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

# Questions tagged [spherical-geometry]

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### 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 the distance of the point to the plane by…

madison

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### 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 $A = \{u \in \mathbb{S}^{n}\mid v\cdot u \geq \epsilon\}$.
Given some point $w \in \mathbb{S}^n$, I would like…

Cain

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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 touch the horizon from where I'm standing.
For…

Tom Collins

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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 started at. Would I need to turn my steering wheel…

Zed

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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 as the cosine of the angle from the…

Dr. P

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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 $S(B,r_1),\ S(C,r_2),\ S(A,r)$. Here by choosing
$r$ suitably we have two…

HK Lee

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### Spherical distance

The spherical distance between two points $P_1=(0,0,1)$ and $P_2=(\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},{-}\frac{\sqrt{3}}{2})$ is $\frac{5\pi }{6}$.
I am at a loss as to how the spherical distance was obtained. My notes just give the answer but…

Nessa

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### How to find the intersection between the great circle and a hyperplane?

Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and $u$ (in the direction from $s$ to $u$) until the…

mkolar

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### 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:
var NORTH = 0;
var WEST = -90;
var SOUTH = 180;
var EAST =…

Alexander Farber

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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

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### 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|}}] & \text{if}\ m<0\\
\displaystyle Y_\ell^0 & \text{if}\…

trbabb

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### The globe, spherical disks and spherical straight lines

Does a spherical triangle with 2 equal sides necessarily have 2 base angles of size $\pi/2$? The reason I think this is that if we have a triangle $ABC$ and $AB=AC$ (in spherical distance), we could view it as $A$ being at the center of a spherical…

sphere

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### 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.

SamHaim

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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

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 this if the radius in the formula of the sphere…

BEBS

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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

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