geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
Questions tagged [spherical-geometry]
889 questions
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Area of a special isosceles spherical triangle
I was asked to calculate the area of the following isosceles triangle.
Let $\triangle ABC$ be an isosceles spherical triangle with $d(A,C)=d(B,C)$. Then $C$ lies on the perpendicular bisector of $AB$. Let $M$ be the midpoint of $AB$. Suppose…
Polymorph
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Locations equidistant from three points on a sphere
A group friends is now split in three different cities on earth. They want to meet anywhere equidistant from all three cities. Where can that be ? The three cities happen to be Paris (France), Stuttgart and Munich (Germany). Where does that actually…
Charles
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how to calculate the 2D visible curve and area on spherical engulfment?
I am currently modelling the dynamic engulfment of a sphere. The sphere is fixed. Here is the model.
The ODE part is very simple. The data are associated to A_2d_psi0 (2D signal for A when there is no rotation, or psi=0). A_3d ODE is the…
pdp10
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Surface integral on a sphere using spherical coordinates
I have a question involving surface integral on a unit sphere. Suppose $s_1$ and $s_2$ are two points on a unit sphere with spherical coordinates $(\theta_1, \psi_1)$ and $(\theta_2, \psi_2)$, respectively. I want to compute
$$\int_{{\bf x}\in…
JACKY88
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Area of of spherical triangle with curved sides
The area of a spherical triangle (sphere radius $a$ ) is given by spherical deficit $ (A+B+C-\pi)$ times $ a^2 $ when enclosed between great circular arcs.
If tangential curvature radii are $ Rg_a, Rg_b, Rg_c$ what is the area of the spherical…
Narasimham
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Rotation to a spherical coordinate vector
To rotate a vector A in x-y plane through longitude $ \theta $ in the same x-y plane we multiply by $ e^{i \theta }$.
By what operation is A rotated through latitude $\phi$ out of the x-y plane?
Narasimham
- 40,495
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Existence of spherical triangles and their uniqueness upto rigid motion
On a 2-dimensional sphere of radius $\frac{1}{\sqrt{k}}$, call it $S_k$, where $k > 0$, we have the metric $d$ that is the great circle distance between any two points. How do I prove the following?
If $a + b + c < \frac{2 \pi}{\sqrt{k}}$, then…
Sayantan
- 3,418
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Area of Spherical Zone
"Let $\mathcal S$={$\mathbf x \in \mathbb{R}^3 : ||\mathbf x||=1$}
Prove that the area of the part of $\mathcal S$ that lies between the two parallel planes given, say, by $x_3=a$ and $x_3=b$, is the same as the area of the part of the…
Aka_aka_aka_ak
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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 the intersection is only a small part of this…
Peter
- 84,454
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How many spherical quadrangles exist with a given ordered sequence of inner angles.
Well, I think the title already explains my question. Given a sphere and an ordered sequence of inner angles ($\alpha$, $\beta$, $\gamma$, $\delta$) how many spherical quadrangles do there exist that have that sequence as angles and the added…
nvcleemp
- 297
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Line-circle intersection in spherical geometry?
How does one calculate the intersections between a "line" (a Great Circle) and a circle in spherical geometry?
i.e. given start point (as lat,lon), heading, circle centre (as lat, lon) and circle radius (as a % of the sphere's radius), there will be…
OJW
- 113
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Lines through a point on sphere intersecting at antipodal points
Consider a line through a point p on a sphere, now take another line passing through that point. It is found that the second line always intersects the first line in antipodal points. How do I write a mathematical proof that this is always the…
tryst with freedom
- 11,538
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Isometry on the sphere
We know that an isometry $A$ on the sphere is an involution if $A^2=I$. My question would be if the product of two involutions is an involution?
I think is not but I do not know how to prove it.
Mark S
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Finding latitude values that bound spherical segments of a desired surface area
Let's say we have a perfect sphere with surface area equal to 1. In the following diagram (not to scale), I need to calculate the latitudes of parallels A through G (based on the following specifications), but I'm having no luck.
A is the north…
mapp3r
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