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Questions tagged [coordinate-systems]

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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Christoffel symbols of the second kind transformation law

We want to show that the Christoffel symbols of the second kind transform like a connection. the Christoffel symbols of the second kind are given by: $$\begin{Bmatrix}a \\ bc\end{Bmatrix} = \frac{1}{...
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2answers
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Points closest to the edge of a square

Let S be the square formed by the four vertices (1,1),(1,-1),(-1,1), and (-1,-1). Let the region R be the set of points inside S which are closer to the centre than to any of the four sides. Find the ...
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1answer
25 views

Find the middle in the path

I have to find a midpoint in a coordinates set. Let's suppose I have the following GeoJSON with coordinates ...
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1answer
21 views

How to find the domain of a function using spherical coordinates?

Hey guys so I was wondering how to find the domain of a function using spherical coordinates.. For example, I take two functions: F(x,y,x)=√x+√y+√z+ln(4-x²-y²-z²) whose domain is D=((x,y,z):x≥0,y≥...
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Sufficient and necessary conditions for a change of coordinates to be locally invertible

Given a 2D coordinate change $(x, y) \mapsto (u, v)$, what are the sufficient and necessary conditions for the map to be invertible on every local neighborhood? For instance, the map, $$ u = x^3 \quad,...
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3answers
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Is there a parabola which is similar to a branch of hyperbola?

Parabola and a branch of hyperbola, visually looks similar. The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an ...
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1answer
25 views

Minimum distance of curve from origin

I have a parabola $(y+5)^2 = 4x$ and I need to find its minimum distance from origin. Scientific calculators aren't allowed. I have tried : 1) Substituting parametric coordinates $(r\cos Q, r\sin Q)$...
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28 views

Average Distances From The Origin

Consider all points $(m,n)$ where $m$ and $n$ are natural numbers, $m\leq10$, and $n\leq5$. What is the average distances from each point to the origin $(0,0)$? I know how to find the distance $d$...
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Line distance between points (p,q) and (q,p)

I'm trying to find the distance between the points (p,q) and (q,p). As far as I can tell, my steps are correct, but I'm getting ...
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2answers
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calculating motion from angles

First, my apologies. This question may have been asked many times before but I do not know the correct terms to search on..... and my school trigonometry is many years ago. Pointing me to an ...
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Calculation of a point coordination from a camera

May you please help me how can I calculate the point A's coordination if the camera is placed in B side (ideal and with a focus distance of 2CM) Please see this image I've tried using the matrix ...
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Something wrong with the parabola

A line is drawn from $A(-2,0)$ to intersect the parabola $y^2 = 4x$ in $P$ and $Q$ in first quadrant such that $$\frac{1}{AP} + \frac{1}{AQ} < \frac{1}{4}$$ then the slope of the line is always .....
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3D Math Primer for Graphic and Game Development problem with Upright Space

In the book in Chapter 3 he mentions Upright Space. I understood everything so far and I did every exercise from Chapter 2 with vectors correctly. But I cannot understand how did he get Upright Space ...
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tangent bundle of homogeneous space

I'm trying to express coordinate of tangent bundle of $S^n$. ( I have been reading a lecture note about homogeneous space. ) For example, in the case of $S^3$, since it is diffeomorphic to $SO(4)/SO(...
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Are the euler angles and inertia angle the same?

while I'm reading this paper. I came to the question, if the integral of the angular velocity in the inertia frame and the integral of the euler angles are the same? Equation (35) describes a ...
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2answers
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2D Coordinate - Plot a line parallel to another line

I am given values of $x_1, y_1, x_2, y_2$, $\theta_{1}$, $\theta_{2}$, $h$, $m$ and $L$. I want to find out the coordinate of the Unknown Point of Interest, which ...
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1answer
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Calculating square meter area with polygonal geographical coordinates (metric - not DMS system)

I'm working on a program but my problem is not on software side but mathematical. I have the following input : ...
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1answer
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Schrödinger Equation in Spherical Coordinates

I am trying to learn how to solve three dimensional Schrödinger Equation in Spherical Coordinates. I was reading a text book and I found that there is a missed step in the solution, seen below: The ...
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1answer
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How to show that the bipolar co-ordinates are othogonal

How to show that the bipolar co-ordinates are othogonal where $x=\dfrac{\sin hv}{\cos hv-\cos u},y=\dfrac{\sin u}{\cos hv-\cos u},z=z$ where $u\in [0,2\pi]$ and $y,z\in (-\infty,\infty)$. ...
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1answer
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How can i Find the Area of a Traingle Formed inside a Triangle?

Here we have Co Ordinates of A ,B ,C and We have to Find the Area of the Triangle PQR Formed Inside the Triangle ABC.triangle ABC. Point D, E and F divides the sides BC, CA and AB into ratio 1:2 ...
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angle of rotation from coordinate system

Let l_{o}:y=x and m_{o}:y=\sqrt{3}x. Find \Omega_{l_{o}}\Omega {m{o}}(X) using rot\theta . What is the angle of rotation ?
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Finding area of a triangle with integration

I have a triangle with coordinates (0,0), (1,2) and (1,0). Is the area of this triangle same as finding the integral of the function $y=2x^2$ and substituting the value of x=1 and y=2? Because what i ...
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2answers
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Two polar curves intersect problem .

Suppose we have two curves given by: $$r=20sin2\theta $$ $$r= 20 cos2\theta$$ Now by solving the equations, we get the solution as $\theta = \frac{\pi}{8}$. However, on graphing the equations, I ...
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1answer
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Third cordinate of a triangle when we know two sides and two other points?

How to find the third cordinate of a triangle , where as other two points are known and we know sides? Let's say, the two points are (3, 30) and (5, 35) and we need to find the third point ...
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1answer
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Modified tangent function

I remember reading once of a modified arctangent function which was suited to be a chart on the circle. It should be defined something like $$\theta(x,y)=\begin{cases} \arctan(\frac{y}{x}) & x>...
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rotation of spherical coordinate system

If I rotate the coordinate system by spherical angles θ and ϕ, and the vector in the new system is v'=(x′,y′,z′), what is its coordinate (x,y,z) in the original coordinate system? Thank you in advance!...
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proof that the covariant or contravariant basis is a basis

Does anyone know how to prove that given a coordinate system Z, the covariant (contravariant) basis is in effect a basis for Z ?
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Calculating curvature and offsets of curve segment

Given a pair of coordinates a linear interpolation between these, or indeed interpolation of a circular profile given a radius, is simple enough, but... how would one calculate the intermittent ...
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34 views

Divergence in Orthogonal Curvilinear Coordinates - Is there a mistake?

I have to prove that, $$\nabla \cdot (A_1\mathbf e_1) = \frac{1}{h_1h_2h_3}\frac{\partial (A_1h_2h_3)}{\partial u_1} $$ My approach: $$\begin{align} \nabla \cdot (A_1\mathbf e_1) &= \nabla \...
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1answer
20 views

Tensor coordinate transformations: cosine matrix vs rotation matrix

I have a tensor $\varepsilon(\vec r)$ and want to get $\varepsilon(\vec r')$. After some googling I've got confused, should cosine matrix to be used or coordinate transformation matrix? Some sources ...
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Can anyone explain to me the math behind this code ? Calculating a bullet's trajectory in 2D

following the book Beggining c++ game programming by John Horton on Chapter 9 the author explains us how our character can shoot a bullet, the thing is that there is very little explanations on what ...
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Projection On Parallelogram

In my research, I encountered the problem of finding the projection of any point on given parallelogram with the following properties : If the point is in region 1, the projection should be vertex A. ...
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1answer
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Coordinates of a point [closed]

There are $2$ points $A$ and $B$. $A = (2, 7)$, $B = (5, 1)$ and $M, N$ are points on $AB$ such that $AM = MN = NB$. Find the coordinates of $M$.
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How to rotate a coordinate system to find the unstable manifold.

I am considering the dynamical system: $u'=v-0.25(v-u)^2$ $v'=u(1+v)+0.25(u+v)^2$ I have calculated the linear stable and unstable manifold as, $E^s=sp(1,1)$ and $E^u=sp(-1,1)$ for eigenvalues $-1$ ...
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Find coordinate from distance and coordinate

I know that this topic has been done already but i believe that my issues is a bit more annoying. I have a circle which I know its center's coordinate that cross a line like this : ...
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4answers
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How to linearize Quaternions?

Based on an answer to one of my questions and the comments exchanged here earlier I noticed that I cannot uniformly sample Quaternion vectors for rotation even though if I know the bounds of each ...
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3answers
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Rational points on a circle with centre as $(\pi,e)$

What is the number of rational points on a circle having centre as $(\pi,e)$?
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1answer
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Physics: Electric field

This question involves maths. I do not think it is inappropriate here. I am reading Purcell and Morin's Electricity and Magnetism 3rd Edition. Equation ($1.22$): $$\vec{E}(x,y,z)=\dfrac{1}{4 \pi ...
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How to find the user position on a 2d floor plan if other coordinates (3 or more points) are given?

Is it possible to find the position of user if the coordinates of A,B and C are known.D1,D2 and D3 can also be provided.With all these details can we find the user's position(Coordinates/points)?
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How to get coordinates in 2D with the given points

I have the following coordinates: $[-20, -20]$, $[-20, 20]$, $[-40, 0]$, $[-40, 20]$, $[-40, 40]$, $[-20, 80]$, $[20, 80]$, $[40, 40]$, $[80, 20]$, $[80, -20]$. My task is to get coordinates starting ...
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4answers
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Right triangle inside a square

Hello I'm trying to prove the following Let ABCD be a square where X is the midpoint of segment AB and BY = (1/4)BC where DXY is a triangle Prove DXY is a right triangle (using coordinate approach) ...
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co-ordinate geometry with constraints on triangles and a square

Triangle ABC is right angled at A,AB = 2AC,A=(1,2),B= (-3,1) ACD is an equilateral triangle. The vertices of two triangles are in anticlockwise sense. BCEF is a square with vertices in clockwise sense....
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Gradient in Orthogonal Curvilinear Coordinates.

Let us consider the transformation of rectangular coordinates $(x,y,z)$ into an orthogonal coordinates system $(u_1,u_2,u_3)$ Let $r = xi + yj + zk$ be the general position vector that can be written ...
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2answers
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Intuitional explanation needed - Determining if a Point is in a turned square

This should be viewed as a turned square. Now, for the Point P(X,Y) why is it that if $|X|+|Y|\leq1$ then the Point is inside the square?
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Integral points on a line

In this post What is the number of integer coordinates on a line segment? , we have the formula for the number of integral points strictly between $(x_1,y_1)$ and $(x_2,y_2)$ (both are integral points)...
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Coordinate systems and derivative operators

Let $\psi$ be a coordinate system and let $\{\partial / \partial x^{\mu}\}$ and $dx^{\mu}$ be the associated coordinate bases. For any smooth tensor field $T^{a_1 \dots a_k}_{b_1 \dots b_l}$ we take ...
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Cartesian coordinates to Cylindrical coordinates

The position vector, as a function of time and in Cartesian coordinates, of a particle is the following: $$\vec{r}(t) = (5t^2-6t)\vec{e}_1 + \cos(\sin(t))\vec{e}_2+ (8t-5)\vec{e}_3$$ I have that ...
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Convention in intercepts. A point or a value?

I started tutoring a few years ago and I just found some foreign secondary textbooks (I live and studied in HK.) use a point $(x,0)$ for $x$-intercept instead of just a single value (the $x$-...
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A interesting and easy looking question that involves colouring of the Cartesian plane

Take the graph of a $f(x)$ which can be any arbitrary function, e.g. $x^2$ . We draw tangents to each and every point on it and colour them. If relative to a fixed points have $z$ times the number of ...
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Equations in trilinear coordinates for the line through two points, and the circle through three

I am looking for the equations which describe, using trilinear coordinates: A line through two given points. A circle through three given non-collinear points. I would also love to get a ...