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
44 views

Quaternion to Euler angles conversion

I have written the following MATLAB code for transforming Quaternion to Euler angles based on the mathematical formula from wikipedia: ...
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1answer
27 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rcosϴ, r = \frac{x}{cosϴ} , \frac{dr}{dx} = \frac{1}{cosϴ} $$ $$\frac{dA(x)}{dx} = \frac{dA(r)}{dx}\frac{...
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1answer
35 views

isomorphism from one vector space to another one

This is from my textbook I don't quite understand what isomorphism means. Greek word "isomorphism" means same structure, but how can we say $P_3$ has the same structure as $R^4$?
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1answer
27 views

Unit base vectors in a new coordinate system

Let's assume we have a function $f:\Omega =R^2 \rightarrow R $ $f(x,y)=x+2xy+x^2y$. Obviously our unit base vectors on $\Omega$ are $e_x=\hat{i}$ and $e_y=\hat{j}$. Now we want to change the ...
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2answers
40 views

How to determine standard equation of a conic from the general second degree equation?

From a given general equation of second degree i can determine the conic by following rules: Given equation: $ax^2+by^2+2hxy+2gx+2fy+c=0$ then if, $abc+2fgh-af^2-bg^2-ch^2$ is not equal to zero ...
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1answer
15 views

Reaching a point B in Cartesian coordinate via Euler angles knows its initial point A Euler angles and B Euler angles

I have a point A: Known it's Cartesian coordinates (X,Y) and its Euler angle Aka head rotation (R,P) respectively Roll (rotation around X axis) , Pitch (rotaion around Y axis). (I'm not using Yaw ...
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1answer
38 views

How to understand rotation around a point VS rotation of axes?

I am puzzled about linear transformation and coordinate transformation, any help will be appreciated. From wiki rotation matrix, we know rotates points in the xy-Cartesian plane counter-clockwise ...
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1answer
23 views

Aligning 2 Coordinate Systems

I have a camera and a table and I want to align the camera to co-exist in the same coordinate system as the table. Here is an image of the setting. What type of mathematical transformations I need to ...
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4answers
30 views

Check if a given coordinate lies in path of a ray (coordinate geometry)

As shown in the image I have two known coordinate pair A and B and few other known coordinate pairs (RED blob) on the graph. I need to know if any of the other given coordinates fall in line of the ...
0
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1answer
35 views

Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
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0answers
44 views

How to use slopes (3 points are given) to prove that they form a right triangle?

Question: Use slopes to show that $A(-3, -1)$, $B(3, 3)$ and $C(-9, 8)$ are vertices of a right triangle. My try at the problem: I know that we can find the slopes of $AB$, $BC$ and $CA$ and then ...
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0answers
21 views

Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
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1answer
33 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...
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0answers
18 views

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 ...
0
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1answer
25 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
22 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 $x,y$...
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1answer
40 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 ...
1
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1answer
15 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 Y-...
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2answers
25 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 ...
0
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1answer
47 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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2answers
90 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
38 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
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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
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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
24 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
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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0answers
25 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
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
26 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
25 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
29 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 ...
-1
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2answers
27 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 ...
0
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1answer
28 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
22 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
47 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 ...
5
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2answers
41 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 ...
0
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1answer
18 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 ...
2
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1answer
28 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
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 ...
3
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1answer
81 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
23 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
35 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
23 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
53 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
37 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: ...