Maths in navigation

By Agata Skupień

Travelling salesman problemThe travelling salesman problem (TSP) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city.  It is a special case of the travelling purchaser problem.The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization.  It is used as a benchmark for many optimization methods. Even though the problem is computationally/ calculatingly difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved.The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips.  In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder.In the theory of computational complexity, the decision version of the TSP belongs to the class of NP-complete problems. Thus, it is likely that the worst-case running time for any algorithm for the TSP increases exponentially with the number of cities.

As a graph problem

TSP can be modeled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's length. It is a minimization problem starting and finishing at a specified vertex after having visited each other vertex exactly once. Often, the model is a complete graph (i.e. each pair of vertices is connected by an edge). If no path exists between two cities, adding an arbitrarily long edge will complete the graph without affecting the optimal tour. 

Symmetric TSP with four cities

Asymmetric and symmetric

In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic collisions, one-way streets, and airfares for cities with different departure and arrival fees are examples of how this symmetry could break down

TSP by Hamiltonian path

DefinitionsA Hamiltonian path or traceable path is a path that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except the vertex that is both the start and end, and so is visited twice). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

TSP by Hamiltonian path

Weighted graph which owns the Hamiltonian’s path. The blue specks are graph’s vertices, the arrows are edges and the

Hamiltonian’s path is marked by red colour.

Exemplary Hamiltonian’s circuit in Mycielski’s graph

A Hamiltonian cycle in a dodecahedron. Like all platonic solids, the dodecahedron is Hamiltonian.

Examples of graphs

Examples of graphs 1) Complete graph - a complete graph is

a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

A drawing of a complete graph, with its vertices placed on a regular polygon, is sometimes referred to as a mystic rose.

2)Circle graph

In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain.

3) Tournament graphA tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of vertices is connected by a single directed edge.

MATHS IN MAPS

MATHS IN MAPS Definition : Map- A set of points, lines, and

areas all defined both by position with reference to a coordinate system and by their non-spatial attributes.

Maps are the world reduced to points, lines, and areas, using a variety of visual resources: size, shape, value, texture or pattern, color, orientation, and shape. A thin line may mean something different from a thick one, and similarly, red lines from blue ones.

How do Maps represent reality? A photograph shows all objects in its

view; a map is an abstraction of reality. The cartographer selects only the information that is essential to fulfill the purpose of the map, and that is suitable for its scale. Maps use symbols such as points, lines, area patterns and colors to convey information.

How are Maps used? To locate places on the surface of the

earth, To show patterns of distribution, and To discover relationships between

different phenomena by analyzing map information.

Mathematics of Cartography

Map scale Definition- The relationship between distances on a map and the

corresponding distances on the earth's surface expressed as a fraction or a ratio.

One unit of measurement on the map -- 1 inch or 1 centimeter -- could represent 10,000 of the same units on the ground. This would be a 1:10,000 scaled map.

  Large scale/Small scale Cartographers talk about large and small scale maps. A large scale map

shows a small area with a large amount of details. A small scale map shows a large area with a small amount of detail. A good way to remember it: when you give a friend a map to your school or home, that's most likely a large scale map.

Think of a 1:1 map. Now that would show some details! 1:2 scales out and shows less, 1:10 less. So 1:100,000 shows less detail than 1:24,000.

The larger denominator (the second number) the smaller the scale. Large scale maps are 1:24,000 and larger. Intermediate scale maps occur in the 1:50,000 to 1:100,000 scale range. Small scale maps are 1:250,000 and smaller.

Coordinate systems Numeric methods of representing

locations on the earth's surface. Latitude and Longitude The most commonly used coordinate

system today is latitude and longitude- angle measures, expressed in degrees, minutes, and seconds.

Equator and Prime MeridianThe Equator and the Prime Meridian are the reference lines used to measure latitude and longitude. The equator which lies halfway between the poles is a natural reference for latitude. A line through Greenwich, England, just outside London, is the Prime Meridian.

Latitude- Parallels that run east-west.Longitude- Meridians that run north-south. Latitude runs from 0 at the equator to 90N or 90S at

the poles. These lines of latitude, called parallels, run in an east-west direction. Lines of longitude, called meridians, run in a north-south direction intersecting at both poles. Longitude runs from 0 at the prime meridian to 180 east or west, halfway around the globe.

More on Degrees, Minutes, and Seconds On the globe, one degree of latitude

equals approximately 70 miles. One minute is just over a mile, and one second is around 100 feet. Length of a degree of longitude varies, from 69 miles at the equator to 0 at the poles. Because meridians converge at the poles, degrees of longitude tend to 0.

Longitude and Time

Since the earth rotates 360 degrees every 24 hours, or 15 degrees every hour, it's divided into 24 time zones- 15 degrees of longitude each. When it is noon at Greenwich, it is 10:00 A.M. 30 degrees W., 6:00 A.M. 90 degrees W., and midnight at 180 degrees on the opposite side of the earth.

Maps Topics – Projections

Map projectionA function or transformation which relates coordinates of points on a curved surface to

coordinates of points on a plane.

A Problem – DISTANCES, ANGLES, AREAS If you cut a cylinder apart lengthwise, you can lay it flat.

Can you cut apart a sphere and lay it out flat? Distortion is inevitable when we try to project the points of

a 3-dimensional earth onto a 2-dimensional piece of paper (or computer screen). Angles, areas, directions, shapes and distances can become distorted when transformed from a curved surface to a plane. Many different projections have been designed where the distortion in one property is minimized, while other properties become more distorted.

For Example The Mercator Projection Angles are preserved, but distances

away from the equator become progressively distorted. (South America is actually nine times as big as Greenland.)

CONTOUR LINE

A contour lineof a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines.

More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value. The gradient of the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the gradient is large: the variation is steep. A level set is a generalization of a contour line for functions of any number of variables.

Contour lines are curved or straight lines on a map describing the intersection of a real or hypothetical surface with one or more horizontal planes. The configuration of these contours allows map readers to infer relative gradient of a parameter and estimate that parameter at specific places. Contour lines may be either traced on a visible three-dimensional model of the surface, as when a photogrammetric viewing a stereo-model plots elevation contours, or interpolated from estimated surface elevations, as when a computer program threads contours through a network of observation points of area centroids. In the latter case, the method of interpolation affects the reliability of individual isolines and their portrayal of slope, pits and peaks.

Four color theorem In mathematics, the four color theorem, or the four

color map theorem states that, given any separation of a plane into contiguous  regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not.

The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property.

Precise formulation of the theorem

The intuitive statement of the four color theorem, i.e. 'that given any separation of a

plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color', needs to be interpreted appropriately to

be correct. First, all corners, points that belong to (technically, are in the closure of) three or more countries, must be ignored. In addition, bizarre

maps (using regions of finite area but infinite perimeter) can require more than four colors.

Precise formulation of the theorem Second, for the purpose of the theorem every

"country" has to be a simply connected region, or contiguous. In the real world, this is not true ,because the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map

Precise formulation of the theorem In this map, the two regions labeled A belong to the same country, and must

be the same color. This map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. If A consisted of three regions, six or more colors might be required; one can construct maps that require an arbitrarily high number of colors. A similar scenario can also be constructed if blue is reserved for water.

An easier to state version of the theorem uses graph theory. The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment. This graph is planar (it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region. Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, "every planar graph is four-colorable" (Thomas 1998, p. 849; Wilson 2002).

http://en.wikipedia.org/wiki/File:Four_Colour_Planar_Graph.svg

Thanks everyone Agata Skupień