How to convert Longitude,Latitude, Elevation to Cartesian coordinates?

Question

I downloaded weather data and it has longitude (in decimal), latitude (in decimal), and elevation (in m) values. There is no information about the coordinate system used. How I can convert it to cartesian coordinates ?. My attempts are below. But, my problem is to find the right formulas

def cartesian(self,longitude,latitude, elevation):
    R = 6378137.0 + elevation  # relative to centre of the earth
    X = R * math.cos(longitude) * math.sin(latitude)
    Y = R * math.sin(longitude) * math.sin(latitude)
    Z = R * math.cos(latitude)

def cartesian3(self,longitude,latitude, elevation):

    X = longitude * 60 * 1852 * math.cos(latitude)
    Y = latitude * 60 * 1852

    Z = elevation

    return X,Y,Z

An answer here by Daphna Shezaf uses different formulas. However, it does not use elevations. I would appreciate if someone could clear my confusion, should elevation be used in converting from long/lat or not ?. What are the right formulas ?. I have tried to compare the result of my codes on this website by using specific long, lat, elev. My both methods above have results that are far from the obtained result from the website

UPDATE

I would like to share the solution to my problem. I have implemented lla2ecef function from Matlab as here in python. It allows to convert radian longitude, latitude, and elevation (height in m) to cartesian. I only need to convert latitude and longitude to radian iff they are in decimal by :

latitude = (lat * math.pi) / 180  #latitude in radian, and lat in decimal

To verify my calculations. I compared the conversion result to the website above (website) and this one as well. Both give me almost same result.

Note: If you consider for simplicity earth is sphere, you can use def cartesian (I updated it; thanks to Sasha for correction). If you consider earth is ellipsoid (WGS 84 Geodetic System), you can implement the conversion as in lla2ecef. def cartesian is for cartographic projection (Thanks for rodrigo)

Answer 1

Elevation is measured from sealevel. Radius connects the center of the earth to you geographic location. This means that R = 6371km + elevation. This fixed point can vary and the exact value should be specified by the data provider. Your first function seems to be correct, just replace the R calculation.

To be blunt: Without radius (elevation), it is not possible to convert from spherical to cartesian coordinates. Least you could do is use the "sea level height", but this will only give you the coordinates on a planet which is a perfect sphere. Which Earth isn't.

For example, on the website you provided, you can select the ellipsoid. For WGS 84 standard, I found the following in wikipedia;

The WGS 84 datum surface is an oblate spheroid (ellipsoid) with major (equatorial) radius a = 6378137 m at the equator and flattening f = 1/298.257223563.[6] The polar semi-minor axis b then equals a times (1−f), or 6356752.3142 m