Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula.

If we know the lat,lng of two points we can calculate their distance. I found the following formulas.

dlon = lon2 - lon1
dlat = lat2 - lat1
a = (sin(dlat/2))^2 + cos(lat1) * cos(lat2) * (sin(dlon/2))^2
c = 2 * atan2(sqrt(a), sqrt(1-a)) 
d = R * c

Let's assume R = 6371 #km

The change I want to make is for only one given point and a given distance to find a point. Now for only those two parameters we create a circle with ofc infinite points around the given point.

Additionally I will create my program to start from the point -90,-180 up to 90,180. So we need to find every time either:

1. For the same latitude from our circle we have two points in the line. We want to find the one with the bigger longitude. So for the starting point of -70,-170 and a given distance of 100km we can have two points that have the same latitude (approximately): -70,-167.37029 and -70,-172.6297. So I will pick the -70,-167.37029

2. For the same longitude, I do the same.

Any possible ways to reverse the formula? It has been some time since my last trigonometric problem... Thanks

Solution 1:

Firstly let us use these variables:

• $\phi_1, \phi_2$: The latitude of points 1 and 2
• $\lambda_1, \lambda_2$: The longitude of points 1 and 2
• $r$: the radius of the sphere (in this case $6371$km)
• $d$: the distance that we want to find

Now the haversine formula is defined as: $$\text{haversin}\left(\frac dr\right)=\text{haversin}(\phi_2-\phi_1)+\cos(\phi_1)\cos(\phi_2)\text{haversin}(\lambda_2-\lambda_1)$$ and the $\text{haversin}$ function is defined as: $$\text{haversin}(\theta)=\sin^2\left(\frac\theta2\right)=\frac{1-\cos(\theta)}2$$ where $\theta$ is some angle in radians.

Note: your angles are presumably in degrees. This may cause problems in the forumla. To fix this just multiply the degrees angle by the conversion factor of $\frac\pi{180}$. If you want to convert back just multiply by $\frac{180}\pi$.

As you may have noticed $d$ is nested within the function so to extract it we need to use the inverse haversin function given by: $$\text{haversin}^{-1}(\theta)=2\sin^{-1}(\sqrt \theta)$$ Now applying this knowledge we can give a full formula for $d$: $$d=2r\sin^{-1}\left(\sqrt{\text{haversin}(\phi_2-\phi_1)+\cos(\phi_1)\cos(\phi_2)\text{haversin}(\lambda_2-\lambda_1)}\right)$$

On the topic of your "circle" description. You want to have to a point on a sphere and connect it to another point finding $d$. Then you want find all points at distance $d$ from your point. To do this you would need to know $d$ by processing your two points using the haversin formula above. Once $d$ has been found you will be able to rearrange the equation such you can find a point. I am warning you though. This is a pretty hard rearrangement.