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Maths in
navigation
By Agata Skupień
Travelling salesman problem
The 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 worstcase running time for any algorithm for the TSP increases exponentially with the
number of cities.
As a graph
problem
Symmetric TSP with
four cities
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.
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, oneway streets, and airfares for cities with different
departure and arrival fees are examples of how
this symmetry could break down
TSP by Hamiltonian path
Definitions
A 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
Exemplary Hamiltonian’s circuit
in Mycielski’s graph
.
colour
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
graph
A 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
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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.
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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 longitudeangle measures, expressed in degrees,
minutes, and seconds.
Equator and Prime Meridian
The 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 northsouth.

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 projection
A function or transformation which relates coordinates of points
on a curved surface to coordinates of points on a plane.
A Problem – DISTANCES,
ANGLES, AREAS

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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

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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 specialpurpose 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
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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).
Thanks
everyone
Agata Skupień