straightening. Using this invariant, it is possible to predict the limit-length along steepest descent in terms of the initial elastic energy. This yields examples where all negative gradient trajectories converge despite the fact that the Palais-Smale condition is violated. The length of the curves grows monotonically along steepest descent. In Euclidean space this behavior is not shared by the curve-straightening flow applied to curves with both endpoints fixed.
1991 Mathematics Subject Classification. Primary 58F25; Secondary 58E10, 53C21.
Key words and phrases. Curve-straightening, Palais-Smale condition, gradient trajectories.
1. introduction
1.1. Gradients. Let be a Hilbert space and a sufficiently smooth function. The directional derivative is denoted by . This family of linear maps is represented by the gradient vector field so that for all and . More generally, if is a Riemannian submanifold, then the projection of onto the tangent space satisfies for all and all . Although elementary, the calculation of is not entirely trivial. Explicit formulas in terms of integrals are derived in Section 2 for the common forms of encountered in the calculus of variations. It is seen there how the form of the gradient changes with the choice of Riemannian structure.
1.2. Steepest descent. When is bounded below, it is known that the negative gradient trajectory , such that
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(1.1) |
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extends indefinitely so that . Due to the presence of the gradient, the trajectories are affected by the choice of Riemannian structure. In particular, it is possible that smoothness may be lost, and initial symmetries may be destroyed along the trajectory. When corresponds to a space of curves, it is of interest to compare the flow in to the flow in the space of velocities. This relationship is tied to the choice of Riemannian structure. Perhaps surprisingly, the two flows and the derivative operator do not always combine to form a commutative diagram.
1.3. Free curve-straightening. By choosing a Riemannian structure where the commutative diagram is satisfied, a flow-invariant is discovered when represents curves where is a fixed given two-dimensional manifold. The endpoints are not subject to any constraints, and hence the term free. The function is here the total squared geodesic curvature. In terms of the length of the curves, the flow invariant is given by . In free curve-straightening along steepest descent. It follows that the limit-length is in terms of the initial data. By considering geodesic segments in of positive integer length, it is clear that the Palais-Smale condition is not satisfied. This provides examples where all negative gradient trajectories converge despite a failing Palais-Smale condition.
1.4. Organization. The gradient formulas are provided in Section 2. There it is also shown how to deal with point-wise and isoperimetric constraints. Section 3 concerns steepest descent and the relationship between the flow in the space of curves and the flow in the space of velocities. An example is given that illustrates preservation of symmetry at the expense of smoothness.
Section 4 examines the free curve-straightening flow. The flow-invariant is the main topic here. Free curve-straightening is illustrated by considering circular arcs in Euclidean, spherical, and hyperbolic geometry, subject to either variable or fixed length. The more routine proofs are available in the Appendix.
2. Gradients
2.1. Building blocks. Consider functionals of the form
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(2.1) |
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Assume , and sufficiently smooth so that is well defined. There are two quantities that simplify the presentation of the gradients:
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(2.2) |
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and
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(2.3) |
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Here denotes the partial derivatives of with respect to the second/third variable. The super and subscripts on , and are there to remind the reader that these quantities depend on and . This formalism anticipates the presence of isoperimetric constraints. The quantity introduced here is the negative of the anti-derivative of the so-called Euler operator; see page 16 in [2]. In Mechanics the quantity corresponds to the work.
2.2. Inner products. Several different inner products will be considered. To be precise, let , and consider
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(2.4) |
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(2.5) |
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(2.6) |
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Here , , , and . To save space, write , , and later .
Theorem 2.1. For functions such that ,
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(2.7) |
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(2.8) |
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(2.9) |
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Proof. See the Appendix. ÿ
The inner product
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(2.10) |
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utilized in [8], [9], [10], [11], and [12], is a standard Sobolev inner product. Note that with and , leads to the much simpler formula
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If the integrand of the functional has no explicit dependence on so that , then
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(2.12) |
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because in this case. Moreover, , so the initial point is fixed. The endpoints are not in general expected to remain fixed under gradient flows. For instance, if and , then .
2.3. Classical calculus of variations. The gradient formalism relates to the classical development of the calculus of variations as follows.
Theorem 2.2. If is twice continuously differentiable and a critical point, then for all and . The statement
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with equality everywhere, is the Euler-Lagrange equation. The statements , correspond to the natural boundary conditions .
Proof. If the gradient vanishes in one metric, then it vanishes in all metrics. In the present context, the most efficient argument is available using the metric . Set in the formula for the gradient and conclude that . The derivative of the gradient formula reveals that for all . Next note that . Finally, since (2.13) implies
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(2.14) |
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it is also true that
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(2.15) |
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and therefore . ÿ
Observe that the statement incorporates the Euler-Lagrange equation as well as the necessary conditions given by the natural boundary conditions. It follows that the equation is more selective than the Euler-Lagrange equation.
2.4. Boundary conditions. To have a unified treatment, let , be given and define by . The idea is to exhibit the functions such that as .
Theorem 2.3. The gradients in the three metrics are given by
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(2.16) |
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(2.17) |
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and depending on if or
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(2.18) |
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(2.19) |
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Proof. See the Appendix. ÿ
Note that the gradient vector fields do not depend on so these are constant vector fields on . If , then
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(2.20) |
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(2.21) |
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(2.22) |
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If , then
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(2.24) |
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(2.25) |
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When the metric is or , the vectors in the vector field are smooth ( ) elements in . Only if or is the same true in the metric .
2.5. Fixed endpoints. Let denote the projected gradient onto the affine subspace
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(2.26) |
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where is equipped with the metric , , and , respectively.
Theorem 2.4. The projected gradients are given by
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(2.27) |
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(2.28) |
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(2.29) |
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Proof. See the Appendix.
ÿ
Note the absence of in the expression for the projected gradients. It
is also obvious that . It is seen
that, if only the initial point is kept fixed and , then the projected gradient is given by
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(2.30) |
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provided . In general,
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(2.31) |
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When only is kept fixed, the general case is given by
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(2.32) |
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It is somewhat surprising that both and are absent in this last formula. Another peculiarity is that if and , then
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and
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On the other hand if and , then
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and
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The formulas (2.33) and (2.36) are possible to interchange in problems where it is irrelevant which of the two endpoints is fixed. The same remark applies to (2.34) and (2.35). Formulas (2.36) and (2.34) have theoretical and numerical advantages.
2.6. Isoperimetric constraints. Assume the constraint is given by a functional of the form
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(2.37) |
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where satisfies the same assumptions as . The constraint is classically known as an isoperimetric constraint. Assume regularity so that implies is onto, and hence . With this assumption, is a closed submanifold of co-dimension 1. Specifically, at each , splits into a one-dimensional normal space and the tangent space
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(2.38) |
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Note that depends on the metric and that it spans the normal space. The projection of a gradient onto the tangent space is here denoted by . Up to this point the tangent space has played only a minor role since and
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(2.39) |
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so in both cases the tangent space is independent of . When the regularity assumption is satisfied, then for each metric there is a unique scalar field defined on such that
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(2.40) |
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The condition determines . In fact, the formula is given by
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(2.41) |
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To use this formula, recall that
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(2.42) |
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For instance, when the metric is , use
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(2.43) |
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and
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(2.44) |
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The special case with and is given by
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(2.45) |
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2.7. More than one isoperimetric constraint. To deal with several constraints at the same time it is necessary to check that the constrained subsets intersect transversally. To guarantee that this is indeed the case, assume that all the gradients of the constraining functionals are linearly independent. The next thing to decide is whether to deal with one constraint at a time or maybe all constraints at once. The following intuitive and elementary lemma implies that each way of dealing with the constraints is theoretically equivalent.
Lemma 2.5. Let be a Hilbert space. Consider two closed linear
subspaces and . Let , , , , and denote the various projections. The following is
always true .
Proof. See the Appendix.
ÿ
With the help of this lemma, the formula for may be applied to gradients that are themselves
the result of projections. This is particularly useful in the presence of
constrained endpoints. If all the
constraints are dealt with at once, or at least all of the isoperimetric
constraints, then is a vector field with the same number of
components as the number of constraints. The formula for this time involves the inverse of a matrix. For
theoretical purposes this may be the preferred approach, but for computations
consideration should be given the iterative approach.
3. Velocity Flows
3.1. Definition. It is assumed that the anti-derivatives are ‘normalized’ so that for some given value , and . Define
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(3.1) |
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and let be given by
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where satisfies the same assumptions as before. Define by
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Note that if , then . Let
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With , let
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The gradient vector field in is given by
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The flow along the negative gradient trajectories of is here referred to as the velocity flow. As seen in (3.6), the gradients are free of derivatives of the function . This fact is important when numerical algorithms are developed. Specifically, the flow along the gradient trajectories of is rarely known explicitly, and hence only discrete approximations of are available. This means is also only available as a discrete approximation. In contrast, the flow of yields discrete approximations of and hence discrete approximations of the anti-derivative . The operation is better behaved numerically than the operation .
3.2. Comparison. The process of differentiation yields a projection from to that restricts to . There is a ‘lift’ from to
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given by
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If and , then the derivative operator projects onto . Conversely, lifts to when . Look at the following diagram
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The upper half depends on the metric in . It is natural to ask if the derivative operator along the ‘missing edge’ turns this into a commutative diagram.
Theorem 3.1. Denote the metric in by and let have the induced metric. Suppose the form of is given by
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(3.10) |
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Consider all and such that both and exist. Now
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if and only if is a positive constant times for all .
Proof. Start with an arbitrary so that . Compute the two directional derivatives and note that they are related by . It follows that
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(3.12) |
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If is a positive constant times , then and
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because , and is an arbitrary element in . Conversely, if is not a positive constant times , then there is a pair such that and but . Let so that , and observe that
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for this , and hence . ÿ
It follows from this that behaves better than the other metrics when the
velocity flow is treated together with the original flow. The positive constant
alluded to is 1 in this case.
3.3. Heuristics. For the metric it is still true that . It follows that . In general, consider a pair of functions such that for all . Use integration by parts to rewrite
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in the form
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An application of the duBois-Reymond’s lemma shows that
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for some . The difference must be twice differentiable and
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From this it is seen that , and in particular
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If and are twice differentiable, then . When is known, this is a non-homogenous linear ordinary differential equation. The homogenous solution may be expressed in terms of hyperbolic functions as seen in the formula for .
3.4. Symmetries and smoothness. If in metric , then it is known to break space symmetries along gradient trajectories; see [13]. It is also shown in [13] that the metrics , and with , preserve symmetries. The velocity flow in metric does not form a commutative diagram with the derivative operator, but the smoothness of the curves along the negative gradient trajectories is maintained. In contrast, the velocity flow of metric forms a commutative diagram with the derivative operator, but in this case smoothness is not always preserved as seen in the following example.
Example 3.2. Suppose in the metric with and . Consider the smooth unconstrained functional
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Consider the initial function given by . The global minimum value of is , and this value is attained at . Click the following link and press Ctrl+A followed by Ctrl+Y, adjust animation speed via Edit|Preferences …|Graphics Options|Animation. The animation simulates the negative gradient trajectory with initial point and limit . The gradient flow slows down as the limit is approached. Smoothness is lost instantly during the descent. The symmetry of reflection is retained.
4. Free curve-straightening
4.1. Background. Several fundamental geometric objects appear naturally as critical points of some ‘energy’. For instance, the generalization of the concept of a straight line to an arbitrary Riemannian manifold yields so-called geodesics. The integral of the speed or its square are natural choices as energy. The trouble is that, with either choice, it is possible to minimize the energy by shrinking the curve down to a point. This undesirable behavior motivates the study of the elastic energy as an alternative. The phrase curve-straightening refers to the flow along the negative gradient trajectories of the elastic energy. As seen in the Euclidean plane, the flow restricted to periodic curves of variable length is always divergent; see [11], and for curves with fixed distinct endpoints some, but not all, trajectories converge; see [12]. The divergence in the periodic case is due to the absence of critical points. When a two-dimensional sphere , with metric , replaces the Euclidean plane, it is not known if there are divergent trajectories. The combined work of Franks [4], Bangert [1], and Hingston [6] show that there are infinitely many geometrically distinct critical points in this case.
4.2. Palais-Smale condition. Free curve-straightening refers to the case when the endpoints are free to move and the ambient two-dimensional space is arbitrary. In this general context the elastic energy is the total squared geodesic curvature. One motivation for the interest in free curve-straightening is its source of convergent flows despite a failing Palais-Smale (P-S) compactness condition. Recall the P-S condition for a nonnegative smooth function where is a ‘locally complete’ Riemannian manifold. The condition asserts that each sequence has a convergent subsequence whenever is non-increasing and . Observe that must be a critical point. With a geodesic segment of length and the elastic energy, since the geodesic curvature is zero, it is clear that the P-S condition fails. Despite this, every negative gradient trajectory converges under free curve-straightening.
4.3. Open problems. As indicated, the choice of boundary conditions is an important factor when analyzing the convergence of negative gradient trajectories. The elastic energy does not satisfy the P-S condition in the space of periodic curves in the standard round sphere. It is not known if there exist divergent trajectories in this case. For periodic curves in the hyperbolic disc the status of the P-S condition is unknown. In contrast, does satisfy the P-S condition when , here is the length of the curve; see [7]. There are indications in [14] that the heuristics that explains divergence in the Euclidean case; see [12] is not present in the hyperbolic case. It is therefore entirely possible that every trajectory converges in the case of periodic curves in the hyperbolic disc.
4.4. Indicatrix. Let be a fixed two-dimensional Riemannian manifold. Represent each sufficiently smooth curve in by the pair , where , , and is the length of . Parallel transport along the curve and let be the angle in the tangent plane from the transported vector to . In this fashion, the pair together with he initial point and direction of uniquely represents a curve in . For the purpose of this article, it is assumed that is absolutely continuous, and . The geodesic curvature is ; see Lemma 25.10 page 585 in [5].
4.5. Setup. The total squared geodesic curvature of a curve in is in terms of the arc-length parameter . A substitution and an additional factor of one half yield the form best suited to what follows. Let be the Sobolev space , and all positive real numbers. With , let be given by
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In the context of curve-straightening, the traditional Riemannian structure on is
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The gradient of relates to the directional derivative via . Using the previous notation, it is straightforward to verify that the two components of the gradient are given by and .
4.6. Fixed length. Things simplify as follows in the case of fixed length. It suffices to let , and so that
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The gradient has only one component, . Along the negative gradient trajectory with flow parameter , the turning angles satisfy
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From the gradient formula, it is clear that that is constant. The initial represents the turning angle between the parallel transport of and . It follows that , and hence . One expects a different choice of Riemannian structure on to yield a different formula for the gradient . There are Sobolev metrics that produce gradients of that do not vanish at the initial point; see equation (2.11). With the current metric, the general solution through some initial is the explicit .
4.7. Fixed length circular arcs in the Euclidean plane. Refer to all curves of the form , with some constant, as ‘circular arcs’. This collection is invariant under the fixed length free flow. In the Euclidean plane, the initial circle of radius evolves through unit length arcs of circles of radii and centers . The turning angle and the length do not completely determine the curve, so it is necessary to also provide the initial point and direction. Here the choice is the origin and the positive direction of the -axis. The free curve-straightening flow does not move the initial point. Figure 1 shows ‘snapshots’ of the flow at with . Figure 1 animation.
Figure 1
4.8. Fixed length circular arcs in the sphere. With the help of the ‘tangential cone’, the parallel transport along a circular arc in the standard sphere of radius 1 is available explicitly; see Example 1 page 243 in [3]. Consider a circle with Euclidean center the distance from the center of a sphere equipped with an outward normal. Let traverse this circle once so that the smaller spherical cap is on the left side. The turning angle of is , the length is , and the geodesic curvature is . A curve of length and turning angle , with , has geodesic curvature and resides on a circle with . The evolution of a unit length circle is through unit length circular arcs along the intersection of planes at an angle with respect to the equatorial plane. The evolution is completely determined by
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Figure 2 shows snapshots of the flow at the same as before. The equatorial geodesic is also indicated. Figure 2 animation.
Figure 2
4.9. Fixed length circular arcs in the hyperbolic plane. In the hyperbolic plane, using the Poincaré model, the evolution is less explicit. The initial curve is a circle of hyperbolic length 1 with its center at the origin of the disc. The coordinate description is
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(4.6) |
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The initial hyperbolic radius is
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According to Mathematica, the initial geodesic curvature, which by the Gauss-Bonnet theorem is the hyperbolic area enclosed plus , is the bewildering
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The turning angle is since the curve has unit length. Figure 3 shows snapshots of the flow at the same . The tangential geodesic is also indicated. Figure 3 animation.
Figure 3
4.10. Variable length and the general case. The free curve-straightening flow in the case of variable length is far less explicit due to the presence of the component . Convergence is in question because the length can diverge to infinity along a negative gradient trajectory as seen in the case of fixed endpoint curves in the Euclidean plane. Let denote the curves along the negative gradient trajectory of originating at the initial curve .
Theorem 4.1. Consider the negative gradient flow trajectory emanating from the initial point defined for by
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where
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It is always true that for all ,
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Proof.
Let
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The derivative is
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The negative disappears since the flow is in the negative gradient direction. The derivative with respect to eliminates the constant in the gradient. The gradient and the operator do not automatically satisfy a commutative law. There are Riemannian structures on based on other selections of Sobolev inner products that violate the commutative law. The structure of the present article is not one of them; see Theorem 3.1. The next step is to use the -component of the gradient. Observe that
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(4.14) |
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Integrate and shuffle the terms to show that this is equivalent to , where is a constant of integration that only depends on the initial . ÿ
Corollary 4.2. The free curve-straightening flow increases the length of the curves in a strictly monotone fashion. Moreover, each initial curve , such that and with turning angle , converges to a geodesic segment of length with initial point and initial direction .
Proof. From the previous proof, it is clear that is positive. Since and , it follows that is bounded from above. Observe that
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(4.15) |
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so if , it must be that converges to a constant function. All that remains is to determine at the initial curve and use the consequence that . ÿ
4.11. Variable length and circular arcs. The free curve-straightening flow specializes to a nonlinear system of ordinary differential equations when the initial curve is a circular arc. Specifically, write and . A quick calculation brings the negative gradient flow into the form
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(4.16) |
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The free flow invariant has the simpler form here, and this leads to the separable equation
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(4.17) |
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According to Mathematica, the inverse of the solution is given by
(4.18)
It is of course possible to approximate the solution numerically by solving the ordinary differential equation. Compare the free curve-straightening flow of fixed length in the Euclidean plane with its variable length counterpart. The ‘snapshots’ are taken at the same flow times.
Figure 4
Here is the free curve-straightening flow in the Euclidean plane followed much longer ( ). Figure 5 animation.
Figure 5
Figure 6 shows the variable length free curve-straightening flow in the hyperbolic disc. The initial curve is the unit length circle centered at the origin. Figure 6 animation.
Figure 6
Figure 7 shows the variable length free curve-straightening flow in the standard sphere. The initial unit length curve is at ‘the bottom’. The sphere is here invisible. The limit curve is part of the great circle that is tangent to the initial curve at its initial point. Click and hold the right mouse button while moving the pointer to examine Figure 7 three-dimensional.
Figure 7
In arbitrary two-dimensional Riemannian manifolds, the turning angle has its -plot in figure 8.
Figure 8
The length is plotted in figure 9.
Figure 9
As suggested by this and the different examples, the length ‘converges faster’ than the turning angle. ‘The free curve-straightening flow prefers to spend the available elastic energy towards lengthening rather than flattening the turning angle.’ The limit of the length is here .
Acknowledgement: Many thanks to Franz Pedit for the invitation to visit GANG and for our many useful and interesting discussions.
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Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, USA
E-mail address: alinner@math.niu.edu