Questions on coordinate systems, a geometric method where a point in n-dimensions is assigned a corresponding n-tuple of numbers.

learn more… | top users | synonyms

7
votes
0answers
426 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
5
votes
0answers
170 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
4
votes
0answers
41 views

Solving a system of equations

I'm trying to prove the existence of a solution to the system of equations $$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$ for $i\in\{1,2,....\}$ where $\sum c_i=1$. I am also ...
4
votes
0answers
186 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
3
votes
0answers
37 views

What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
3
votes
0answers
67 views

Why does a figure look the same in every coordinate system?

After reading Maximilian M. Answer here: Gauss' Theorem - Can't understand a parameterization I'm trying to figure out why does a figure look the same in every coordinate system I choose. For ...
3
votes
0answers
71 views

Changing coordinate system with non standard definitions

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$ The question is whether I ...
3
votes
0answers
47 views

Can all 2-surfaces be “coordinated” using 2 numbers

Consider a 2-dimensional surface embedded in 3 dimensional euclidean space. ex: A plane, a sphere, a hyperboloid of 2 sheets (or 1 sheet), the graph of sin(x + y), 2 parallel planes etc... If we ...
3
votes
0answers
73 views

How to solve a distance problem inside of a picture?

sorry for my bad english. I have the following problem: In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y). Now i want ...
3
votes
0answers
60 views

Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
3
votes
0answers
101 views

Translating coordinates on a Riemann surface

Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the ...
3
votes
0answers
250 views

Transforming triangular coordinate system from angle to another

I need to set a coordinate system fro a triangular grid so I did this: ...
3
votes
0answers
116 views

Local Coordinate Systems under Integral Extension

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be an integral extension of regular local rings of dimension $d$ (of course, $\varphi$ is a local homomorphism). Furthermore, assume that $A$ contains ...
2
votes
0answers
16 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
2
votes
0answers
31 views

Transition Functions for Cartesian Coordinate Systems

This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ...
2
votes
0answers
25 views

Ellipsoidal Coordinates Geometrically

Is there a quick, geometric, way of writing down (the square root of?) the Cartesian coordinates $$\begin{align} x^2 &= (a^2+\xi)(a^2+\eta)(a^2+\zeta)/(b^2-a^2)(c^2-a^2)\\ y^2 &= ...
2
votes
0answers
77 views

Get 2D coordinate transformation matrix based on points in a system and their angles in the other?

I'd like to get the parameters (rotation angle,$\Theta$, and translation coefficients, $x_0$ and $y_0$)) of a transformation for translating and rotating points in a coordinate system to another. As ...
2
votes
0answers
148 views

how to calculate the volume of irregular shape by the Cartesian coordinates of its corners?

I've 4 points in a plane "A" and another 4 in another plane "B". is there a way to automatically calculate the volume contained into this irregular box? The automation is important as this set of 8 ...
2
votes
0answers
2k views

Converting an equation from cartesian to cylindrical coordinates

This is going to seem pretty basic, but I'm trying to figure out if there is a problem in my homework's text or if it's just not clicking for me. I have to find the volume for the paraboloid $$z = 6 ...
2
votes
0answers
75 views

Evaluation of oscillating gaussian integral

I've problems to evaluate the following integral $$\int_{\mathbb{R}^3} dx \, dy \, dz \, \frac{e^{-i \Gamma |\vec{r}-\vec{r_0}|}}{|\vec{r}-\vec{r_0}|} \frac{e^{i \Upsilon |\vec{r}|}}{|\vec{r}|} ...
2
votes
0answers
159 views

Proper name for Inverted Cartesian coordinate system?

In most 2D computer graphic rendering applications (HTML Canvas, Flash, etc...), the coordinate system used is like this: My question is, what is the mathematical/technical name for this kind of ...
2
votes
0answers
219 views

Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$: $$ \int_{\Omega} \nabla u \nabla v d x = ...
2
votes
0answers
310 views

Lines in coordinate system of Hyperbolic Plane

An orthogonal coordinate system of the hyperbolic plain can be set up by fixing an orgio $O$, an $x$-axis, a $y$-axis (intersecting each other at $O$ in angle $90^\circ$), and, from any point $P$ ...
2
votes
0answers
269 views

Gradient when changing coordinate system

A change of variables from $\vec{r_1}$, $\vec{r_2}$ to $\vec{r}$, $\vec{R}$ is given by: $$ \vec{r} = \vec{r_1}-\vec{r_2}\text{ , }\vec{R}=c_1 \vec{r_1} + c_2 \vec{r_2} $$ I'm supposed to find ...
2
votes
0answers
177 views

How to calculate rotated global location coordinates (Long, Lat)?

Given the current global location coordinate system: -180 <-> 180 Longitude -90 <-> 90 Latitude A rotation of the globe 90 degrees counter-clockwise around the Y-axis would bring the north ...
1
vote
0answers
9 views

Add a rotation to latitude/longitude -> screen coords?

I found an algorithm which allows to convert latitude/longitude to (x, y) of a screen. The problem is a picture on a map is rotated. If it is not rotated then I use the following calculations: ...
1
vote
0answers
22 views

Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model ...
1
vote
0answers
10 views

Solid angle subtended in latitude-longitude maps

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles? Thank you very much
1
vote
0answers
76 views

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points?

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points? I don't quite the understand the solution which is in ...
1
vote
0answers
14 views

Radians : negative and positive values

Recently I have been reading books on DSP where I came across Polar co-ordinates. I understand that on Polar graph (4 quadrants) we have 0,pi/2,pi,3/2pi and 2pi radians as we move from one quadrant to ...
1
vote
0answers
26 views

Getting coordinate vector in linear algebra

I know how to get the coordinate vector of single matrices by just joining them and doing a gauss jordan. But these are a 2x2, I don't know how to go about this, apparently no elimination can take ...
1
vote
0answers
29 views

How do points change in a curved surface?

In the middle picture it shows a row of sticks at certain points along a flat surface. Now in the outer left picture (never-mind the outer right one), when the surface becomes curved the points ...
1
vote
0answers
15 views

Vocabulary of line coordinates

We can represent a line in 2 and 3 dimensions using 2 and 4 parameters respectively. For example, in 2 dimensions, we can represent a line using the angle $\theta$ of the normal and orthogonal offset ...
1
vote
0answers
29 views

Locus of point moving with circle

Consider the circle of radius $1$ unit with its centre at the point $(0,1)$. From the initial position, the circle is rolled along the positive $x$- axis without slipping. Find the locus of the point ...
1
vote
0answers
9 views

What does the phrase “uncoupled across coordinate directions” mean in this text?

The following paragraph is from a paper about comparison of maneuvering target tracking models.In the paragraph it talks about constant acceleration models. The above models are simple but crude. ...
1
vote
0answers
22 views

Parabolic Coordinates Radius

Given Cartesian $(x,y,z)$, Spherical $(r,\theta,\phi)$ and parabolic $(\varepsilon , \eta , \phi )$, where $$\varepsilon = r + z = r(1 + \cos(\theta)) \\\eta = r - z = r(1 - \cos( \theta ) ) \\ \phi ...
1
vote
0answers
39 views

Fit cartesian coordinate system to point cloud

I have a cloud of points that initially lie in a plane and have a coordinate system attached to them. I then displace the points slightly, and I want to find how a 'best fit' of the coordinate system ...
1
vote
0answers
30 views

Finding coordinates of nodes in a graph

I have a complete graph in which the edges represent the euclidean distance between the nodes which is known. Assuming a node to be (0,0), I want to find (approximately) the coordinates of other ...
1
vote
0answers
16 views

A method of calculation coordinates in order to implement it to a code language!

lets say that we have three points A(xa,ya,za), B(xb,yb,zv), C(xc,yc,zc) with known coordinates in 3d space. Is there a method to calculate the coordinates (x,y,z) of another point D for which the ...
1
vote
0answers
87 views

Change of basis matrix notation confusion

I've got strange notation of change of basis matrix in my book and I'd like to have it explained a little bit. It says, if: $M _{\mathcal A}^{\mathcal B}(id) \cdot \vec{v} _{\mathcal B} = ...
1
vote
0answers
176 views

Reverse rotation back to original coordinates (Euler Angles)

so in the program I'm trying to write (still, it's a mathematical question) I have a set of coordinates and angles (Euler angles) which represent the place and orientation of an object in space, ...
1
vote
0answers
111 views

Number of classes of K-sets

I am having a plane in N dimension. Th distance between 2 points (a1,a2,...,aN) and (b1,b2,...,bN) is max{|a1-b1|, |a2-b2|, ..., |aN-bN|}. I need to to know how many K-sets exist(here K-set refers to ...
1
vote
0answers
32 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
1
vote
0answers
30 views

equation of a refracted straight line

we have got a line $$x-y=1$$ which after refracting from $x$-axis bends at angle $\pi/6$ from normal what's eqation of that line? what i did was that putting $y=0$ to get points on $x$-axis which ...
1
vote
0answers
76 views

Indexing Goldberg (0,n) polyhedron faces

I would to know how to uniquely identify a face of a Goldberg (0,n) polyhedron: http://en.wikipedia.org/wiki/Goldberg_polyhedron#Icosahedral_G.280.2Cn.29_polyhedra It's possible to uniquely assign ...
1
vote
0answers
65 views

Change of variables in Fokker-Planck equation

I have the following Fokker-Planck equation: $$\frac{\partial\psi}{\partial t} + \nabla_{r_1} \cdot \left[ u(r_1,t)\psi + \frac{1}{\zeta} \textbf{F} (r_2 - r_1) \psi \right] + \nabla_{r_2} \cdot ...
1
vote
0answers
19 views

Differential system in cylindical coordinates

How do exactly I pass in cylindrical coordinates a system of this kind? $ \partial_t\phi=\partial_z(\beta\phi^2\sin\theta+\lambda u \sin\theta) $ ...
1
vote
0answers
47 views

Linejoin for fat lines?

I draw a figure with 2 fat lines. I need to draw a join between these lines correctly. Long red lines are in a middle of each fat line. What I know: coordinates of white points. the angle between ...
1
vote
0answers
73 views

Parameterizing a spherical cap in cylindrical coordinates

In calculating the mean curvature for a surface of the form $(f(z)\cos(\theta), f(z)\sin(\theta), z)$ I need to use some checks to ensure I haven't made any mistakes along the way. The one I would ...
1
vote
0answers
42 views

What's the mathematical formulation for frames for reference?

In this lecture I've come across an interesting formulation of kinematics and reference frames. When I studied kinematics, the idea of what is meant by a 'frame of reference' was never quite fully ...