geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
11
votes
1answer
441 views
What's the name of a parabola mapped onto a sphere?
It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle ...
11
votes
1answer
120 views
Relation between area of a triangle on a sphere and plane
We know area of a plane triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\frac{a+b+c}{2}$.
I was just thinking: let we have a triangle with arc length $a,b,c$ on a sphere of radius $r$, do we have ...
8
votes
1answer
279 views
Navigating though the surface of a hypersphere in a computer game
People in StackOverflow seems not so into this theme, so I thought I could have better luck in here.
I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
8
votes
1answer
149 views
Circle on sphere
Foreword
This question was inspired by initial mistakes in this question. I wanted to explore the strange circle with $A>\pi r^2$ and got lost into geometrical jungle.
A spherical cap is usually ...
7
votes
1answer
331 views
Spherical geometry: Arbitrary point between two points
If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path?
Points are in radians
longitude = 0 to 2pi
...
6
votes
2answers
636 views
Is an equilateral triangle the same as an equiangular triangle, in any geometry?
I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry?
I know they are the same in ...
6
votes
1answer
608 views
How to calculate a heading on the earths surface?
Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
6
votes
3answers
520 views
What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?
So the problem that I'm trying to solve is as follows:
Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
5
votes
2answers
1k views
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 ...
5
votes
1answer
217 views
Elementary arguments concerning the stereographic projection
How does one give a proof that is
short; and
strictly within the bounds of secondary-school geometry
that the stereographic projection
is conformal; and
maps circles to circles?
5
votes
2answers
2k views
Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry
I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
5
votes
2answers
603 views
How do I map a spherical triangle to a plane triangle?
My goal here is to make my own custom "polyhedral map" of Earth. If you print out something from the "Map Fold-outs" page, you will have something almost exactly like what I'm trying to make.
I have ...
4
votes
4answers
757 views
How to find the distance between a point and line joining two points on a sphere?
How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below.
In the above ...
4
votes
3answers
346 views
Article or book about the history of spherical geometry?
I teach a course on non-Euclidean geometry to high schoolers. I'm looking for an article or book that gives a thorough and interesting history of spherical geometry and trigonometry. I'm looking for ...
4
votes
1answer
119 views
Why do derivatives of certain equations relating to circles yield other similar equations? [duplicate]
Possible Duplicate:
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
We all know that the volume of a sphere is:
$V = \frac{4}{3}\pi r^{3}$
and its ...
4
votes
1answer
478 views
Deriving the Surface Area of a Spherical Triangle
A triangle on a sphere is composed of points $A$, $B$ and $C$.
The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle:
The Girard's theorem states that the ...
4
votes
1answer
123 views
spherical triangle inequality for multiple edges
Consider a spherical triangle, with side lengths $0 \leq a,b,c \leq \pi$. In texts on spherical geometry we see proofs that $a \leq b + c$ but this inequality is unnecessarily weak, in that some sets ...
4
votes
1answer
351 views
approximating geodesic distances on the sphere by euclidean distances of a transformed sphere
Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing
$$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$
, where $d$ stands for ...
3
votes
5answers
329 views
How do you parameterize a sphere so that there are “6 faces”?
I'm trying to parameterize a sphere so it has 6 faces of equal area, like this:
But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice").
I ...
3
votes
3answers
2k views
Area of a spherical triangle
A triangle on a sphere is comprised of points A, B and C. How to determine its area?
I know the formula:
A = E * R^2, where R is radius of sphere, and E is the excess angle of (a + b + c - pi), but ...
3
votes
2answers
88 views
How do I apply a digital filter to points on a sphere
Given a set of points on a sphere, how can I implement a higher order low pass filter on them?
At the moment, I am just multiplying the vectors from the input and output set by their weights and ...
3
votes
2answers
2k views
Napier's Rules applied to spherical distance calculations
I was in the middle of writing the same old geographic distance calculation using the Haversine formula when it occurred to me: shouldn't there be simpler way to do this? Haversine is of course ...
3
votes
2answers
40 views
Given a latitude how many miles is the corresponding longitude?
OK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.)
So given a ...
3
votes
2answers
133 views
Volume of a sphere using geometry
How to derive the formula for the volume of a sphere using geometry?
$V = (4/3) \pi r^3$
Edit: How did Archimedes calculate the volume of a sphere? Integration wouldn't have been there at his ...
3
votes
2answers
202 views
What is the average rotation angle needed to change the color of a sphere?
A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at ...
3
votes
1answer
93 views
A property of Hilbert sphere
Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than ...
3
votes
1answer
36 views
problem or doubt regarding visualizing angles of spherical triangle
I must confess that I am not able to visualize or understand what is the angle of a spherical triangle say $ABC$ where $A,B,C$ are vertices of the triangle which is formed by intersection of three ...
3
votes
1answer
134 views
Geodesics on a 2-sphere
I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
3
votes
1answer
84 views
Integration on a sphere
I have an integral at hand which has the form of
$$I = \int_{u\in \mathbb{S}^2} f(\mathbf{u}\cdot \mathbf{s}_1) f(\mathbf{u}\cdot \mathbf{s}_2) d\mathbf{u}$$
where $\mathbb{S}^2$ is the unit sphere ...
3
votes
1answer
294 views
How to construct the midpoint in spherical geometry?
I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler.
...
3
votes
1answer
108 views
Formula for the fourth side of a spherical quadrilateral
Given two sides $a,b$ of a spherical triangle and the angle $C$ between them, the spherical law of cosines gives an elegant formula for the missing edge length $c$:
$$\cos c = \cos a \cos b+ \sin a ...
3
votes
0answers
39 views
Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$
I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question.
In $\mathbb{E}^2$, ...
3
votes
2answers
384 views
Lat/Long grid points covered by projecting rectangle onto sphere
Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view.
Suppose we have a ...
2
votes
1answer
88 views
Real numbers mapped onto a sphere
We can compare real values if they were greater, lesser but we cannot do same for complex numbers.
What if we map real values(within some small range) onto a sphere and declare each one of them as ...
2
votes
1answer
166 views
Finding the area of a spherical triangle
I am asked to calculate the area of a spherical triangle of points $(0,0,1),(\frac{1}{\sqrt2},0, \frac{1}{\sqrt2})$ and
$(0,1,0)$.
I know I will have to use Gauss Bonnet formula , after having found ...
2
votes
2answers
218 views
How to calculate the area of a circle ( given: origin, radius ) on a sphere ( Earth )?
I know that the Earth isn't a sphere, not even an ellipsoid, but for my measurements, its an acceptable approximation.
Assuming I have a coordinate(lat,lon) and a distance( e.g.: 1000km ), what is the ...
2
votes
4answers
404 views
Latitude and longitude of points on a line
How could you get the latitude and longitude of four points (equal distance apart) on a line from $(27,-82)$ to $(28,-81)$? The four points should split the line into 5 parts.
2
votes
2answers
260 views
Coordinates and distance in higher dimensional spherical and hyperbolic space
For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
2
votes
3answers
152 views
Spherical projection
An image of a square is projected onto a sphere (radius $R$) as above (the dot is the centre of the sphere, and the red projection is marked out where the line from the centre-dot to a point on the ...
2
votes
1answer
287 views
Arc length of a great circle which is the hypotenuse of an isoceles right triangle on the sphere
I am doing a problem which requires me to find the arclength of the hypotenuse of an isosceles right triangle. (The book calls it a 2 Dimensional Sphere but I hope that is a typo)
I start at the ...
2
votes
1answer
71 views
A different way of calculating the surface area of the sphere
I just don't understand this. I know $A_{sphere} = 4\pi r^2$ and the circumference $C_{sphere} = 2\pi r$, so why can't I just sum up (integrate) all the circumferences to get the area? That is, why ...
2
votes
2answers
216 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) ...
2
votes
1answer
890 views
Angle between GPS coordinates
I realize GPS Coordinates are spherical coordinates. However I know the earth is more of an ellipsoid. I need to compute with a fairly high degree of accuracy the pitch and yaw between two objects ...
2
votes
1answer
51 views
Spherical geometry - relating angles of lunes and segments of great circles
Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated ...
2
votes
2answers
51 views
What's wrong with an irregular digon?
I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a ...
2
votes
1answer
103 views
Ratio of geodesic segments on the sphere
Let $\mathbb{S}^2$ be the unit sphere. Let $0<\lambda<1$ be fixed. What is the smallest number $0<\mu<1$ (depending on $\lambda$) such that for any three points $A,B,C\in \mathbb{S}^2$, ...
2
votes
1answer
282 views
quaternion representation of the rotation of a sphere into plane displacement
I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZs. The sphere can be rotated and displaced ...
2
votes
2answers
139 views
Simplest form for locus of latitudes/longitudes equidistant from two given latitudes/longitudes?
Given two latitudes/longitudes (th1,ph1 and th2,ph2), I want to find a
simple formula for the locus of th3,ph3 that are equidistant from
th1,ph1 and th2,ph2.
Mathematica happily spits out an answer ...
2
votes
1answer
36 views
Geometry Question - Trihedral angles, planar geometry, spherical geometry
Two rays, $OX$ and $OY$, are drawn in the horizontal plane $\pi$, and the third ray, $OZ$, is drawn in space so that the rays $OX$, $OY$, and $OZ$ form a trihedral angle $OXYZ$. The planar angles ...
2
votes
1answer
185 views
how to calculate distance from a given latitude and longitude on the earth to a specific geostationary satellite
As the title suggests, I would like to know how to calculate the straight-line distance from a given latitude+longitude point on the earth to a given satellite in the geostationary belt. Perhaps a ...
