Concept# Length

Summary

Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre.
Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.
Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object.
Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squared) and vo

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Metre

The metre (or meter in American spelling; symbol: m) is the base unit of length in the International System of Units (SI).
The metre was originally defined in 1791 as one ten-millionth of the dista

International System of Units

The International System of Units, internationally known by the abbreviation SI (for Système International), is the modern form of the metric sys

Imperial units

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The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

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