Implement Great-circle distance

[peter279k]

Please refer this: https://en.wikipedia.org/wiki/Great-circle_distance

[DivineOmega]

I believe this is implemented already under the name Haversine distance.

https://github.com/DivineOmega/php-distance/blob/master/src/Types/Haversine.php

Please feel free to double check this though.

[peter279k]

@DivineOmega, thank you for your reply. After checking these two distance formulas, they're different and the Haversine class doesn't implement Great-circle distance formula well.

According to this wiki reference, it has two different formulas in the different condition.

The first formula is as follows:

And the distance is: (r is the $greatCircleRadius value)

On computer systems with low floating-point precision, the spherical law of cosines formula can have large rounding errors if the distance is small.

The haversine formula is numerically better-conditioned for small distances:

And the second formula is same as Haversine distance.

I think we don't have to implement the first formula I mention because this formula will have the rounding errors on the computing system, right?

[DivineOmega]

Interesting. This is not something I have considered before.

The wiki also states that:

For modern 64-bit floating-point numbers, the spherical law of cosines formula [...] does not have serious rounding errors for distances larger than a few meters on the surface of the Earth.

[peter279k]

@DivineOmega, thank you for your reply.

I think we can get the two points are as follows:

• if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle comes out 0.99999999 (means the cos value is very closed to 1.), The haversine formula is numerically better-conditioned for small distances.

• The first formula I mention will be implemented and we can write some explanations about that in README.

What do you think about that? Thanks.

[peter279k]

@DivineOmega, any thought about this :)?