From Geographical coordinates to Cartesian coordinates

• I
• Aleoa
In summary, the conversation discusses the calculation of the angle with respect to the north pole using two points expressed in latitude and longitude. The speaker proposes using the arctan function to calculate the angle from the equator and then adding it to -90° to get the bearing. However, another speaker suggests using the cosine formula instead and taking a direct measurement of the length between the two points. The conversation also mentions the use of Cartesian coordinates and the need to use a Mercator projection to get the correct y coordinate. Ultimately, it is determined that longitude is not needed in the calculations.

Aleoa

I have 2 points expressed in (latitude,longitude) and I want to calculate the angle with respect to the north pole.

Since the two points are very near (like hundred of meters), is it possible to consider the two points in the carthesian system simply as:

x=longitude
y=latitude

Then applying $\arctan(y/x)$, I get the angle from the equator and so, summing this angle to (-90°) i get the bearing (angle from north pole).

Is this correct?

Locally, it is possible to express coordinates on the surface of a sphere with Cartesian coordinates and work with those, but it seems unnecessary. There is an easier solution. Perhaps try drawing a diagram. Also, the angles in a triangle on the surface of a sphere don't add up to 180°...

Matternot said:
Locally, it is possible to express coordinates on the surface of a sphere with Cartesian coordinates and work with those, but it seems unnecessary. There is an easier solution. Perhaps try drawing a diagram. Also, the angles in a triangle on the surface of a sphere don't add up to 180°...

I wasn't clear in the main post. I simply need to calculate the direction of a moving car given two points in the map and i want to express this direction using the angle from the north pole.
Is the solution i proposed correct ?

In this case, it seems clear that they expect you to assume that locally, coordinates are Cartesian and hence, a bearing is probably what they expect. i.e. North is y axis, East is x axis.

Use the arctan, but use it for the difference of the coordinates: The length north/east from one point to the other. This is not directly the difference in coordinates but it is quick to calculate from them.

This is only going to work near the equator. Transforming latitude to the y-axis in this way is called the “equirectangular projection” and does not preserve bearing. You need to use a Mercator projection.

## Cosθ = \frac { Cos(a/R) - Cos(b/R).Cos(c/R)} { Sin(b/R).Sin(c/R)} ##

R is the radius of the Earth. You find b and c directly from Latitude expressed in Radians, viz. b = (π/2-Latitude)R etc.

Since the points are very close together you said, you expect the line joining them to be a geodesic, so take a by direct measurement. The angle θ is what you need.

Longitude is not needed in the calculations.

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## Cosθ = \frac { Cos(a/R) - Cos(b/R).Cos(c/R)} { Sin(b/R).Sin(c/R)} ##

R is the radius of the Earth. You find b and c directly from Latitude expressed in Radians, viz. b = (π/2-Latitude)R etc.

Since the points are very close together you said, you expect the line joining them to be a geodesic, so take a by direct measurement. The angle θ is what you need.

Longitude is not needed in the calculations.

Hi, in which way the cosine formula you writed has to be used ?

take a by direct measurement.
Of what?
Longitude is not needed in the calculations.
It is needed to calculate a, if using your formula. Personally I think transforming to Cartesian coordinates is more intuitive, you just need to use the Mercator transformation to get the right y coordinate. Google will help here.

1. What is the difference between geographical coordinates and Cartesian coordinates?

Geographical coordinates, also known as geographic coordinates, are a set of numbers used to specify a location on the Earth's surface. They consist of latitude and longitude values, which measure a location's distance from the equator and prime meridian, respectively. Cartesian coordinates, on the other hand, are a set of numbers used to represent a point on a plane or in space. They consist of x and y values, which measure a point's horizontal and vertical distance from a reference point.

2. How are geographical coordinates converted to Cartesian coordinates?

To convert geographical coordinates to Cartesian coordinates, we use a mathematical formula that takes into account the Earth's curvature and the chosen reference point. This formula involves converting the latitude and longitude values to radians, and then using trigonometric functions to calculate the x and y values. The resulting Cartesian coordinates can then be plotted on a map or used in mathematical calculations.

3. Why are Cartesian coordinates often used in scientific and mathematical applications?

Cartesian coordinates are commonly used in scientific and mathematical applications because they provide a simple and standardized way to represent points and locations. They also allow for easy calculation and manipulation of distances and angles, making them useful for tasks such as navigation, mapping, and data analysis.

4. Are there any limitations to using Cartesian coordinates?

One limitation of using Cartesian coordinates is that they are based on a flat plane, which does not accurately represent the curved surface of the Earth. This can lead to inaccuracies in mapping and navigation. Additionally, Cartesian coordinates only provide a two-dimensional representation of a location, so they are not suitable for representing locations in three-dimensional space.

5. How do we know which reference point to use when converting geographical coordinates to Cartesian coordinates?

The reference point used in converting geographical coordinates to Cartesian coordinates can vary depending on the application. In some cases, the origin (0,0) is chosen to be the center of the Earth, while in others, it may be a specific point on the Earth's surface. It is important to clarify which reference point is being used in order to accurately convert between coordinate systems.