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

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How to make the standard change-of-variables in the plane-parallel radiative transfer equation?

This is a basic technique used frequently in going from the general coordinate-free radiative transfer equation (RTE) to the RTE formulated for the plane-parallel atmosphere geometry (see Liou, ...
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2answers
30 views

Optimization of three right-angled vectors

In my case, I have three given vectors $\vec{a}, \vec{b}, \vec{c}$ with $$\vec{a}= \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} $$ and these vectors span a ...
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24 views

Is the Divergence of Curl equal to Zero for All Coordinate Systems?

Is the divergence of curl equal to zero for all coordinate systems? Even a curvilinear coordinate system such as double spheroidal coordinates?
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8 views

Third order partial derivatives in cylindrical coordinates

Do you know, where I can find formulas for third order partial derivatives in cylindrical coordinates? All I can find are second order partial derivatives. Thanks!
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2answers
24 views

The expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$.

Using vectors I tried obtain the expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$. The point of intersection is $$X=\frac{bc'-cb'}{ab'-...
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1answer
24 views

complex number multiplication by a real number [closed]

I'd like to multiply a complex value by a real integer. I know that multiplication of complex numbers is similar in the polar form, but the way I know and have been taught is to multiply the two real ...
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1answer
22 views

Cartesian to spherical coordinate :MATLAB program

I am a beginner to MATLAB. I have written this function, but don't understand what is wrong. I have used a if statement to correct the phi. Say if i use (x,y,z) = (0,-4,3) i should get (5,270,53.13) ...
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1answer
28 views

Recognising patterns and turning it into a formula

On a coordinate plane lets name a move dot A. The dot A moves each day. On day 1, it moves 1 in the x- axis direction. On day 2 it moves $2^2$ in the y- axis direction. On day 3 it moves -$3^2$ in the ...
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2answers
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Coordinates and formulas

Say there is a moving dot on the coordinate plane. It starts on the coordinates of (0,0). On the 1st day it moves to (1,0) the next, (1,4) then (-8,4) then, (-8,-12), then, (17,-12) and so on. Now I ...
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1answer
25 views

Explicit non-singular coordinate system for $S^3$

Define a "non-singular" coordinate system on a manifold as a continuous, everywhere differentiable set of coordinates such that the determinant of the metric tensor $g_{\mu\nu}$ is everywhere ...
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21 views

How to calculate coordinates $(x,y)$ of rotated polygon?

I have coordinates $x,y$ of a point (red point on the image). If I rotate the image with a specific angle (for example 30 degrees) how can I get the coordinates $x,y$ in the new polygon (which is ...
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2answers
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Interesting Locus problem

A variable line passes through $P(2,-1)$ and cuts the co-ordinate axes at $A$ and $B$ respectively. $Q$ lies on line AB such that $$\frac{2}{PQ} = \frac{1}{PA} + \frac{1}{PB}$$ Find the locus of ...
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2answers
37 views

Diffeomorphism group of product manifold

For a given differentiable manifold $M$, the diffeomorphism group $\mathrm{Diff}\left( M \right)$ of $M$ is the group of all $C^\infty$ diffeomorphisms of $M$ to itself. Consider a product manifold of ...
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1answer
41 views

Is there a solution to the equation $tan({\phi})=\frac{0}{0}$

I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as: $\tan({\phi})=\frac{y}{x}$ My ...
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1answer
16 views

Position of point between 2 points in 3D space

I need to find the position v3 between the given points v1, and v2 and a given distance d in 3D space. I came across this post: Position of point between 2 points which is basically what I need but ...
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1answer
26 views

Informations about the cut-locus of a closed geodesic

Let be $(M^2,g)$ a closed riemannian manifold and $c:[0,L]\to M$ a simple closed geodesic on $M$. For each $s\in [0,L]$, let be $n(s)$ a unit normal vector field along to $c(s)$ and $\beta(s)$ the cut ...
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0answers
43 views

Is it possible to calculate a surface integral of a vector field when the vector field is described in non-cartesian coordinates?

Every note and book I read about surface integrals of vector fields only show how to solve these integrals when the vector field is in Cartesian coordinates. I'm curious about what would be the right ...
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1answer
77 views

A parallelogram between two points on a hexagonal lattice containing all the shortest paths

For any two points on a hexagonal grid with integer coordinates there is a unique parallelogram which contains all of the shortest paths (in terms of taxicab norm) between these points. See the ...
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1answer
22 views

Understanding Coordinate transformation under tensor calculus

I am reading the book "Tensor Calculus" by Schaums Outlines and I came across this paragraph. Suppose that in some region of $\mathbb{R^n}$ two coordinate systems are defined and these two systems ...
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1answer
42 views

Differential forms should be invariant under coordinate transformations

I am wondering why, if we transform the following differential form, it does not seem to be invariant under the coordinate transformation. The $1$-form on $\mathbb R^2$ is $$ \omega = \sqrt{x^2 + y^2}...
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1answer
28 views

2D coordinates of rotating a “bent line”?

I have this problem, when I am given a point A an an XY plane, and I need to find the coordinates of a point B that is of a constant distance of my point A, and my OAB angle is fixed (O being the ...
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0answers
39 views

I'm looking for a rotation matrix for following transformation

I'm working with a 3D camera and I found out the formula to transform the camera measurements to real world coordinate system when you have a rotation around x and y (no z rotation). http://i.imgur....
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2answers
34 views

Prove that an Equilateral cannot have natural number points

Let $ OAB $ be an equilateral triangle with $O(0, 0),\ A(m, n),\ B(x, y)$, where $m, n \in \mathbb{N}^{\ast}$ and $x, y \in \mathbb{R}_{+}$. Prove that $B$'s coordinates can't be both natural numbers....
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1answer
21 views

find the coordinates of the point that divides the join of A(-1,-7) & B(1,2) internally, in 2:1.

What I wanted to ask was that after finding the coordinates of the point my answer was (1/3, -1) now since the ordinate is -ve doesn't that make this an external division? How can it divide the line ...
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1answer
49 views

How to convert 2D coordinates to 3D coordinates?

I am writing some software for image processing where a user can just draw something (e.g. a cube) in paint and the software will give you the 3d coordinates of the corners on that drawing. What would ...
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2answers
35 views

How do I convert an index for a one-dimensional array into x and y?

Given a flat array of values as such, representing a Sudoku board: ...
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1answer
56 views

Christoffel symbol in 2D Euclidean-Space

In 2D Euclidean space straight line in $(x,y)$ coordinate $x=x(s)$ and $y=y(s)$ satisfy $$\frac{d^2x}{ds^2}=\frac{d^2y}{ds^2}=0$$ is the Christoffel symbol $\Gamma^a_{bc}=0$ in $(x,y)$ coordinate? ...
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1answer
45 views

Coordinates on a Riemannian manifold given by a distance function

I am currently studying the book "Riemannian Geometry" by Petersen. Defintion: Let $(M, g)$ be Riemannian manifold and let $U \subset M$ be an open set. A function $r : U \to \mathbb{R}$ is said ...
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0answers
15 views

Collision point of trajectory

I'm wondering how you could determine which object a trajectory hits first. This is quite hard to explain, but I'll try. I only want to consider the initial velocity, initial angles and gravity. I ...
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1answer
32 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
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21 views

Transforming points between two polar coordinate systems

I have 2 dimensional points (r, theta) defined in a polar coordinate system A, and a second polar coordinate system B with a known homogeneous transform T transforming between A and B in a Cartesian ...
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1answer
27 views

Let $BD$ be the internal angle bisector of $\Delta ABC$ with $D$ on $AC$. The incentre of $\Delta ABC$ is $(0,4)$ and $D$ is at $(1,3)$

Let $BD$ be the internal angle bisector of $\Delta ABC$ with $D$ on $AC$. The incentre of $\Delta ABC$ is $(0,4)$ and $D$ is at $(1,3)$. If $a,b,c$ are in arithmetic progression, find the point $B$. ($...
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2answers
78 views

Converting $r=\sec^2(\theta)$ to Cartesian

I encountered this problem on my Calculus test today and am struggling to figure it out: Write $r = \sec^2(\theta)$ as a Cartesian equation. I have tried using all sorts of tricks on it ($x^2 + y^2 =...
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3answers
148 views

Why is it bad to pick basis for a vector space?

Reading `This Week's Finds', http://math.ucr.edu/home/baez/week247.html, I'm informed that one should avoid picking coordinate systems and I'm unsure why that is the case. Any help on the matter is ...
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0answers
16 views

eccentricity of the conic

I'm given this question to find the eccentricity of this conic : $x^2 + ky = 0, k>0$ The given equation can be written as $x^2 = -ky$ now we can say compare this with the equation of parabola. But ...
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2answers
24 views

d3x - Cartesian to Cylindrical Coordinates

Given is $d^3x = dxdydz$ and I need to convert it to cylindrical coordinates (given through: $x = r\cos\varphi$ and $y = r\sin\varphi$). The expected result is: $(dz)(dr)(r)(d\varphi)$ and I cannot ...
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5answers
41 views

Devide line to 3 points

I have two point in a coordinates system, let's say $(x_1,y_1)$ and $(x_2,y_2)$, and I want to find the coordinates of the point that separates the line into 3 parts Like this I want to know the ...
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0answers
45 views

Vector Calculus in Curvilinear Coordinates and Index Notation

I am trying to understand how would I use the index notation in curvilinear coordinates. Checking out this reference, I got until this point $$\vec{\nabla} = \sum_a \vec{e}_ah_a^{-1}\partial_a $$ ...
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2answers
20 views

Finding the coordinates of the last point to form a parallelogram

Points: $$A(-2,6),\quad B(-5,0),\quad C(1,0).$$ Find the coordinates of $D$ such that $ABCD$ is a parallelogram. My workings: Midpoint of $AD$ = Midpoint of $AC$ Letting $D$ be $(X,Y)$ $$\left( \...
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1answer
37 views

Why is the equation for its distance from the $x$ axis is twice that of its distance from the $y$ axis $y = 2x$?

I have an understanding problem that I need to clarify.. The equation of line $L$ is $2y = -5x + 10$. A point $P$ lies on the line such that its "distance from the $x$ axis is twice that of its ...
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1answer
76 views

Calculate direction vector of azimuth and Elevation in NED

I need to calculate the unit direction vector of given azimuth and elevation angles. My target coordinate System is a NED (North east down). https://de.wikipedia.org/wiki/Roll-Nick-Gier-Winkel#/media/...
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1answer
23 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
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1answer
41 views

Quaternion from global space to local space

I've searched but have not found a response for this question specifically. I have a smartphone with a sensor that gives me a quaternion representing its absolute rotation relatively to the following ...
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1answer
54 views

Divergence theorem in curvilinear coordinates

Suppose I have a tensor \begin{gather} \stackrel{\leftrightarrow}{A} = \begin{bmatrix} a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\ a_{21}(\vec{r}) & a_{22}(\vec{r}) & a_{23}(...
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1answer
58 views

How to rotate a line segment around one of the end points?

I am given x1, y1, x2, y2 and θ. How can I find x3 and y3? By the way, there can be another line segment on the other side of AB (as if the line was rotated counter-clockwise). How to find that too?
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3answers
165 views

Show that $\nabla\cdot\left(\dfrac{\mathbf{e}_r}{r^2}\right)=4\pi\delta(\mathbf{r})$ using the divergence theorem.

The book answer goes as follows: By the divergence theorem, in spherical coordinates we find $$\color{red}{\iiint_\limits{\large\text{volume}\,\tau}\nabla\cdot\...
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1answer
18 views

Are Cartesian coordinates considered to be curvilinear coordinates?

In the wikipedia page on curvilinear coordinates it is said: "Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R^3) are Cartesian, cylindrical and spherical ...
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2answers
37 views

What is the locus of the points of intersection of the lines as shown in the figure?

I don't know if it should matter but the sum of the intercepts the lines make with x and y axis is constant I think. It looks like a hyperbola to me.
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Loci of intersection of lines with positive intercepts?? [closed]

How does one go about proving the equation of the curve that is formed by tracing the intersection of the lines with each other?
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2answers
43 views

How to draw a curve through every point. [closed]

I have x,y coordinates. They are arranged at fixed intervals of 1 unit along the x axis. The Y values are arbitrary. I want to draw smooth curvy line that passes through all of them. Or rather, I want ...