Convex Plane Curves Research Articles

Anisotropic flows for convex plane curves

where V and k are, respectively, the normal velocity and curvature of the interface. Equation (1) is sometimes called the “curve shortening problem” because it is the negative L2-gradient flow of the length of the interface. By drawing pictures, one can be easily convinced that a simple closed curve stays simple and smooth along (1) and shrinks to a point in finite time, with the limiting shape of a circle. However, a mathematical treatment of (1) turned out to be rather delicate. In fact, a rigorous study did not exist until the early 1980s, when differential geometers considered (1) as a tool in the search for simple closed geodesics on surfaces. It was also regarded as a model case for more general curvature flows, which are believed to be important in the topological classification of low-dimensional manifolds. As a first attempt, Gage [10] proved that the isoperimetric ratio decreases along convex curves. Then in Gage and Hamilton [12], it was shown that a convex curve stays convex and shrinks to a point in finite time. Moreover, if one normalizes the flow by dilating it so that the enclosing area is constant, the normalized flow converges smoothly to a circle. Finally, Grayson [14] completed this line of investigation by showing that a simple curve evolves into a convex one before shrinks to a point. For an alternative approach to this result, one may consult Hamilton [18]. Recently Mullins’s theory was generalized by Gurtin [15], [16] and by Angenent and Gurtin [4], [5] (see also the monograph of Gurtin [17]) to include anisotropy and the possibility of a difference in bulk energies between phases. Anisotropy is indispensable in dealing with crystalline materials. For perfect conductors, the temperatures in both phases are constant. The equation becomes

Bifurcations of affine invariants for one-parameter family of generic convex plane curves

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions. We describe how affine vertices and affine inflexions are created and destroyed.

The centre symmetry set

A centrally symmetric plane curve has a point called it's centre of symmetry. We define (following Janeczko) a set which measures the central symmetry of an arbitrary strictly convex plane curve, or surface in $R^3$. We investigate some of it's properties

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

On the affine heat equation for non-convex curves

In this paper, we extend to the non-convex case theaffine invariant geometric heat equationstudied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-knownEuclidean geometric heat equation.

Introducing projective billiards

We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

Asymptotic dynamics of the dual billiard transformation

Given a strictly convex plane curve, the dual billiard transformation is the transformation of its exterior defined as follows: given a point x outside the curve, draw a support line to it from the point and reflect x at the support point. We show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a Hamiltonian flow associated with the curve.

Integral geometry of plane curves and knot invariants

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves, and it is related with the invariants of generic plane curves recently defined by Arnold, with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.

Geometric expansion of convex plane curves

Geometric expansion of convex plane curves

The mean lattice point discrepancy

Consider a sufficiently smooth simple closed convex plane curve enclosing the origin, expanding linearly with time. The root mean square of the discrepancy (number of lattice points minus area) from time t = M to t = M + 1 is almost as small as the root mean square discrepancy from time t = 0 to t = M, so the discrepancy has no memory.

On Affine Plane Curve Evolution

An affine invariant curve evolution process is presented in this work. The evolution studied is the affine analogue of the Euclidean Curve Shortening flow. Evolution equations, for both affine and Euclidean invariants, are developed. An affine version of the classical (Euclidean) isoperimetric inequality is proved. This inequality is used to show that in the case of affine evolution of convex plane curves, the affine isoperimetric ratio is a non-decreasing function of time. Convergence of this affine isoperimetric ratio to the ellipse′s value (8π 2), as well as convergence, in the Hausdorff metric, of the evolving curve to an ellipse, is also proved.

Asymptotic probability measures of zeta-functions of algebraic number fields

We prove the existence of asymptotic probability measures connected with the value-distribution of the Dedekind zeta-function ζ K ( s) of an arbitrary algebraic number field K. This is a generalization of Bohr-Jessen's classical result, which has shown the existence in the case of the Riemann zeta-function. We note that, to prove our theorem in the non-Galois case, we must take a way which is independent of the properties of convex plane curves. Also, we obtain a quantitative estimate in case K is Galois, which is an improvement of the author's former result.

Convex sets and plane curve singularities

Convex sets and plane curve singularities

The heat equation shrinking convex plane curves

Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point

Dynamic analysis of thin elastic noncircular conical shells

The geometry of the middle surface lines of curvature of a thin conical shell, whose cross-section is bounded by a certain closed convex plane curve, is studied. Then, for such a shell, several sets of linear and nonlinear equations of motion are derived in terms of its middle surface orthogonal line-of-curvature coordinate system. As an application of the presented analysis, the free vibration problem of thin circular and elliptical frustums is investigated by means of linear Donnell-type equations of motion. These equations are expressed in terms of the shell middle surface displacement components and they are solved approximately by means of Galerkin's method. Numerical results are presented for frustums with clamped both edges.

On the area of binary tree layouts

The binary tree is an important interconnection pattern for VLSI chip layouts. Suppose that the nodes are separated by at least unit distance and that a wire has unit width. The usual layout of a complete binary tree with n leaves takes chip area ( nlogn), but it can be arranged that all the leaves are on the boundary of the chip. In contrast, the “recursive H” layout has area of order n, but has only O( p n) leaves on the boundary. Thus, the recursive H layout enjoys a small area at the expense of a small number of possible I/O ports. This note shows that it is not possible to design a complete binary tree layout with area O(n) and all leaves on the boundary. More precisely, if the boundary of the chip is a convex plane curve and the leaves on the boundary are separated by at least unit distance, then area of order nlogn is necessary just to accomodate all the wires.

The number of lattice points on a convex curve

If C is a strictly convex plane curve of length l, it has been known for a long time that the number of integer lattice points on C is O(l 2 3 ) and the exponent is best possible. In this paper, it is shown that the exponent can be decreased by imposing suitable smoothness conditions on C; in particular, if C has a continuous third derivative with a sensible bound, the best possible value of the exponent lies between 3 5 and 1 3 inclusive.

A gradually turning curve must come near a lattice point

Leo Moser conjectured that given ε > 0 there is a δ > 0 such that any closed convex plane curve whose curvature is bounded by δ must come within ε of a lattice point. In this note we prove this conjecture and indeed we show that the “correct” order of δ is around ε 2.

Geometric theory of differential equations. II. Analytic interpretation of a geometric theorem of Blaschke

According to Blaschke, a C2 closed convex plane curve admits at least three pairs of points with parallel tangents and equal curvature radii. We generalize the result to theorems on HIill equations with either first or second intervals of stability collapsed. We also prove a four-extrema theorem for second order linear differential equations without periodicity properties. 1. We consider a second order differential equation (1) x + (q(t) + X)x = 0, Xo Xo. Following Bor'uvka, the dispersion 4(to, X) is defined as the value t1 of the first zero following to of any solution x(t) that vanishes at to without being identically zero (x(to) =0, x'(to) FO). In other words, X is the first eigenvalue of (1) for the problem x(to) =x(o(to, X)) =0. Higher dispersions are defined by 4) =ck1, qk(t, X) = 0(4k-1l(t, X)). A special case of interest are the Hill equations, for which q(t+T) = q(t), T constant. The kth interval of instability of a Hill equation is the set of X for which there exists at least one T so that ok(T, X) =r+T, i.e., for which X is the kth eigenvalue for the problem x + (q(t) + X)x = 0, x(r)=x(r + T) = O, r < t < r + T. We put Xk = inf X, Ak = SUp X. It is well known that Ak-1 <Xk Ak <Xk+l. The kth interval of instability collapses (Xk =Ak) if there exists a X* so that (2) 4k(t X*) = t + T, co < t < co. It follows from the theorem on p. 123 of [2] that if (2) holds for an equation (1) we also have q(k(t)) =q(t) and, hence, a relation (2) is possible only for a Hill equation. 2. W. Blaschke has indicated [1] the following theorem: Any C2 closed, convex curve admits three distinct pairs of points with parallel Presented to the Society, June 8, 1970; received by the editors May 28, 1970. AMS 1970 sutbject classifications. Primary 34A30; Secondary 52A10.

On a class of locally convex closed curves

For a certain type of locally convex plane curves some relations are shown between the numbers of double points and double tangents and the order and class of the curve. It is proved that the curve has an equal number of double points and double tangents and at least v-1 where 2v denotes the order (and class) of the curve. At last it is shown how a curve with more that v-1 double points (tangents) may be deformed into a curve with the minimum number.