Latitude Longitude -> Polar Form -> Cartesian Coordinates

In summary, the conversation discusses the conversion of latitude and longitude coordinates into Cartesian coordinates, taking into consideration the Earth as a spherical object. The solution provided by one person is (-767.18, -4350.91, 4575) and they request clarification on the signs and step-by-step process for setting up the Cartesian coordinate system. The respondent explains the setup with the north pole as the origin and provides instructions for correctly handling the co-latitude. They also mention the effect of longitude and latitude on the positive or negative values of the x, y, and z coordinates.
  • #1
jwj11
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[SOLVED] Latitude Longitude -> Polar Form -> Cartesian Coordinates

Homework Statement



I need to convert 46 Degrees North 80 Degrees west into Cartesian coordinates, based on the assumption that the Earth is a sphere (althought it's not).

Homework Equations



http://en.wikipedia.org/wiki/Spherical_coordinate_system

The Attempt at a Solution



I've attempted to convert and I got ( -767.18, -4350.91, 4575 )
Not sure if this is correct. I'd like to know the North and West signs factor into the equation when plugging in. Can anyone help, or show me a step by step basis please??
 
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  • #2
HOW have you set up your cartesian coordinate system? I assume the origin is at the center of the Earth and the positive z-axis runs through the north pole. Is the positive x-axis pointing toward the Greenwich meridion (i.e. longitude= [itex]\theta[/itex]= 0)? And be sure you handle [itex]\phi[/itex] correctly. In spherical coordinates [itex]\phi[/itex] is the "co-latitude": measured from the north pole rather than from the equator.

If so, then 80 degrees West longitude means that x will be positive (beyond 90 degrees W longitude would make x negative) but that y will be negative (any west longitude makes y negative). Since latitude is north, z will be positive but be sure to use [itex]\phi[/itex]= 90- 46= 44 degrees.
 

Related to Latitude Longitude -> Polar Form -> Cartesian Coordinates

1. What is the purpose of converting latitude and longitude to polar form?

Converting latitude and longitude to polar form allows for easier visualization and calculation of distances and angles on a two-dimensional map or globe. It also simplifies mathematical equations involving coordinates.

2. How do you convert latitude and longitude to polar form?

To convert latitude and longitude to polar form, you use the following equations:
r = √(x² + y²) where r is the distance from the origin and x and y are the Cartesian coordinates, and
θ = arctan(y/x) where θ is the angle from the positive x-axis.
Alternatively, you can use online converters or specialized software.

3. What are the advantages of using Cartesian coordinates over polar coordinates?

Cartesian coordinates are easier to work with in terms of calculations, as they involve simple addition and subtraction. They also allow for more precise measurements and are better suited for representing three-dimensional objects. Additionally, most mapping and GPS systems use Cartesian coordinates.

4. Can you convert Cartesian coordinates back to latitude and longitude?

Yes, you can convert Cartesian coordinates back to latitude and longitude using the following equations:
φ = arctan(z/√(x² + y²)) where φ is the latitude, and
λ = arctan(y/x) where λ is the longitude.
Alternatively, you can use online converters or specialized software.

5. What is the importance of understanding latitude, longitude, polar form, and Cartesian coordinates in scientific research?

These concepts are crucial in various fields of science, such as geography, geology, astronomy, and navigation. They allow scientists to accurately locate and map objects and phenomena on Earth and in space, as well as analyze and interpret data. Understanding these coordinate systems is essential for conducting research, creating maps and models, and communicating findings to others.

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