Coordinate system 1st

Co-ordinate Geometry

Mr. HIMANSHU DWAKAR

Assistant professor

Department of ECE

JETGI

Mr. HIMANSHU DIWAKAR JETGI 1



General DefinitionA co-ordinate system is a system designed to establish positions with

respect to given reference points.

The co-ordinate system consists of one or more reference points, the styleof measurement (linear or angular) from those reference points, and thedirections (or axes) in which those measurements will be taken.


1. POLAR CO-ORDINATE SYSTEM: It isa two-dimensional coordinate system inwhich each point on a plane indetermined by a distance from a fixedpoint and an angle from a fixed direction.(r,θ)

2. CYLINDRICAL CO-ORDINATESYSTEM: It is a three-dimensionalcoordinate system, where each point isspecified by the two polar coordinates ofits perpendicular projection onto somefixed plane, and by its distance from theplane. (ρ,Ф,z)


CLASSIFICATIONS

4. CARTESIAN CO-ORDINATE

SYSTEM: It specifies each point

uniquely in a plan by a pair of

numerical coordinates, which are

the signed distances from the point

to two fixed perpendicular directed

lines, measured in the same unit of

length.

3. SPHERICAL CO-ORDINATE SYSTEM: It is athree-dimensional coordinate system, wherethe position of a point is specified by threenumbers: the radial distance of that pointfrom a fixed origin, its elevation anglemeasured from a fixed plane, and theazimuth angle of its orthogonal projection onthat plane. (r,θ,Ф)


Learning Co-ordinate geometry is not just to clear your present class but also helps your understanding in various ways. Like–1. Geometry is applicable in computers or cell phones.2. The text file or PDF file which we open is itself an example of coordinate plane. 3. In these, the words or images are written or modified with the use of coordinate geometry. 4. Any PDF file, which contains text, images and different shapes, are placed according to the 2-dimentional coordinate (x, y) system. 5. All the concepts like distances, slopes and simple trigonometry are also applicable here.

USES OF CO-ORDINATE SYSTEM


Describing position of any objectCoordinate system can be used to find the position of any object from its original place (called origin) to its present location

APPLICATIONS IN REAL LIFE

Location of Air TransportWe all have seen the aero-planes flying in the sky but might have not thought of how they actually reach the correct destination. Actually all these air traffic is managed and regulated by using coordinate geometry.


http://blog.askiitians.com/wp-content/uploads/2016/03/331.png




Map ProjectionsMap Projection is a technique to map any 3D curved object on a flat 2D surface.


The Global Positioning System (GPS):

GPS is a space based satellite navigation system

that provides location and time information in all

weather conditions. In a GPS, the longitude and

the latitude of a place are its coordinates. The

distance formula is used to find the distance

between 2 places in a GPS.






In real life, when weather forecasters are tracking hurricanes, they note the absolute location on a periodic basis to see the path of the storm and try to predict the future path based partially on these findings.


A latitude measurement indicates locations at a given

angle north or south of Earth’s equator. So, latitude

measurements range from 90° North at the North Pole

to 0° at the equator to 90° South at the South Pole.

A longitude measurement indicates locations at a

given angle east or west of an imaginary north-south

line called the prime meridian, which runs through

Greenwich, England. Longitude measurements begin

at 0° at the prime meridian and extend 180° both to

the west and to the east.Mr. HIMANSHU DIWAKAR JETGI 10





Now imagine what if coordinate system doesn’t exist. Pilots, aircraft controller, passengers in the flight, persons waiting for the flight all will not be able to get the location or position of aircraft. These will also definitely increase the chances of aircraft crushes. So from here we can easily say that coordinate system is one of the most important parts of air transport.

WHAT WILL HAPPEN IF IT DOES NOT EXISTS?




Circular Cylindrical Co-ordinate System


An overview

Cylindrical coordinate surfaces. The three orthogonal components, ρ(green), φ(red), and z (blue), each increasing at a constant rate. The point is at the intersection between the three colored surfaces.


Circular Cylindrical Co-ordinate (𝜌, ∅, 𝑧)

Cylindrical coordinates are a simple

extension of the two-dimensional polar

coordinates to three dimensions.


http://mathinsight.org/polar_coordinates

Definition

The three coordinates (ρ, φ, z) of a point P are defined as:

• The radial distance ρ is the Euclidean distance from the z-axis to the

point P.

• The azimuth φ is the angle between the reference direction on the

chosen plane and the line from the origin to the projection of P on

the plane.

• The height z is the signed distance from the chosen plane to the

point P


https://en.wikipedia.org/wiki/Rho_(letter)

https://en.wikipedia.org/wiki/Phi

https://en.wikipedia.org/wiki/Euclidean_distance

Point P and unit vectors

in the cylindrical

coordinate system.

𝜑 is azimuthal angle

0 < 𝜌 < ∞

0 < 𝜃 < 2𝜋

−∞ < 𝑧 < ∞


• A vector A in cylindrical coordinates can be written as

𝐴𝜌, 𝐴∅, 𝐴𝑍

𝐴 = 𝐴𝜌𝑎𝜌 + 𝐴∅𝑎∅ + 𝐴𝑧𝑎𝑧

𝐴 = 𝐴𝜌2 + 𝐴𝜌

2 + 𝐴𝜌2


Cartesian to spherical & vice versa

• The relationships between the variables (x, y, z) of the Cartesian coordinate system and those of the cylindrical system (p, ∅, z) are easily obtained from Figure.

𝑥 = 𝜌 cos∅𝑦 = 𝜌 sin∅

𝑧 = 𝑧

• And by solving above eq’ns

𝜌 = 𝑥2 + 𝑦2

∅ = tan−1𝑦

𝑥𝑧 = 𝑧


Cont’d

• The relationships between (ax, ay, az) and (𝑎𝜌 , 𝑎∅ , 𝑎𝑧) are obtained

geometrically from Figure.


Cont’d


SPHERICAL CO-ORDINATE SYSTEM


Introduction to Spherical Co-ordinate System

• Spherical coordinates are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid.

The coordinate ρ is the distance from P to the

origin.

θ is the angle between the positive x-axis and the

line segment from the origin to Q.

ϕ is the angle between the positive z-axis and the

line segment from the origin to P.


Location of a Point

• The spherical coordinate system extends polar coordinates into 3D by using an angle ϕ for the third coordinate. This gives coordinates (r,θ,ϕ) consisting of

Co-ordinate Name Range Definition

r radius 0≤r<∞ distance from the origin

θ azimuth −π<θ≤πangle from the x-axis in the x–y plane

ϕ elevation −π/2<ϕ≤π/2 angle up from the x–y plane

The location of any point in spherical is (r,θ,ϕ)


Relationship b/w Spherical and Cartesian coordinate System

x = ρ sinϕ cosθ

y = ρ sin ϕsinθ

z = ρ cosϕ.


Cont’d

• The space variables (x, y, z) in Cartesian coordinates can be related to variables

• (𝑟, 𝜃, ∅) of a spherical coordinate system. From Figure

𝑟 = 𝑥2 + 𝑦2 + 𝑧2

𝜃 = tan−1𝑥2 + 𝑦2

𝑧

∅ = tan−1𝑦

𝑥


The unit vectors ax, ay, az and ar, 𝑎𝜃, 𝑎∅ are related as follows


SPHEROIDS AND SPHERES

• The shape and size of a geographic coordinate system’s surface is defined by a sphere or spheroid

The assumption

that the earth is a sphere is

possible for small-scale

maps (smaller than

1:5,000,000)

To maintain accuracy

for larger-scale maps (scales

of 1:1,000,000 or larger),

a spheroid is necessary to

represent the shape of the

Earth

A sphere is based on a circle, while a spheroid

(or ellipsoid) is based on an ellipse


Latitude & Longitude

• A Geographic Coordinate System (GCS) uses a 3D spherical surface to define locations on the Earth

• GCS uses the azimuth and elevation of the spherical coordinate system

• A point is referenced by its longitude and latitude values

• Longitude and latitude are angles measured from the earth’s center to a point on the Earth’s surface.


Latitude • Horizontal line

• It is the angular distance, in degrees, minutes, and seconds of a point north or south of the Equator.

• Often referred to as parallels.

• The coordinate ϕ corresponds to latitude

• On the Earth, latitude is measured as angular distance from the equator.

• In spherical coordinates, latitude is measured as the angular distance from the North Pole


At the North Pole,Φ=o

At the equator,Φ=

At the South Pole,Φ=

𝜋 2

𝜋

Latitude


Longitude

• Vertical line

• It is the angular distance in degrees, minutes and seconds of a point, East or West of the Prime (Greenwich) Meridian

• Often referred to as Meridians

• Each longitude line measures 12,429.9 miles

• The coordinate θ corresponds to longitude

• θ is a measurement of angular distance from the horizontal axis.


Longitude

At the North poleΘ=

At the equatorΘ=0 or 𝜋

At the south poleΘ= -

𝜋 2

𝜋 2


Latitude & Longitude

Distance between Lines

If we divide the circumference of the earth (approximately 25,000 miles) by 360

degrees, the distance on the earth's surface for each one degree of latitude

orlongitude is just over 69 miles, or 111 km.


GPS (Global Positioning System)

• Space-based satellite navigation system

• Developed in 1973 to overcome the limitations of previous navigation systems

• Provides location and time information in all weather conditions, anywhere on or near the Earth


GPS• Any desired location can be found by entering its coordinates in our GPS

device.

• We only need to know the latitude and longitude of that location to know exactly where it is.

• Today GPS is a network on 30 satellites


THANK YOU


Engineering

Coordinate system 1st