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I have some earth-centered coordinate points given as latitude and longitude (WGS-84).
How can i convert them to Cartesian coordinates (x,y,z) with the origin at the center of the earth?
I have recently done something similar to this using the
"Haversine Formula" on WGS-84 data, which is a derivative of the "Law of Haversines" with very satisfying results.
Yes, WGS-84 assumes the Earth is an ellipsoid, but I believe you only get about a 0.5% average error using an approach like the "Haversine Formula", which may be an acceptable amount of error in your case. You will always have some amount of error unless you’re talking about a distance of a few feet and even then there is theoretically curvature of the Earth… If you require a more rigidly WGS-84 compatible approach checkout the "Vincenty Formula."
I understand where starblue is coming from, but good software engineering is often about trade-offs, so it all depends on the accuracy you require for what you are doing. For example, the result calculated from "Manhattan Distance Formula" versus the result from the "Distance Formula" can be better for certain situations as it is computationally less expensive. Think "which point is closest?" scenarios where you don’t need a precise distance measurement.
Regarding, the "Haversine Formula" it is easy to implement and is nice because it is using "Spherical Trigonometry" instead of a "Law of Cosines" based approach which is based on two-dimensional trigonometry, therefore you get a nice balance of accuracy over complexity.
A gentleman by the name of Chris Veness has a great website that explains some of the concepts you are interested in and demonstrates various programmatic implementations; this should answer your x/y conversion question as well.