# Questions tagged [spherical-trigonometry]

For geometric questions about solving spherical triangles and spherical polygons on spheres.

250 questions
Filter by
Sorted by
Tagged with
19 views

### What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals? [closed]

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
19 views

### Conversion from NED to 'flat earth' coordinates.

I have a flat earth problem of a missile that needs to return to launch pad. The solution to this problem (using convex optimization in case you are interested) is then meant to be fed to a simulator ...
• 25
1 vote
35 views

### Formula to calculate number of sunsets per year based on lattitude

How can I create a formula to turn latitude into number of sunsets per year? Let's keep it relatively simple. Assuming the Earth is a smooth ball, assuming the sun is a single point, not getting ...
1 vote
11 views

### Calculating Diameter of Metric Space Built from Spherical Polygons

$\DeclareMathOperator{diam}{diam}$ Suppose we have a convex spherical polygon $P$ and suppose that we've figured out the two vertices which are the greatest distance apart, say $u$ and $v$. Consider ...
• 868
1 vote
32 views

### Spherical right triangles identities

We have a spherical triangles with angles $\alpha,\beta,\gamma$ that have opposite sides of length $a,b,c$. So our triangle is right angled as we have that $\gamma = \frac{\pi}{2}$. I have to prove ...
58 views

### Worded spherical triangle problem

We consider a sphere with a radius of 4000 metres. We start at point A and travel on a spherical line segment to point B, turn 60$^{\circ}$ to our left then travel on a spherical line segment to point ...
• 23
27 views

### Spherical triangle problem: find the normal vector of the third edge

The following could be a problem that has already been answered to, I've looked at similar questions but I can't figure out the answer. Consider the sphere $S^2$ embedded in $\mathbb{R}^3$ whose ...
13 views

### Shape of largest convex spherical polygon

Are all convex spherical polygons of diameter equal to the diameter of the sphere identical in shape to a circle of the same diameter, or am I thinking about this wrong?
• 167
47 views

### Rotate a sphere about an arbitrary axis using 3 angles relative to coordinate axes?

I have a globe in 3D Euclidean space with the center of the globe at the origin, only the globe is tilted off axis by $\phi$ degrees so it doesn't rotate around the z-axis anymore, but an arbitrary ...
• 157
37 views

### Minimum Sight Distance of Incident Headlight on Vertical Curvature

I am working on the problem of calculating the distance at which the vehicle's headlight beam hits the ground, if the vehicle is traveling on a Vertical curvature. However, I am bit confused if using ...
58 views

50 views

### Explanation of the formula for horizontal sundials

I am researching sundials and the maths behind them. For horizontal sundials (ones that stand perpendicular to the equator) there is a formula to compute the hour angles (the angle between the shadow ...
• 89
1 vote
86 views

120 views

### Analytic formula for integral $I_p(\theta) := \int_0^{2\pi}\cos(t)^p\cos(t-\theta)^pdt$

Let $p$ be a nonnegative integer and $\theta \in [0, \pi]$. Question. What is an analytic formula for the integral $I_p(\theta) := \int_0^{2\pi}\cos(t)^p\cos(t-\theta)^pdt$ ? Note. My ultimate goal ...
• 8,309
52 views

### What happens with the convex hull of $6$ random points on a sphere?

Given a collection of points on the sphere, we can consider their spherical convex hull: add all points on the shortest path between two points in the set, repeat until the resulting set does not ...
• 11.6k
24 views

### Formulas to find the end point of an arc given the start point and midpoint on the unit sphere?

So the radius and the center of the arc are $1$ and $(0, 0, 0)$ respectively because the points and the rac is on the unit sphere $S^2$. Given the start and the midpoint of the arc, is there any ...
• 711
173 views

### The distribution of areas of a random triangle on the sphere - what are the second, third, etc. moments?

Suppose that we choose three points independently and uniformly at random on the surface of a unit sphere as the vertices of a triangle, and consider the area of this triangle. Call this random ...
• 11.6k
1 vote