# Tagged Questions

geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect

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### Analytical solution for volume when a plane cuts a hemi-sphere

I need to find the analytical solution when the plane $P: z = grad\cdot y + z_{cut}$ cuts the hemi-sphere $S: x^2 + y^2 + z^2 = r^2;\:y \leq 0$. I constructed two 3D images in MatLab of the ...
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### Sun position at sunrise & sunset

There are many many references telling me what time the sun will rise and set. There are also references telling me the sun's latitude on a given day. But... I want to find out where the sun will ...
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### Test to know if a vector is inside a spherical triangle

Given a spherical triangle defined by $3$ unit vectors on a sphere, how can we test if a vector is contained inside the spherical triangle?
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### Spherical Triangle

I know that the area for a spherical triangle is calculated as Area $= r^2(a+b+c-\pi)=r^2E$ where $E= (a+b+c-\pi)$ is the spherical excess I was wondering why do you have to multiply by $r^2$ (the ...
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### Fit circle between points located on unit-sphere

Suppose I have a sphere of points with two coordinates (two angles), all points are located on a unit sphere, so radius of the sphere is one. Now my problem is, I want to find empty circles, or ...
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### In an equilateral spherical triangle, show that SecA=1+Seca

Q. In an equilateral spherical triangle, show that $SecA=1+Seca$ So A is the vertex or the angle of the triangle and a is the side of the equilateral spherical triangle. I started off the proof by ...
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### If I wanted to drive due west around the earth would I need to turn my steering wheel?

Assume I found a land route around the earth that followed a single line of latitude and was perfectly smooth. I want to drive my car due west around the earth and return to the same point that I ...
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### Continuity of maximum distance between geodesics on a smooth manifold

I am working on my own version of a proof of the Jordan Separation Theorem (just for fun - I know it's been proved countless times) and in the course of so doing I use the apparently fairly obvious ...
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### Finding latitude and longitude

Suppose that P is the north pole and points X and Y in the northern hemisphere are 45◦ apart and form a triangle P XY with angles 60◦ at X and 80◦ at P. Find the latitude of Y . Can you determine the ...
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### Find the sum of the sides in a spherical right triangle

In a spherical triangle the angles at α, β and γ are π/5, π/3, π/2. Find the sum of the sides, we shall call the sides a,b,c So I'm looking at the formulas and I see one of Napier's rule which ...
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### Find the intersection point of a great circle arc and latitude line

In spherical geometry, I need to know at what longitude λ a great circle arc φ1,λ1-φ2,λ2 has intersected a line of latitude φ. I have found the equivalent equation for solving latitude φ for an ...
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### Intensity distribution of a Lambertian LED as a function of angle

I have a practical spherical geometry problem that I'm having trouble cracking. I'm illuminating a planar surface with an LED that has a Lambertian intensity distribution, i.e. the intensity drops off ...
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### Intersection point is in the triangle

On $X={\bf R}^2$ or $S^2(1)$, we have a triangle $\triangle ABC$ whose perimeter is small. On $D\in \overline{BC}$, let $$r_1:=|BD|,\ r_2:=|CD|$$ Consider spheres $S(B,r_1),\ S(C,r_2),\ S(A,r)$. ...
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### n-by-n degree grid on a sphere?

I've been trying to generate an evenly spaced grid centred at a given point on a sphere, such that the angular separation between any neighbouring pair of points is the same (e.g., 1 degree). The grid ...
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### Spherical Triangle properties

In a spherical triangle ABC do the following properties hold? (a) If AB = AC are the base angles at B and C equal? Yes (b) If the angles at B and C are equal is it true that AB = AC? Yes (c) Do the ...
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### Point within a spherical triangle given areas

Consider a spherical triangle like this: where $A_1, A_2, A_3,$ and $P$ are points on the sphere and $t_1, t_2, t_3$ are the proportion of the area of the large triangle contained within the small ...
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### Spherical distance

The spherical distance between two points (P1=(0,0,1) P2=($\frac{1}{2\sqrt{2}}$,$\frac{1}{2\sqrt{2}}$,$-\frac{\sqrt{3}}{2}$) ) is $\frac{5\pi }{6}$ I am at a loss as to how the spherical distance was ...
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### Find the missing sides and angles in following case for a spherical triangle ABC:

$\bf{QUESTION}$: Find the missing sides and angles in following case for a spherical triangle ABC: $$a)a=60°,\beta=90°, \gamma=75°$$ So, if I am right my book says sides are denoted by lowercase ...
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### Arc length in spherical triangle [closed]

A spherical triangle has angles of 120◦, 60◦ and 45◦. Find the cosines of the (arc) lengths of the sides. How many sides have an arc length larger than 90◦?
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### Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter [duplicate]

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter. I know that diameter is equal to the twice of radius. How can you possibly solve ...
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### Find the length of any great circle on $\mathbb{S}^2$

I am speculating that the length is $2\pi$ because the circumference of a unit circle in $\mathbb{R}^2$ is $2\pi$. From my understanding, a great circle in $\mathbb{S}^2$ would be a circle centered ...
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### Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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### Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...