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Mathematics Cartography

•Cartography (from Greek khartēs, "map"; and graphein, "write") has been an integral part of the human story for a long time, possibly up to 8,000 years.

•Cartography: The Art and Science of Map Making

•Beauty vs. usefulness

•A good map makes it easy for a reader to acquire your intended information by:

Depicting data effectively – convenient reduction through Scale factor

Reflecting the relative importance of features

Mathematical principles of transformation of 3D Surface onto a 2D plane

A ‘warming up’ exercise ’A picture tells a thousand words’

(百闻不如一见)

How do you tell people directions to somewhere? Have you ever used the terms ‘north’, ‘south’, ‘east’ or ‘west’ to describe

directions?

Do you use reference locations? For example, It is two blocks from the metro station? Or, It will be right in front of you when you get off the bus. It is right on the river Vltava or ....

Describe to someone how you would get to University of West Hungary from the Airport or from train station

Have you ever drawn a map to illustrate to someone how to get somewhere?

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GIS - Geographic Information System

Cartography Today

Today, Smart phones made our life

more geography-friendly using

location based services.

Digital databases are replacing the

printed map as a means of storage

of geographic information

Cartography Tomorrow?

Vision 2030

Information is available anytime and anywhere

In its provision and deliver it is tailored to the user’s context and needs

In this, the location is a key selector for which and how information is provided

Persons would feel spatially blind without using their map, which enable them to see

o who or what is near them, get supported and

o do searches based on the current location

o collect data on - site accurately and timely

The current mobile technologies have demonstrated their huge potential and changed :

o How we work, how we live

o how we thing and imagine

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Geographic Information Systems (GIS)

A system for the input, storage, manipulation, and output

of geographic data

Elements od GIS:

Database with spatially-coded data (latitude/longitude)

GIS Application Software (ArcView, ArcInfo)

Video Map Display

Scanners, Digitizer

Functions of GIS

Street grid navigation

Municipal water supplies, sewers

Hydrology (rivers, streams, lakes)

GPS

Global Positioning System(GPS) Reveal the geometric enigma

Control station

Receiver

Control station

GPS Satellites

Receiver

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GPS

Global Positioning System(GPS) Reveal the geometric enigma

Atomic clocks on the GPS satellites all

show the same time.

GPS receiver receives the signal and

compares the time at which it received

the signal with the time at which it was

transmitted and thus work out traveling

time.

distance = time . c

Ideal situation: In order to get your

position on the Earth you would need to

know your distance from three satellites.

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Problem: GPS receiver does not have an accurate atomic clock so it cannot

know the current time as accurately as the GPS satellites.

Solution: Using the time signal from a fourth satellite to work out the clock

error (time offset).

Four distances from satellites represent radiuses of four spheres. The unknown

time offset gives the inaccuracy of distances. Thus we find the sphere tangent to

four given spheres – your position is in the center.

Global Positioning System(GPS) Reveal the geometric enigma

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GPS system currently consists of 31 working satellites on six separate orbits.

Each of these six orbits is at a height of 20,200 km above the Earth’s surface.

Basic characteristics of all maps:

Location

Attribution (приписывание)

Reduction of reality

Scale

Geometrical transformation/projection

Abstractions of reality

However, this class will help you to start discover mathematics essentials of maps.

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Reference surfaces

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, the truer figure of the Earth.

Sphere – for the small area – diameter max 200 km scale < 1:1 000 000

Plane – topographic maps - diameter max 40 km

a

baf

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Geographics coordinates

The position of a point on the Earth surface is determined by geographic (geo-graphical) coordinates: latitude and longitude

Latitude –angle between a normal at a given point to a spherical reference surface with a plane of the equator of a reference sphere. <0°, 90°> North + South –

Longitude -angle between a plane of the prime meridian (Greenwich - 0°) and a local meridian, passing through any given points.<0°, 180°> East + West –

Geographics net - Parallels and Meridians . The locus of points having a constant latitude/longitude

,X

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Measurement of Direction

Direction (azimuth)

By using the meridians, direction from one point to another can be measured in degrees, in a clockwise direction from true north. To indicate a course to be followed in flight, draw a line on the chart from the point of departure to the destination and measure the angle which this line forms with a meridian. Direction is expressed in degrees

Variation

Variation is the angle between true north and magnetic north. It is expressed as east variation or west variation depending upon whether magnetic north (MN) is to the east or west of true north (TN), respectively.

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Maps are flat, but the Earth is not

M

x

y

a

r M

x

y

x

y

, ,x y

,

,

x f

y g

Cartesian Polar

ˆ ,

ˆ ,

f

g

r

a

, , r acos

sin

x

y

r a

r a

Cartesian and Polar

point on 3D sphere

point on 2D map

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M1

Spherical coordinates

x y

z

M

x

y

1OM d

cos

sin

x

y

d

d

cos

sin

R

z

d

R

cos cos

cos sin

sin ;

0, 2 ; ,2 2

x R

y R

z R

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Important curves on the sphere

Great Circles

Intersection with plane through the center.

Circular arc on great circles:

z

x

A

0

B

r

P

P

S

J

AB R

Distance on meridians

Distance on parallels:

A[1,1]

B[2,1] 1 2 AB R

1 1 2cos AC R

C[1,2]

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Geodesics on the Sphere – Great Circles

the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere– orthodromic distance

Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles

Suppose two points X1(1, 1), X2(2, 2) on the sphere Orthodromic distance is R ., where

1 2 1 2 2 1cos sin sin cos cos cos

z

x

A

0

B

r

P

P

S

J

Clairaut Theorem:

maxcos sin cosA

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Rhumb Line - Loxodrome

A rhumb line is a curve which intersects all of the meridians at the same angle – course A

A = 0° meridians A = 90° parallels

costan

R dA

Rd

tan

cos

dd A

tan ln tan4 2

A c

d

A R d

cosR d

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A rhumb line is a curve which intersects all of the meridians at the same angle – course A

Work out parametric equations for a Rhumb line by substituting the relation for .

tan ln tan4 2

A c

cos cos

cos sin

sin ;

,2 2

x R

y R

z R

Rhumb Line - Loxodrome

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Rhumb Line in the maps

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Great circle Rhumb line (loxodrome)

Mercator Navigation Techniques Gnomonic Projection shows great circles as straight line. Mercator Projection shows constant compass headings (azimuth) as straight lines When steering a ship across an ocean, a navigator will plot a grat circle to minimize distance, but he will then approximate the great circle with rhumb line segments to set up an azimuth.

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Rhumb Lines

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No flat representation of the Earth is entirely accurate

There is no such thing as BEST projection.

Innumerable projection have been developed, each suited to a particular purpose

Equidistant (true distance in specific direction) traffic, military

Equal - area – preserve areas

Conformal (true shape) - preserve angles navy and air navigation

True in area ✕ true in angle

Conformal x Equivalent

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Maps can be either equivalent or conformal, but cannot emphasize

both characteristics.

The map-maker must decide which property is most important and

choose aproect on base on that.

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Distortion in maps

Distance distortion m = distance in map : true distance

Area distortion = area in map : area on the globe

Angular distortion

Distortion in a map differs from point to point on the sphere, and from map to map. Therefore all distortion are function of longitude and latitude. The visual geometrical representation of distance distortion is schematic net of images of discs – distortion ellipses.

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Tissot’s indicatrix – distortion ellipse

projecting a circle of infinitesimal radius

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Choosing a projection surface

Normal (polar)

Equatorial(transversal)

Oblique

P

P

S

J

O

PJ

PS

P

P

S

J

O

Developable surfaces – cone, plane, cylinder A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking

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AZIMUTHAL – projection onto a plane

Preserves Azimuth from the Center – great circles through the central point are represented by straight lines on the map.

Best for Polar Regions

P

P

S

J

P

P

S

J

P PS J

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CYLINDRICAL

Meridians and Parallels intersect at right angle

Often Conformal

Least distortion along Equator

Universal Transverse Mercator works well for narrow strips of the globe

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CONICAL

Most accurate along standard parallels

Meridians radiate out from vertex (often a pole)

Poor in polar regions – just omit those areas

Used in most USGS topographic maps

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Map on a cuboctahedron, gnomonic projection in two hemispheres

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Gnomonic projection on Icosahedron

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Copyright © 2009 C.A.Furuti – All rights reserved – www.progonos.com/furuti

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Gnomonic projection on dodecahedron

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PS=PJ

r

p0

Orthographic projection View from an infinite distance.

1) Normal orthographic projection

Parallels map as a congruent circles without distortion Meridians map to a segment lines.

P

P

S

Jr

p

M(,)

Equations of the map projection

cosRr

a

, cos ,R a

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Distance distortion of orthographic normal projection

0

cos coslim sinp

R Rm

R

Distance distortion for paralels mr = 1

Distance distortion for maridians

distance in the mapDistance distortion

true distance

Maping Equatio

cos

ns

Rr

a

r

R

Př: Determine the N-S distance distortion in Prague (50°s.š, 14°v.d)

distortion =sin 50 0,77

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Central projections

Gnomonic – projection center is in the globe center

Stereographic – the center of the projection is a point on the sphere Hipparchos from Nicee, 180-125 B.C

P

P

S

J

P

P

S

J

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Stereographic projection

The center of the projection is a point on the sphere.

Stereographic projection is conformal, meaning that it preserves the angles at which curves cross each other. On the other hand, stereographic projection does not preserve area.

Circles map to the circles or straight lines

When the projection is centered at the Earth's north or south pole, it sends meridians to rays emanating from the origin and parallels to circles centered at the origin.

S2=

V

V2

V1 S1

k1

k2

k

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Stereographic projection Transversal orientation

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Gnomonic projection

projection center is in the globe center

Image of great circle is straight line.

1. Normal gnomonic projection – it

sends meridians to rays emanating from

the origin and parallels to circles centered

at the origin (except equator).

P

P

S

J

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P PS J

Gnomonic projections

2. Transversal.

3. Oblique

P

P

S

J

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Normal azimuthal projections

Equal distance on meridians

Equal area (Lambert’s p.)

Conformal

- stereographics

2R

r

a

1r pm m 22 sin

2R

r

a

r pm m

1pm

tan4 2

R

r

a

For the polar aspect of all azimuthal projections, the only difference is the spacing of parallel arcs. In this diagram, with parallels in 10° steps, polar regions are blue, the Equator red.

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Loxodrome – Rhumb line

1. Stereographic projection 2. Orthographic projection

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Cylindrical projections (normal orientation)

Lines of latitude and longitude

are parallel intersecting at 90°.

Meridians are equidistant

Forms a rectangular map

Scale along the equator or

standard parallels is true.

The poles are represented as

lines

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Normal cylindrical projections

Equal distance for equator, The identical distortion along the parallel. 1

cosrm

1) Equal distance on meridians (Marinus of Tyre, Erasthostenes)

1, 1,

cosr pm m

P

P

S

J

x

y

x R

y R

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2) Mercator’s projection = equal angle (conformal)

Rhumb line projects as a straight line

1ln tan

cos

x R

y R

1

cosp rm m

3) Equal-area projections (Archimedes)

sin

x R

y R

P

P

S

J

x

y cos

1

cos

p

r

m

m

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Mercator projections – map for marine navigation

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Conical projection

O

PJ

PS

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Conical normal projections

Normal conical projection sends meridians to rays emanating from the origin and parallels to circles centered at the origin

1. Equal distance on meridians

2. Equal area

3. Equal angle

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