# What is Spherical geometry: Definition and 20 Discussions

Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sphere" are used for the surface together with its 3-dimensional interior.
Long studied for its practical applications to navigation and astronomy, spherical geometry bears many similarities and relationships to, and important differences from, Euclidean plane geometry. The sphere has for the most part been studied as a part of 3-dimensional Euclidean geometry (often called solid geometry), the surface thought of as placed inside an ambient 3-d space. It can also be analyzed by "intrinsic" methods that only involve the surface itself, and do not refer to, or even assume the existence of, any surrounding space outside or inside the sphere.
Because a sphere and a plane differ geometrically, (intrinsic) spherical geometry has some features of a non-Euclidean geometry and is sometimes described as being one. However, spherical geometry was not considered a full-fledged non-Euclidean geometry sufficient to resolve the ancient problem of whether the parallel postulate is a logical consequence of the rest of Euclid's axioms of plane geometry. The solution was found instead in hyperbolic geometry.

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1. ### B Method of images and spherical coordinates

I am finding the potential everywhere in space due to a point charge a distance 'a' on the z-axis above an infinite xy-plane held at zero potential. This problem is fairly straight forward; place an image charge q' = -q at position -a on the z-axis. I have the solution in cartesian coordinates...
2. ### I Triangles on Spheres: Isosceles and Shortest Distance Inferences

If I have a triangle on a sphere with two of its angles 90 degrees each, do I conclude that it's isosceles and that the shortest distance (on the sphere) beteeen the base and the vertix of the thid angle is 1/4 the circumference of a great circle on the sphere? This is the picture I have in...
3. ### I What does the equation dΩ32 = dr2 + r2dΩ22 mean in terms of flat space?

A 1-sphere exists in 2D space. It is a circle in flat space. A 2-sphere is a 1-sphere embedded in 3D space. Its surface is non flat and 2 dimensional. A 3-sphere is a 2-sphere embedded in 4D space. Its surface is non flat and 3 dimensional. What does this last sentence mean in...
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### I On spherical geometry and its applications in physics

Hello. Questions: do you know any applications of spherical geometry in physics? Are there any relations between spherical geometry and hyperbolic geometry? Why does Riemannian geometry use sphere theorems so much? Thank you.
5. ### I Deriving the area of a spherical triangle from the metric

The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.
6. ### Coriolis Drift of Discrete Objects

I am writing a program aimed at 'gun nuts' designed to display Coriolis Drift of bullets (ie, 'discrete objects in free-fall', and not large fluid masses). Using the 2 equations below, I am able to calculate and display the values of Coriolis Drift (in terms of X & Y (vertical and horizontal))...
7. ### B Spherical Geometry (Two dimension ) Defining a metric

I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says, On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...
8. ### I Spherical Geometry Equilateral Pentagon

Hey PF! I'm going through a textbook right now and it just said "obviously, you can't have an equilateral pentagon with 4 right angles in spherical geometry (Lambert quadrilaterals). However, I am not able to make the connection. can somebody help me understand why this is?
9. ### Topology Find the Best Spherical Geometry Book for You!

I am in need of a spherical geometry book.can some one suggest a good one?
10. ### Geometry Spherical Geometry: Astronomy Books for Study

I am studying spherical astronomy can some suggest good books on spherical geometry.
11. ### Area of triangle on sphere problem.

Homework Statement What is the area of a triangle on Earth that goes from the North Pole down to the equator, through the prime meridian, across the equator to 30 degrees east longitude, then back up to the equator? The radius of the Earth is about 6378 km. Homework Equations alpha + beta +...
12. ### Particle density in spherical geometry

Homework Statement Neutrons are emitted uniformly from the inner surface of a thin spherical shell of radius R at a velocity V. They are emitted normal to the inner surface and fly radially across the volume of the sphere to be absorbed at diametrically opposed points. The neutrons are non...
13. ### How to find an angle in spherical geometry.

Hi I am not familiar with spherical geometry. I am working with elliptical polarization that involves using poincare sphere that present the latitude and longitude angle in spherical geometry. I need to find the great circle angle if given two points that each specified by their longitude angle...
14. ### Area of square in spherical geometry

Homework Statement Please see the attached. It is a badly drawn sphere :-p By common sense,the area of the shaded region in the sphere = area of square = r^2 But can anyone show me the mathematical proof? Moreover,does it apply to the reality? Imagine when you bend a square sheet with...
15. ### What is the difference between superposition and intersection?

The spherical geometry thought that in the surface does not have the parallel line, actually this idea is wrong, should say: In the spherical surface does not have the straight line parallel line, but in the spherical surface has the curve parallel line. Is very actually good about the curve...
16. ### Playing with Coordinate Systems (Spherical Geometry)

I'm working on a problem that involves two Earth stations that scan the skies. I'm writing a simulation program (no physics involved) that simply finds the az/alt of an event observed simultaneously by each station. At this point, I'm warming up to the mathematics, spherical geo, etc. to pull...
17. ### Solving Spherical Geometry Homework: Area, Angles, Sides, Dual Triangle

Homework Statement The area of an equilateral triable is Pi/2. 1) Find the magnitude of its angles 2) Find the length of its sides 3) Find the area of its strict dual Homework Equations Area + Pi = sum of 3 angles cosa=cosbcosc + sinbsinccosu cosu=cosvcosw + sinvsinwcosa for...
18. ### Why does a universe with a spherical geometry have to be finite?

I have read that "A universe with a spherical geometry is called a closed universe because a universe with this geometry must be finite" but even after looking up different sources i cannot find an decent explanation of it is finite. I know that a flat universe is just an unbound 3d grid that...
19. ### What is the Volume to Surface Area Ratio of a Sphere?

hi all could anyone tell me the ratio of the volume of a shere to the surface area of the same sphere, i can't seem to sort this one out in my head, thx
20. ### Spherical geometry, some simple things

So, I'm working my way through "Geometry from a Differentiable viewpoint" (or, trying to get through section 1.1, anyways). right now, it's spherical geometry. so far, a great circle has been defined as the set of points on the sphere that intersect with a plane that intersects the origin of...