Abstract:
The interplay between selective forces and the developmental processes that produce selectable variation is the focus of much attention in evolutionary biology. In a new article, Adams and Nistri [1] pursue this line of investigation by examining the evolution of growth processes in the foot of European cave salamanders. Foot morphology is particularly important in this group of salamanders because the degree of webbing between the toes has been related to their ability to cling to rocks or other substrates. Adams and Nistri use the methods of geometric morphometrics to quantify foot shape in general and to derive a measure of the degree of webbing of the feet.For the clade of salamanders included in the study, an isometric growth trajectory, where the degree of webbing of the foot is constant over ontogeny, appears to be the ancestral condition [1]. There were at least two evolutionary changes of allometry: one lineage evolved an allometric growth pattern, where the degree of foot webbing increases with size, and a species in this lineage later reverted to an isometric mode of growth. Adams and Nistri [1] interpret the switch to allometric growth as a possible adaptation for climbing. Moreover, the ontogenetic trajectories of the different species resulted in a clear convergence from different juvenile foot morphologies toward a shared adult morphology with extensive webbing. This convergence suggests an adaptive explanation, where the common morphology corresponds to a functional optimum [1]. Overall, there appears to be a considerable degree of ontogenetic plasticity that provides opportunities for adaptive evolution.This paper is one of a growing trend for studies at the interface of evolution and development to use morphometric methods to quantify shape [2]. Allometry and its role in evolution have long been recognized as factors that potentially can influence evolutionary processes [3-5]. In recent years, a variety of studies have provided evidence that this i

Abstract:
We generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis.The framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.Morphological symmetry results from the repetition of parts in different orientations or positions and is widespread in the body plans of most organisms. For example, the human body is bilaterally symmetric in external appearance because the same anatomical parts are repeated on the left and right sides. Likewise, many flowers are radially symmetric because sets of petals and other organs are repeated in circular patterns. The evolution of morphological symmetry is of interest in its own right [1-8] and variation among repeated parts, such as fluctuating asymmetry, has been widely used for research in evolutionary biology [9-12]. For instance, fluctuating asymmetry can be viewed as a measure of developmental instability [13] and has been related to measures of environmental stress [14], hybridization [15,16], or fitness [17]. In a different context, fluctuating asymmetry can also be used to investigate the develop

Abstract:
Recent advances in twistor theory are applied to geometric optics in ${\Bbb{R}}^3$. The general formulae for reflection of a wavefront in a surface are derived and in three special cases explicit descriptions are provided: when the reflecting surface is a plane, when the incoming wave is a plane and when the incoming wave is spherical. In each case particular examples are computed exactly and the results plotted to illustrate the outgoing wavefront.

Abstract:
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of $T{\Bbb{P}}^1$ are investigated and a number of theorems are proven that generalise results for closed convex surfaces in ${\Bbb{R}}^3$.

Abstract:
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in ${\Bbb{R}}^3$) with the aid of the natural complex structure on the space of all oriented affine lines.

Abstract:
The total space of the tangent bundle of a K\"ahler manifold admits a canonical K\"ahler structure. Parallel translation identifies the space ${\Bbb{T}}$ of oriented affine lines in ${\Bbb{R}}^3$ with the tangent bundle of $S^2$. Thus, the round metric on $S^2$ induces a K\"ahler structure on ${\Bbb{T}}$ which turns out to have a metric of neutral signature. It is shown that the isometry group of this metric is isomorphic to the isometry group of the Euclidean metric on ${\Bbb{R}}^3$. The geodesics of this metric are either planes or helicoids in ${\Bbb{R}}^3$. The signature of the metric induced on a surface $\Sigma$ in ${\Bbb{T}}$ is determined by the degree of twisting of the associated line congruence in ${\Bbb{R}}^3$, and we show that, for $\Sigma$ Lagrangian, the metric is either Lorentz or totally null. For such surfaces it is proven that the Keller-Maslov index counts the number of isolated complex points of ${\Bbb{J}}$ inside a closed curve on $\Sigma$.

Abstract:
We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{TS}^2\to {S}^2$ induced by the neutral K\"ahler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${TS}^2$. This flow is shown to be completely integrable when the sections are symplectic and the behaviour of the geodesics is described.

Abstract:
We review the complex differential geometry of the space of oriented affine lines in ${\Bbb{R}}^3$ and give a description of Hamilton's characteristic functions for reflection in an oriented C$^1$ surface in terms of this geometry.

Abstract:
We study the neutral K\"ahler metric on the space of time-like lines in Lorentzian ${\Bbb{E}}^3_1$, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric on ${\Bbb{E}}^3_1$. In addition, we give a new characterisation of Weingarten surfaces in Euclidean ${\Bbb{E}}^3$ and Lorentzian ${\Bbb{E}}^3_1$ as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our construction to the classical Weierstrass representation of minimal and maximal surfaces in ${\Bbb{E}}^3$ and ${\Bbb{E}}^3_1$.