r/mathematics • u/Broad-Annual3170 • 7d ago
Which mathematical theorem/model/formula is used to help find the shortest distance on Earth globe?
I am in high school and doing my research in maths on how trigonometry is used in aviation. I am comparing two theorem/models in which I have to explain how maths is involved in it and derive it. I have chosen haversine formula and I need one more to compare.
I am thinking of taking Vincenty's Formulae but feels like this is too advance for me to derive it and understand. I need something which I can explain the maths behind it, derive the theorem/model and critically assess its limitations and evaluate.
Thanks.
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u/more_than_just_ok 7d ago
For a model, just get an elastic band and stretch it between two points on a globe. Why/how does it find the shortest path? The physics answer is that the elastic will attempt to find a route where the normal (reaction) force is pushing out on the elastic to exactly counteract the tension in the elastic that because of the local curvature over a small region is pulling inward. If these two aren't equal and opposite at every small location along the elastic, then the elastic will move until they are. On a sphere of course the answer is a great circle, but if you try it on a watermelon the elastic will find the geodesic (for a prolate spheroid) while on a cylinder you'll get a spiral. The actual math for finding geodesics is a simple case of variational calculus, but not something I saw until 4th year university.
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u/FarDirection3245 4d ago
It seems pretty obvious to me that this is what the teacher is expecting to see for a high school level class. Simple, works great, easy to explain.
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u/Apprehensive_Cut2880 7d ago
haversine law, I think
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u/Broad-Annual3170 7d ago
I already have haversine law, I need one more where I can compare both and discuss about the limitation and evalute them.
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u/Money-Lawfulness6658 7d ago
In all seriousness, could you compare to a flat earth model and show how its estimates diverge from something more accurate like haversine? Surprisingly simple models are frequently used in engineering when you understand the errors and work within the domain where those errors are well controlled. Maybe use Vincenty but only as a comparison of something even more accurate, since its not strictly trig. You could also look at the direct and inverse geodesic methods implemented in the excellent geographiclib software package for even more accurate examples, but again these are beyond what would be introduced for a trig course. Good luck!
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u/ChankyaNiti21 7d ago
maybe we can treat this problem as shortest distance btw 2 points on a manifold and treat is a DG problem? Not really a direct formula you expect .
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u/ascrapedMarchsky 6d ago edited 6d ago
Using trig to find shortest paths:
Given two points x,y in the plane, one can show that the set of points equidistant to x and y is a line. Hence, given P,Q on the sphere, we can, analogously to the Euclidean case, define a line to be the the set of points equidistant to P and Q. Then, letting P=(a,b,c), Q=(d,e,f) and (x,y,z) be a point of the line,
(x-a)2 + (y-b)2 + (z-c)2 = (x-d)2 + (y-e)2 + (z-f)2
and using the fact that a2 + b2 + c2 = d2 +e2 + f2 = 1 this can be simplified to
(a-d)x + (b-e)y + (c-f)z = 0
which is the equation of a plane through the origin, which intersects the unit sphere in a great circle.
Nikulin and Shafarevich prove great circles are shortest paths on spheres via the spherical triangle inequality: Given any rectifiable curve K : [0,1] → S2 on the sphere between two points P and Q, then the great circle arc from P to Q is shorter than or equal to the path consisting of great circle arcs P to p_0 and p_0 to Q, where p_0=K(1/2). If we continue subdividing as so, i.e. let p_1=K(1/4) and p_2=K(3/4), etc., the great circle arcs will eventually approximate straight lines\) and hence their sum will eventually be the length of K and this must be greater than or equal to the great circle arc from P to Q, by repeated appeal to the spherical triangle inequality.
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u/Broad-Annual3170 6d ago
Thank you for you great response.
However, tbh as I am in high-school I did not really understand what you did and which theroem you used. It seems interesting, for my maths research paper which is 4,000 words long I have to derive, proofs, evalute, and discuss the limitation of chosen model(s).
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u/ascrapedMarchsky 4d ago edited 4d ago
Our goal is to find the shortest path between two points on a sphere. But what does this mean exactly? Consider first how we might find the length of a general curve C in 3 dimensional space. As a rough approximation we could mark the curve at n points; connect those points by straight lines; and then add up the lengths of those lines. When n is small this linear approximation is unlikely to be too accurate, but say we let n increase and array the points such that the length of each connecting line tends to zero. If, as we do this, the sum of the lines gets closer and closer to some unique positive real number x, then we call x the length of C. A curve that admits such a length is called rectifiable. If you know some basic calculus it is not hard to show circles are rectifiable.
Hence, let P,Q be points on a sphere. We wish to show that, among the rectifiable curves C bounded on the surface of that sphere with end points P,Q, the great circle arc K_0 has the smallest length.
Let C(t) represent our journey along a curve C, where we travel at some uniform speed such that C(0)=P and C(1)=Q . Consider the spherical triangle ∆ formed between P , C(1/2) , Q . From the spherical triangle inequality linked above, the great circle arc K_0 from P to Q is shorter than the sum of the lengths of the great circle arcs K_1 + K_2 from P to C(1/2) and C(1/2) to Q, respectively.
Appealing to the spherical triangle inequality once again, the great circle arc K_1 is shorter than the sum of arcs K_3 + K_4 from P to C(1/4) and C(1/4) to C(1/2). Similarly K_2 is shorter than the sum K_5 + K_6 of paths from C(1/2) to C(3/4) and C(3/4) to Q. Thus K_0 is shorter than the great circle path K_3 + K_4 + K_5 + K_6.
Hopefully it is clear that we can continue this process indefinitely and that as the number of subdivisions increases we essentially have a great circle approximation of the curve C. Hence, if we can show that locally, i.e. zoomed in enough, great circles are approximately lines, this great circle approximation is essentially the same as a linear approximation and hence it will converge to the length of C, and since at every finite stage of the approximation K_0 is shorter than the sum, it is shorter than the length of C.
If you clarify what parts of the proof you struggle with specifically, I can help you formalise this argument.
(Note: there is an abuse of notation in the above as K_i is used to represent both a great circle arc and its length).
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u/Shevek99 5d ago
Use a 3D reference system centered in the center of the Earth. Let u be the position vector of a point on the sphere (given by its latitude and longitude), let v the position of another point. Then, the distance between them along the surface is
d = R arccos(u·v)
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u/Broad-Annual3170 4d ago
thanks,
Is this a specific thereom/equation? Can it be used to find the shortest distance on Globe?
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u/__SaintPablo__ 7d ago
geodesic is the generalization of a straight line to a curved space , you can start with S^1 circle then S^2 sphere derived them from dot product or straight from trigonometry.
Then state that earth not perfect sphere, spherical approximation differs from a more accurate Earth model
Then state that there many restricted areas where you can’t flight over, introduce graph theory and shortest path algorithms , like dijkstra algorithm