Converting latitude/longitude to Cartesian coords?

[kronchev]

Does anyone have a quick method to do this?

 

[cookiemonster]

It's just a straight application of the spherical coordinate transformation.

$$x = \rho \sin{\theta} \cos{\phi}$$
$$y = \rho \sin{\theta} \sin{\phi}$$
$$z = \rho \cos{\theta}$$

Where $\phi$ is the longitude, $\theta$ is the latitude, and $\rho$ is the radius of the Earth.

cookiemonster

 

[deltabourne]

It's just a straight application of the spherical coordinate transformation.

$$x = \rho \sin{\theta} \cos{\phi}$$
$$y = \rho \sin{\theta} \sin{\phi}$$
$$z = \rho \cos{\theta}$$

Where $\phi$ is the longitude, $\theta$ is the latitude, and $\rho$ is the radius of the Earth.

cookiemonster

I believe $$z = \rho \cos{\phi}$$ and $$x = \rho \cos{\theta} \sin{\phi}$$

 

[HallsofIvy]

Though I suspect that our original poster wanted 2 dimensional coordinates: like a flat map. Of course, you can't do that: no flat map of the world can be an isometric representation of the sphere. You would need to specify how that is to be handled.

 

[uart]

I believe $$z = \rho \cos{\phi}$$ and $$x = \rho \cos{\theta} \sin{\phi}$$

All you're doing there deltabourne is interchanging the values of theta and phi in cookiemonsters definition. Theta is normally used to denote the angle from the positve z axis and with that definition of theta cookiemonsters equations are correct.

 

[HallsofIvy]

Maybe this is an "America against the rest of the world" thing but every text I've ever seen defines φ to be the angle the straight line from (0,0,0) to the point makes with the positive z axis while θ is the angle the projection of that line onto the xy-plane makes with the positive x-axis.

 

[uart]

Maybe this is an "America against the rest of the world" thing but every text I've ever seen defines φ to be the angle the straight line from (0,0,0) to the point makes with the positive z axis while θ is the angle the projection of that line onto the xy-plane makes with the positive x-axis.

Yes it looks like both conventions are in common use unfortunately. Here is what Mathworld has to say about it.

A system of curvilinear coordinates which is natural for describing positions on a sphere or spheroid. Define $$\theta$$ to be the azimuthal angle in the xy-plane from the x-axis and $$\phi$$ to be the polar angle from the z-axis with ...

Unfortunately, the convention in which the symbols and are reversed is frequently used, especially in physics, leading to unnecessary confusion. The symbol $$\rho$$ is sometimes also used in place of r. Arfken (1985) uses $$(r, \phi, \theta)$$, whereas Beyer (1987) uses $$(\rho, \theta, \phi)$$. Be very careful when consulting the literature.

 

[HallsofIvy]

Aha! So instead of "America against the world", it is "Physicists against Mathematicians"!

 

[Richardg]

Does anyone have a quick method to do this?

The 'quick and dirty' method (assuming the Earth is a perfect sphere):

x = longitude*60*1852*cos(latitude)
y = latitude*60*1852

Latitude and longitude must be in decimal degrees, x and y are in meters.
The origin of the xy-grid is the intersection of the 0-degree meridian and the equator, where x is positive East and y is positive North.

So, why the 1852? I'm using the (original) definition of a nautical mile here: 1 nautical mile = the length of one arcminute on the equator (hence the 60*1852; I'm converting the lat/lon degrees to lat/lon minutes).

 

[palsy2001]

Does anyone have a quick method to do this?

X = (N+H) cos(phi) cos(lambda)
Y = (N+H) cos(phi) sin(lambda)
Z = [N(1-e^2)+H] sin(phi)

I have solved the inverse problem analytically as well as with other better methods.
This involves solving a complicated quartic eqation. See Vanicek & Krakiwsky, Geodesy.
There are other less efficient methods online. See Mathworks, e.g.

Ben Palmer