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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1answer
56 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
38 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
60 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? ...
4
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1answer
47 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
16 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
38 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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0answers
22 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
85 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
152 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
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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
25 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
50 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
96 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, ...
2
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1answer
57 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 ...
2
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1answer
57 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
66 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
166 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
42 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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2answers
50 views

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?
2
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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 ...
2
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1answer
80 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
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1answer
25 views

Get two specific coordinates on a linear function.

I've been trying to build an algorithm for my Unity3D project to get some specific coordinates, but I got stuck with some math problem. I have two coordinates of ...
2
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2answers
66 views

Angle of rotation based on direction cosines

I have a question which is bothering me for days! Suppose that we have a fixed frame $XYZ$ and a moving frame $xyz$ in 3D. The moving frame is orthonormal and is defined based on the fixed one using 9 ...
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0answers
20 views

Relative Motion between two rotating frames

I am looking for mathematical relations between equations of motion between two rotating frames. Relating the said motion by first going to a globally fixed frame is not an issue but how to approach ...
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0answers
11 views

The existence of a specific kind of coordinate system

Suppose a massive, perfectly spherical ball is fixed on a point in space, and that there is a second free particle of negligible mass. I have noticed that the magnitude of the force exerted on this ...
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2answers
29 views

Solving vector equations of planes

Find the line of intersection of two planes denoted by: $r=\overrightarrow{b}+\lambda(\overrightarrow{b}-\overrightarrow{a})+\nu(\overrightarrow{a}+\overrightarrow{c})$ $r=\overrightarrow{c}+\alpha(\...
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2answers
25 views

Determine whether two segments P1Q1 and P2Q2 have a common point if the (x,y) coordinates of their end points is known?

Does this question have a solution? I think it's impossible to know if line segments P1Q1 and P2Q2 intersect at all with just the information about their end points Q1 and Q2. Thanks.
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0answers
15 views

Cylindrical Coordinate System

I am to write equations of each side for a cylinder.The problem is stated below: There is a cylinder...One cuts it from somewhere near the center point ( height wise ) . So the cylinder is now split ...
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1answer
114 views

How can you find the distance between the center and edges of a rectangle - a line from centre to a edge at an angle $\theta$?

I have a case where I know the coordinates $(x,y)$ of the center of the rectangle and its edges where the line is dropped anywhere on the edges $a(x_1,y_1),b(x_2,y_1),c(x_1,y_2),d(x_2,y_2)$. Say I ...
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0answers
19 views

Cartesian coordinates conventions

Is there any historical account of how did the Cartesian coordinate system get its current conventions of orientation and representation? Are there any mathematical reasons for these conventions?
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0answers
67 views

Vector Laplacian operator in orthogonal curvilinear coordinates

I'm looking for a simple expression for the vector Laplacian $\nabla^2\mathbf{A}$ in orthogonal curvilinear coordinates. Actually, I don't require the whole thing, just the part of $\mathbf{u}_i\...
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1answer
31 views

Matrix to represent transformation $T(a_0+a_1t+a_2t^2) = a_1+2a_{2}t$ with respect to a certain basis

I'm having so much trouble understanding this very concept.. This is a problem from the book. The Mapping $T:P_2\to P_2$ is defined by $T(a_0+a_1t+a_2t^2)=a_1+2a_2t$ is a linear transformation. ...
3
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0answers
22 views

Single atlas implies global normal coordinates?

Let $M$ be a class $C^k$-Riemannian manifold and suppose there exists an atlas $\langle U,\psi\rangle$ for $M$ containing only one global chart. Does this imply that the Riemmanian $Log_p\,(:=Exp_p^{...
2
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1answer
77 views

Integral Calculus: Plane Areas in Rectangular Coordinates

Find the area bounded by the given curves: $y^2+2x-2y-3=0$ and the $y$-axis (using horizontal & vertical strip)
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2answers
26 views

Linear Algebra Coordinate Systems Isomorphism

This is an excerpt from the book. Let $B$ be the standard basis of the space $P_3$ of polynomials; that is let $B=\{1,t,t^2,t^3\}$. A typical element $p$ of $P_3$ has the form $p(t) = a_0 + a_1t+ ...
0
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1answer
21 views

formula to locate x and y coordinates from two point with x and y known and distance to third point known as well

updated pictureI am trying to find the formula or at least the name of this operation to locate x and y coordinates for a point from two other reference points with x and y known and distance to third ...
2
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1answer
116 views

Pullback metric, coordinate vector fields..

I'm doing this computation on $\mathbb{R}^3$ with cylindrical coordinates $(r, \theta, z)$, (which aren't defined on the whole of $\mathbb{R}^3$, but I don't care about that) and I seem to get a ...
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0answers
15 views

Limiting points

For a system of coaxial circles why are there only 2 limiting points? Shouldn't there be infinite limiting points? After all system of coaxial circles are pairs of circles which have same radical axis,...
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0answers
29 views

Direction angle of Line segment in polar coordinates

I have a line segment given by two points $A$ and $B$, that are $(r_1,\theta_1)$ and $(r_2,\theta_2)$ in Polar coordinates. I know that the direction angle of the line segment is given by: $$\...
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1answer
77 views

Coordinate transformations and interpreting what the Jacobian determinant describes

Apologies for a perhaps rather trivial question, but I really want to get the concept cleared up in my head. I understand that when one changes from one coordinate system there is an appropriate ...
0
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1answer
56 views

Why aren't area of triangle not same when calculated by different methods in this case

I came across a question today. Two mutually perpendicular straight lines through the origin forms an isosceles triangle with the line $2x + y = 5$. Then the area of the triangle is ? I know ...
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0answers
26 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi \sin\...
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4answers
114 views

Calculation of $\min$ distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$

Calculation of $\min$ distance between $x^2+y^2=9$ and $2x^2+10y^2+6xy=1$ $\bf{My\; Try::}$ Using the fact that the distance between two curve is independent of shifting. So put $x=u+v$ and $y=u-v$ ...
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2answers
33 views

Can anyone provide a good explanation to converting a vector in cartesian coordinates to cylindricals?

I am sorry about the perhaps trivial question, but for some reason I am really struggling to do this. I was recently taught about curvilinear coordinates, which I believe provides a system for ...