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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4 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
13 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: $$x = rsinϴ, r = \frac{x}{sinϴ} , \frac{dr}{dx} = \frac{1}{sinϴ} $$ $$\frac{dA(x)}{dx} = ...
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1answer
30 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
24 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
32 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
24 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
10 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
22 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
23 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 ...
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1answer
33 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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1answer
13 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
22 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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2answers
43 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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0answers
18 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
19 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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0answers
16 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 ...
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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 ...
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1answer
31 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
2k views

Abscissa, Ordinate and ?? for z-axis?

Like x-axis is abscissa, y-axis is ordinate what is z-axis called? It is one of basic doubts from my childhood.
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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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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
17 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
43 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 [closed]

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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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
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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1answer
579 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
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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
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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
19 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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0answers
54 views

Tangent undefined for polar curves ($r^2=a^2\sin(s\theta)$)?

I am considering the function $r^2=a^2\sin(2\theta)$ and am trying to find tangents perpendicular to the initial line, so $\frac{dx}{d\theta}=0.$ However, when I take the derivative by implicit ...
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1answer
23 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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0answers
40 views

Parabolic Coordinates and the Normal Derivative

I would like some help with the following problem. Thanks for any help in advance. Determine the largest region $\Omega'$ in $\mathbb{R}^2$ spanned by $(u,v)$ such that the transformation $T$ that ...
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5answers
2k views

How to find coordinates of reflected point?

How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis? Example Question: If P is a reflection (image) of point (3, -3) in the line $2y = ...
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5answers
1k views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
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1answer
769 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
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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
18 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
440 views

New x coordinate of a rotated line

I need help finding the equation to find $x$ I work in GIS and I'm working on a script that uses the new x coordinate of a rotated line. I havent work with trigonometry in a long long time so I ...
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1answer
17 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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2answers
376 views

Find parametric expression of an arc given its start point, end point and central angle in 3D cartesian coordinate system

In a 3D cartesian coordinate system, the coordinates of start point and end point have been given as $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$. If the central angle of the two points (the one smaller ...
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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 ...