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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Determine coordinates of rotated line segment

I am trying to determine coordinates (x,y) of Point B, given Point A and Point C and the rotation angle. Point C is the rotation pivot of the line segment. enter image description here I am ...
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1answer
15 views

Is this definition of a Euclidean frame well-defined?

Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ...
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2answers
19 views

Calculate point coordinates from other points

As in the image below i have four points. $P_1,P_2,P_3$ are known distinct points ( i know the $x,y$ of each of them ) also the angles $a_1,a_2$ are known. Can i calculate the coordinates of ...
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1answer
27 views

Can we use slopes in order to find the missing point in coordinate geometry?

Question: Plot the points $P(0, 3)$, $Q(2, 2)$, and $R(5, 3)$ on a coordinate plane. Where should the point $S$ be located so that the figure $PQRS$ is a parallelogram? Write a brief description of ...
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1answer
14 views

elementary question about coordinates directions - can we choose which direction on the coordinate axes is positive?

What determines the positive/negative direction of a coordinate system? It is pre-defined that in the case of the XY plane for example, for $x\geq 0$, the X-axis is positive and for $y\geq 0$, the ...
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2answers
16 views

Can I treat latitude/longitude as (x,y) coordinates to find closest point?

I have a list of coordinates L(lat, lon) and a specific position X. I am interested in finding the nearest location from the list L to the position X. Can I treat the lat, lon as x, y and implement ...
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1answer
41 views

$ax^2+by^2+2gx+2fy+2hxy+c=0$ : Understanding the equation

Given any second degree equation in $x$ and $y$, $ax^2+by^2+2gx+2fy+2hxy+c=0$ is it possible to find out the centre and/or the axis of the conic section it represents? What information can I ...
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0answers
12 views

Great circle distance on an ellipsoid [on hold]

Let's say I have a set of latitude and longitude (B,L on a reference ellipsoid WGS-84) and I also know the great circle distance (both in radians and meters) from my point to some point X on a sphere ...
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2answers
48 views

Find the fourth missing coordinate of a square in a Cartesian plane.

Question: Plot the points $P(5, 1)$, $Q(0, 6)$, and $R(-1, 1)$ on a coordinate plane. Where must the point $S$ be located so that the quadrilateral $PQRS$ is a square? Find the area of this square. ...
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0answers
35 views

HYPERBOLA : Problem [duplicate]

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} -\frac{y^2}{b^2} = 1$ whose centre is $C(0,0)$ are such that $CP$ is perpendicular to $CQ$ , $a<b$ , then prove that $$\frac{1}{(CP)^2} ...
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0answers
15 views

Proof of alternate cartesian to polar transformation of theta

My vector calculus lecturer has claimed that rather than the angle $\theta$ in the transformation from cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$ can not only be given by: $$ ...
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0answers
14 views

Unit Real Space?

Is there a formal name for a space that is the positive orthant of $\mathbb{R}^n$ where each of the $n$ dimensions is bounded to lie between zero and unity? The 1D representation would be a line ...
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0answers
23 views

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
29 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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0answers
22 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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0answers
6 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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1answer
18 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 ...
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1answer
22 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
15 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
26 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
26 views

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
16 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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0answers
19 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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1answer
37 views

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
30 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
40 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
14 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
25 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 ...
2
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0answers
40 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
71 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
20 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
39 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 + ...
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1answer
25 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). ...
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2answers
33 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 ...
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1answer
16 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
38 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
28 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
36 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
39 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
13 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
27 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
19 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 ...
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2answers
58 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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0answers
20 views

if a curve passes through point (0,1) and cuts the curve y=cx^2 orthogonally the curve also passes through which point?

I tried to use the slope of both the curve but i didnt get any conclusion.i.e finding the slope of (0,1) and (x,y) and then equating to 2cx.please help
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3answers
133 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
13 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
23 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
40 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 ...