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

0
votes
0answers
46 views

Constructing coordinate maps on manifolds

I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on ...
1
vote
3answers
64 views

How to “rotate” points through 90 degree?

I am trying to do some intersection tests and so the math gets weird if two certain points have the same $x$ coordinate and so infinite slope. The points can be anywhere in any quadrant. I want to ...
0
votes
2answers
47 views

calculate the volume

There is a triangular prism with infinite height. It has three edges parallel to z-axis, each passing through points $(0, 0, 0)$, $(3, 0, 0)$ and $(2, 1, 0)$ respectively. Calculate the volume within ...
1
vote
0answers
19 views

Converting vector in cartesian to cylindrical coordinates

This seems like a trivial question, and I'm just not sure if I'm doing it right. I have vector in cartesian coordinate system: a⃗ =xi⃗ −2xj⃗ + yk⃗ And I need to represent it in cylindrical coord. ...
0
votes
0answers
26 views

Looking for a formula to map a 2d pixel coordinate to a region within a grid.

I am given a pixel bounding box of the form: (x1, y1), (x2, y2) Where (x1, y1) is the bottom left coordinate and (x2, y2) is the top right coordinate of the ...
0
votes
2answers
35 views

How do I calculate the angles between a point on a sphere and each unit vector in $\Bbb R ^3$?

Given the Cartesian coordinates of any point $p$ on the surface of a sphere in $\Bbb R ^3$, how do I calculate the angles between each axis $(x, y, z)$ and the vector $n$ defined by origin $o$ and ...
3
votes
1answer
64 views

Coordinate systems on manifolds

I am fairly new to differential geometry and something I can't get my head around is, if an $n$-dimensional manifold is locally homeomorphic to $\mathbb{R}^{n}$, i.e. Euclidean space, then isn't it ...
4
votes
2answers
108 views

Geometry: What is being calculated here?

Context: I am a computer graphics programmer looking at a code-implementation. I need help understanding a function that has neither been documented properly nor commented. Given a circle with ...
2
votes
0answers
15 views

Bipolar toroidal coordinates - position vector, velocity and acceleration

Bipolar toroidal coordinates: $x = a \frac{\sinh\tau \cos\phi}{\cosh\tau-\cos\sigma}$ $y = a \frac{\sinh\tau \sin\phi}{\cosh\tau-\cos\sigma}$ $z=a \frac{\sin\sigma}{\cosh\tau-\cos\sigma}$ Would ...
1
vote
0answers
23 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a 2-d figure has maximum distance from a fixed coordinate. Let me demonstrate: 3 points: (1,3), (2,2), (5,0) ; Fixed point: ...
0
votes
2answers
19 views

Getting the coordinates of a point on section [duplicate]

What is the endpoint $P(x,y)$ of a line segment , if I know its starting point: $C(x(1), y(1))$, the gradient $G$, and its length $L$?
2
votes
4answers
110 views

Notation of a function in coordinates

I have a question about some notation which puzzles me a lot. Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$. Then people often write or say that if we choose coordinates $x=(x^1,…,x^n)$ ...
4
votes
1answer
47 views

Coordinates on the sphere not global?

I'm reading a book on differential geometry and some part of the introduction I do not understand but I'm curious to understand it. Maybe someone can try to explain those parts to me. "Each point on ...
2
votes
2answers
22 views

unique number from 3D coordinates

Is it possible to get a unique real number from $(x,y,z)$ coordinates? I need to sort a list of coordinates so i am looking for a simple function that generates one unique number with which to sort. ...
1
vote
0answers
19 views

How can we express $(\hat{r}, \hat{\phi})$ in respect of $(\hat{i}, \hat{j})$ ? [closed]

We have the following: How can we express $(\hat{r}, \hat{\phi})$ in respect of $(\hat{i}, \hat{j})$ ??
0
votes
1answer
37 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
0
votes
0answers
20 views

How to find the location of a point in a global coordinate system from a local coordinate system

I was wondering if you would be able to help guide me on a solution involving rotation matrices. In-terms of data, I have the global coordinate system a $3\times 3$ matrix, the local coordinate ...
3
votes
1answer
55 views

Is there an equation to describe translation and rotation?

Suppose a free rod, $l=2r$, is hit on a tip T and translates with $v= 1r/s$ and at the same time rotates with angular velocity $\omega= 1rad/s$. Is there an equation that can determine the position of ...
3
votes
3answers
103 views

Why do we assume the complex plane is curvey at infinity?

In at least a few areas (i.e., those that I have happened across) when we have a need to capture a half-plane we do so by taking a semi-circle of radius $r$ in that half, and taking the limit as ...
1
vote
1answer
31 views

rectangular coordinate system vs Cartesian coordinate system?

Is there any difference between a rectangular coordinate system and a Cartesian coordinate system? Is one of them a subtype of the other? My book mentions a rectangular Cartesian coordinate system. ...
2
votes
0answers
30 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
0
votes
0answers
26 views

Euler Angle Transformation from righthanded to lefthanded cartesian coordinate system

I have a righthanded and a lefthanded cartesian coordinate system defined as follows: I have Euler angles (x, y, z) defined in the righthanded system and want to transform them to the lefthanded ...
0
votes
2answers
54 views

Fix the radius when drawing a circle.

I found this function and it draws an oval rather than a circle. What do I need to do to fix the calculations to make a circle? Thanks. ...
1
vote
0answers
57 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
6
votes
1answer
1k views

What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
9
votes
4answers
550 views

Check if a point is inside a rectangle (not knowing the coordinates, but knowing distances to vertices)

I have to solve the following problem: I have 4 points (A, B, C, D) which form a rectangle, but I do not know their coordinates. I have another point (X), I do not know its coordinates either, but I ...
3
votes
2answers
30 views

Find the number of possible points $R$.

$P(3,1),Q(6,5)$ and $R(x,y)$ are three points such that the angle $\angle PRQ=90^{\circ}$ and the area of the triangle $\triangle PRQ=7$.The number of such points $R$ that are possible is . $a.)\ ...
5
votes
1answer
64 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
2
votes
2answers
40 views

Find the equation of line in new co-ordinate system.

A line is represented by equation $4x+5y=6$ in the co-ordinate system with the origin $(0,0)$.You are required to find the equation of the straight line perpendicular to this line that passes ...
0
votes
0answers
19 views

Find the length of the intercept cut by the side $BC$ on the y-axis .

The equation of two equal sides $AB$ and $AC$ of isosceles triangle $\triangle ABC$ are $x+y=5$ and $7x-y=3$ respectively.What will be the length of the intercept cut by side $BC$ on the y-axis? ...
1
vote
2answers
32 views

Find the area of $\triangle POQ$ .

If $P$ and $Q$ are two points on the line $3x+4y=-15$ such that $OP=OQ=9$, then the area of $\triangle POQ$ will be ? $\color{green}{a.)18\sqrt2}\\ b.)3\sqrt2\\ c.)6\sqrt2\\ d.)15\sqrt2$ The ...
4
votes
3answers
28 views

Find the relationship between $a$ and $b$.

If the medians $PT$ and $RS$ of a triangle with vertices $P(0,b),Q(0,0)\ \text{and}\ R(a,0)$ are perpendicular to each other,which of the following satisfies the relationship between $a$ and $b$? ...
2
votes
1answer
9 views

Compound map in manifolds

In the description of a manifold, we often start with the mathematical definition that $M=\cup M_i$ and if $m\in M_i \subset M$, where m is a point on the manifold, then it is mapped by a one-to-one ...
4
votes
4answers
167 views

What will be the equation of side $BC$.

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively . What will be the equation of the side $BC$ if the area of the triangle ...
0
votes
2answers
21 views

Which of the following point is outside the triangle?

If $P(6,7),Q(2,3)\ \text{and}\ R(4,-2)$ be the vertices of the triangle , then which of the point is not contained in the triangle? $a.)(4,3)\quad \quad \quad \quad b.)(3,3)\\ c.)(4,2)\quad ...
0
votes
0answers
18 views

What's the relation between 2 points from 2 different planes?

I'm trying to find the relation between my "text" objects, and my "world" objects. This may be related to development, but I thought this question was better fit for this exchange. I have two ...
0
votes
0answers
14 views

Definition of curvilinear coordinates?

Please can someone give me a formal definition of curvilinear coordinates, preferably with as source. The once that I have found don't seem to be very formal.
2
votes
1answer
35 views

Conversion from 2-dimensional parabolic coordinates to cartesian and cylindrical

I have been looking at the Wolfram Mathworld page on parabolic coordinates here: http://mathworld.wolfram.com/ParabolicCoordinates.html and I'm having trouble grasping how to convert between parabolic ...
1
vote
0answers
38 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
0
votes
1answer
13 views

Scalar functions and manifolds

This paragraph is taken from Supergravity book by Freedman and Van Proeyen.he simplest objects to define on a manifold $M$ are scalar functions $f$ that map $M \rightarrow \mathbb{R}$. We say that ...
1
vote
0answers
19 views

Change of Basis Matrix: Cartesian to Spherical Laplacian

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of ...
1
vote
1answer
168 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
0
votes
0answers
38 views

Transforming 2d coordinate system while keeping points placement

I don't know if my question is too simple. Hopefully I can make it clear. I want to find a matrix that I could multiply point with in one coordinate system into another. The first coordinate system ...
1
vote
0answers
29 views

Finding bounds of integration

Let $S$ be the region in the first quadrant of the $xy$-plane bounded by the $x$-axis and the parabolas $$x=1-\dfrac{1}{4}y^2,$$ $$x=\dfrac{1}{4}y^2-1$$ and $$x=4-\dfrac{1}{16}y^2.$$ Use the ...
0
votes
1answer
12 views

Change of Coordinates matrix.

If Q is the change of coordinates matrix From some basis B to B', then Q inverse is the change of coordinates matrix from B' to B? Is this true? I think/ know it is the, but don't know how to prove ...
0
votes
0answers
19 views

Build an orthogonal coordinate transformation?

I am supposed to build an orthogonal coordinate transformation $Q:G\to \Omega$ whose inverse is not orthogonal $Q^{-1}:\Omega\to G$. I am allowed to make the coordinate transformation as simple as ...
0
votes
2answers
61 views

find which two points an arbitrary point is nearest to

I have a line segment of connected points (a path in 2D), and a point $P$ that is not calculated based on this segment, although I can guarantee that the point will be placed along the path. Based on ...
0
votes
0answers
13 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
1
vote
0answers
28 views

Realistic Bounce (Using Trig?)

background: I am making a graphics program where the major purpose of it is to have a ball (traveling on an arbitrary slope) to bounce realistically off of a line (which is also at a arbitrary slope). ...
0
votes
0answers
10 views

How can I find the domain of this diffeomorphism (coordinate transformation)?

I have been struggling with this coordinate transformation in $R^2$. $Q:\begin{bmatrix}\rho\\\phi\end{bmatrix}\to \begin{bmatrix}\cosh(\rho)cos(\phi)\\sinh(\rho)sin(\phi)\end{bmatrix}$ I am ...