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The term generalised coordinate means that they are the basic particles of a system, maybe a natural or a minimal set of parameters that are actually used to specify the configuration of that system as per the Lagrange method.
This can be differentiated with Cartesian coordinates wherein cartesian coordinates we use parameters like x,y,z to represent a system but here with respect to generalised coordinates we use curvilinear parameters like spherical or cylindrical coordinates.
There are many advantages in using this type of coordinate to represent a system since they can be perpendicular to a corresponding constraint of force.
Generalised Coordinates
These generalised coordinates are actually time derivatives of a system where they can be paired with momentum to provide canonical coordinates. When compared to the vibration of a system, they are represented as the number of DOF (i.e)., degree of freedom associated with the motion of the corresponding body.
What is a Coordinate System?
A system that uses one or more numbers or it can also be called coordinates which uniquely determine the position of a position vector basically a location vector on a manifold or a topological space is called a coordinate system
About the order of coordinates in this system, they are very much significant that they are identified by their position in the finite ordered list of elements
Types of Coordinate Systems
There are different types of coordinate systems of which some of them are listed below:
Number line
Cartesian Coordinate system
Polar coordinate system
Cylindrical or Spherical Coordinate
Homogeneous Coordinate system
Different Types of Coordinate Systems
Curvilinear system
Orthogonal coordinates
Skew coordinates etc…
Why Coordinate Systems are Used?
Usually, coordinate systems are used to represent or specify the position of a point, Moreover, coordinate systems are used to specify some more complex figures like graphs, planes, lines or circles spheres in space.
These are more significant because even the locations of the earth’s surface are also measured or determined using coordinates.
Actually, a coordinate is a set of two or more numbers that are used to represent a point or a figure or a line with respect to a reference system. An example is a Geographical coordinate system, which is sometimes represented in degrees, seconds or minutes.
Cartesian Coordinate System and Spherical Coordinate System and Cylindrical Coordinate System
Cartesian Coordinate System
♦ It is a system which is a two dimensional coordinate system with x and y coordinates that are used to represent a system also they are the signed coordinates perpendicular oriented lines that have the same length
♦ Here each reference line is called a coordinate axis and the point where these two lines meet is called the origin which always has the values of x and y as (0,0)
♦ They can also be three dimensional with perpendicular lines which are meeting at the origin
They can also be further higher dimensional also like with x,y,z coordinates also
Spherical Coordinate System
Other different types of coordinate systems include a spherical coordinate system where it is a coordinate system that has coordinates like r,θ,φ
Where r is the points distance from the origin
θ is the angle of rotation and
φ is the angle from the polar axis
Comparing the spherical coordinates with polar coordinates, polar coordinates have an angle between the axis and the line from the origin to that point which is determined
Cylindrical Coordinate System
It is a three dimensional coordinate system where it has three coordinates which are ρ,φ,z
These coordinates are determined in a three dimensional plane and pass through an origin where all its values become zero
This type of coordinate system is used in rotational geometry and also in longitudinal geometry and also in electromagnetic fields
They are also called cylindrical polar coordinates or cylindrical polar coordinates based upon their application
Coordinate Systems used in Surveying
Three types of Coordinates used in surveying
Graphical grid
UTM(Universal Transverse Mercator) and
SPCS( State plane coordinate system)
According to the coordinates in a grid, the perpendicular distance from the axis is termed as Y coordinate and the other axis is called the X-axis.
In the military, the coordinates are used in a polyconic way of projection.
In the case of UTM(Universal Transverse Mercator), the coordinates are measured in latitudes and zones in a specified system.
And so types of coordinate systems in surveying have many applications in the physical world as mentioned and also have their own different coordinates to give the specified function in a system.
Applications
In real life, the coordinate system has a lot of applications that includes placing furniture in a room or determining locations on earth etc.
They are also used in real-life construction in building bridges, digital music roller coaster design and also more significantly in building satellites also automotive design satellites navigation and more specifically in computer gaming. In the case of medical fields, they are used in MRI scanning and also in Tomography.
Also, it is a very useful alignment in determining the position or location of an airport etc.
Conclusion
Coordinate systems are of many types and have many real time applications which include application in real life like how types of coordinate systems are used in surveying etc. The types of coordinate systems include Cartesian, spherical and cylindrical and so on which also have different types of parameters.
