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

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how many Poincare dodecahedrons fill Poincare dodecahedral space?

I was reading Jeffrey's Weeks "shape of space" and that made me wonder: Every spherical 3d manifold (3d Sphere) has a finite volume, The Poincare dodecahedral space is a 3d Sphere. this manifold ...
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16 views

Finding the northernmost latitude in a great circle that passes through two points on the sphere [duplicate]

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on page 23 of the 6th ed.): $A$ and $B$ are two places in the earth's surface with the same ...
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2answers
36 views

Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same ...
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1answer
40 views

How to calculate a solid angle (in Steradians) given only Horizontal Beam angle and Vertical Beam angle data.

I would like to convert a rectangular beam shape given in Horizontal and Vertical beam angle, into solid angle representing the surface area in steradians of projected light. For example a light ...
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2answers
25 views

Finding the shortest distance to the north of the sphere

I found these problems in Alan F Beardon's Algebra and Geometry: Verify that any point with latitude α is a spherical distance R(π/2−α) from the north pole. Suppose that an aircraft flies on the ...
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17 views

Question about trig identities in a celestial sphere problem

So the problem goes: The heavenly body X sets when passing through point F on the horizon. If is the hour angle of X at the time and D the declination, show that : cosH=-tanbtanD cosA=sinDsecb ...
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Cyclindrical coordinates and Spherical coordinates [closed]

Consider the solid body which lies above the upper half of the cone $x^2 +y^2= 3z^2$ and below the sphere $x^2 + y^2 + z^2 = 4z$. Assume this body is of constant density. (a) Use cylindrical ...
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How can I find the center of gravity of a hollow spherical cap?

I am looking to find the center of gravity for a hollow spherical cap. Could I use that point as the point at which the entire mass of the spherical cap is for newtonian gravity problems?
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71 views

Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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1answer
50 views

Can the intersection of two balls be described?

Suppose, two spheres intersect. Subtracting the equations of the speheres, a linear equation appears which indicates the plane conataining all points belonging to the intersection of the spheres. But ...
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1answer
56 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
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19 views

Using azimuthal and polar angles in ECEF coordinate system

I have a physical cone, which its vertex located in some point (x,y,z) in ECEF coordinates, and I want to check if another point is inside this cone. In order to do it, I have to take into ...
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3answers
74 views

Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
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2answers
2k views

Determine depth of a partially filled hemisphere

Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer. The question goes like this: A bowl ...
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1answer
30 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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1answer
41 views

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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2answers
57 views

Four circles touching one another on a spherical surface

The diagram above shows four identical circles, each having a flat radius $r$ (i.e. flat area $\pi r^2$), touching one another at six different points (i.e. each of four identical circles touches ...
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Why does every direction at the north pole point south?

Why does every direction at the north pole point south? Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical ...
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1answer
38 views

Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
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1answer
19 views

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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3answers
587 views

Geometry of spherical triangle

Using the formula for the area of a spherical triangle, find and prove a formula relating the angle sum of a spherical polygon to its area Thought: Area (spherical triangle) ...
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33 views

What is the mathematics behind the two animations?

I found two animated GIFs from a designer's website, which looks very impressive: My questions are: what is the mathematics behind them? How to obtain the mathematical formulas and equations of ...
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1answer
30 views

Spherical and Hyperbolic lines in the Extended Complex Plane.

We work in the Extended Complex Plane: $ \mathbb{C} \cup (\infty)$. Basically, say we have two points, $z_1$ and $z_2$. It can be shown that, on stereographic projection of the Riemann Sphere onto ...
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3answers
239 views

Equal-area sparse spherical shell partitioning

I'm trying to solve a particular problem that arose in a computer graphics context, but can be generalised to a bigger problem as well. I'm not entirely sure if this question belongs to MathExchange ...
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1answer
39 views

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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1answer
45 views

Area of Spherical Polygon

It appears to me that after repeated applications of Girard's theorem on the area of spherical triangles that we can obtain the surface area of a spherical polygon with interior angles ...
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175 views

How to calculate the area covered by any spherical rectangle?

Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a sphere with a radius $R$? Note: Spherical ...
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On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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21 views

Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
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1answer
25 views

Stereographic Projection from an Arbitrary Point

Let $p \in \mathbb{S}^{n}$, then the stereogaphic projection is a diffeomorpshim $h:\mathbb{S}^{n} \setminus \{p\} \to \mathbb{R}^{n-1}$. Suppose that $p$ is the 'north pole' ($p = (0,0,..,1)$), then ...
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2answers
34 views

find the equation of a sphere with endpoints A and B where B is the point of tangency of the sphere and the plane

Find the equation of a sphere with a diameter that has endpoints $A(1, 8, −2)$ and $B$, where $B$ is the point of tangency of the sphere with the plane $−9x +6y + 2z = 2$. Now i know that i can get ...
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58 views

Proof for spherical polar law of cosine

I'm reading my textbook and for some reason, it does not present the proof for the spherical polar law of cosine which is: $$ \cos(a)=\frac{\cos(A)+\cos(B)\cos(C)}{\sin(B) \sin(C)}$$ It does present ...
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1answer
23 views

The locus of points forming a right angle, in nonzero curvature

Given a line segment $AB$ in the Euclidean plane, the locus of points which form a right angle with $A$ and $B$ is known to be a circle, with $AB$ as a diameter. Is this also true for a geodesic ...
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1answer
32 views

Given angles and area, how to find sides of a spherical triangle?

So, given angles and area, how to find the sides of a spherical triangle? I only know that the angles uniquely determine the sides, but what is the relation?
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60 views

Nearest point on Spherical Cap

Let $A \subset \mathbb{S}^n$ be a spherical cap. More specifically, there exists a point $v \in \mathbb{S}^n$ and $\epsilon > 0$ such that $A = \{u \in \mathbb{S}^{n}\mid v\cdot u \geq \epsilon\}$. ...
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2answers
40 views

Solve trigonometric equations with sin and cos

I have two equations for two unknowns $u_k$ and $v_k$: $\tan{u_1} \tan{u_k} + \cos{(v_1 - v_k)} = 0$ $\tan{u_2} \tan{u_k} + \cos{(v_2 - v_k)} = 0$ where $u_1$, $v_1$, $u_2$ and $v_2$ are ...
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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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distance in n-dimensional space

According to answer of this question : Distance between 2 points in 3D space (in spherical polar coordinates) The distance between 2 points in 3 dimensional space is : $$ ...
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49 views

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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1answer
29 views

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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1answer
38 views

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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1answer
21 views

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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1answer
44 views

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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1answer
52 views

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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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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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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1answer
77 views

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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1answer
664 views

How to calculate the number of latitude/longitude coordinates in a specific area at a given precision?

I am looking into ways of grouping large sets of latitude/longitude coordinates and am wondering if there is an easy/standard way to roughly calculate the number of different coordinates in a given ...
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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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1answer
25 views

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 $S(B,r_1),\ S(C,r_2),\ S(A,r)$. ...