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

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62 views

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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77 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
56 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
35 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 ...
4
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1answer
55 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
25 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
46 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 ...
2
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63 views

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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2answers
73 views

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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61 views

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
65 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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1answer
63 views

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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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 $S(B,r_1),\ S(C,r_2),\ S(A,r)$. ...
2
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1answer
174 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 ...
2
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1answer
80 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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1answer
25 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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2answers
21 views

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 ...
2
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1answer
66 views

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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1answer
32 views

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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1answer
74 views

Finding angles in spherical triangle using law of cosines

Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices ...
3
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1answer
63 views

Spherical triangle vertices to spherical coordinates

Problem: Assume that the earth is a sphere of radius 5280 miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices A(70°N,10°E),B(10°S,100°E) ...
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3answers
46 views

Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
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1answer
454 views

Drawing ellipse as google.maps.Polygon with 8 points

In a web page using Google Maps JavaScript API v3 (including Geometry library) I currently draw an ellipse as a "diamond" with 4 corner points by the following JavaScript code: ...
2
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1answer
60 views

Finding the length of the sides, the measure of the angles and area of spherical triangles?

I'm trying to understand this problem in the textbook but I got lost in one part: Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the ...
3
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1answer
47 views

Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
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1answer
31 views

Lift of isometries of spherical space forms

If we have an isometry between two spherical space forms, then it is said that it lifts to an isometry of the sphere. Why is that?
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0answers
80 views

Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
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2answers
75 views

In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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0answers
80 views

Spherical coordinates and the Law of Cosines

I have one question on my project. I am assuming earth is a perfect sphere. How can I get from the Law of Cosines $$\cos(c)=\cos(\operatorname{lat} A)\cos(\operatorname{lat} ...
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1answer
156 views

How to cut a sphere in 3 parts of equal volume?

I ran across this problem when working on an architecture design project. I know this probably involves integral math but I'm not very familiar with it. Any help would be appreciated.
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1answer
69 views

Reflections On Sphere Surface / Getting Great Circle from two 3D points

I'm trying to calculate the reflection of a point across another point, both of which are on the surface of a sphere. I believe I could do this by getting the formula for the great circle of the ...
4
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3answers
307 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 ...
3
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0answers
65 views

How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere of radius $R$?

Question How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S $ of radius $R \quad (R \gg a)$? What I have thought so far Since the area of the ...
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1answer
71 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
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1answer
246 views

Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= ...
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2answers
90 views

Non congruent (spherical) quadrilateral with same angles

I want to construct two quadrilaterals on the unit sphere with same interior angles $\alpha_1,…,\alpha_4$ and the same perimeter, but which are not congruent to each other. Is that possible? How can ...
2
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1answer
122 views

Computing distance on a sphere

Let's say I want to compute the distance between two far points on Earth, say Toronto and Brazil. I can do this by getting in my car, setting my odometer to zero and then driving to Brazil. For me, ...
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1answer
70 views

Does there exist a spherical quadrilateral with all angles pi/2?

Does there exist a spherical quadrilateral with all angles pi/2? I do not think so but I am not sure. I am unable to really visualize this. Please offer suggestions. Thank you.
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1answer
87 views

Formula for Determinant of Vectors given in spherical coordinates

In 2D, one has an easy formula for the determinant of two vectors given in spherical coordinates, i.e. $\begin{vmatrix} \cos(\phi_1) &\cos(\phi_2)\\ \sin(\phi_1) &\sin(\phi_2)\end{vmatrix} ...
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1answer
35 views

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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0answers
43 views

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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2answers
269 views

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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0answers
60 views

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 ...
0
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1answer
71 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
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1answer
55 views

Line-of-Sight Angle on Sphere

I'm trying to calculate the angle (in degrees) between two latitude/longitude pairs, but with a twist. Most calculations I see use the Great Circle / bearing method, but this does not seem correct ...
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48 views

Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
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1answer
136 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
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1answer
106 views

Great circles on a unit sphere

Any hints as to how I can find the equation of all great circles passing through a given point (polar angle $\theta$, azimuthal angle $\phi$) on the surface of a unit sphere? Thanks.
3
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2answers
164 views

Volume of a sphere “corner”

I would like to find the formula of the volume of the "corner" of a sphere of radius R, more specifically the volume delimited in a sphere by the intersection of two perpendicular planes, one parallel ...
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1answer
40 views

Is there any special name for a $n$-torus made by products of hyperspheres?

I was wondering if there exist an accepted name for an $n$-torus made by the product of hyperspheres $\mathbb{S}^d$, that is for the following set: $$ ...