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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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How to convert from left-handed coordinate system to right-handed?

I need to convert coordinates and rotations from left-handed coordinate system (used by Unity) to right-handed (used by camera calib. toolbox in MATLAB\Octave) While converting point coordinates ...
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Track object in space and time

I'm currently struggling to solve the following problem: The goal is determine the movement of an object by tracking an object in 2d-space and time (third dimension). I always know the $x$ and $y$ ...
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Modelling a drone in 3D: finding the z component of the thrust vector knowing pitch (about x) and roll (about y) angles.

I have been trying to create the model of a drone for fun and when calculating the linear dynamics I need to find out the Z component of the thrust vector. Basically, in 3D space the thrust vector is ...
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If $4\alpha^2–5\beta^2+6\alpha+1=0$.Prove that $x\alpha+y\beta+1=0$touches a Definite circle. Find the centre and radius of the circle.

If $4\alpha^2–5\beta^2+6\alpha+1=0$. Prove that $x\alpha+y\beta+1=0$touches a Definite circle. Find the centre and radius of the circle. I tried to solve this question by taking a General equation of ...
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Show that the equation of a straight line meeting the circle$x^2+y^2=a^2$in two points at equal distances$'d'$from a point$(x_1,y_1)$

Show that the equation of a straight line meeting the circle$x^2+y^2=a^2$in two points at equal distances$'d'$from a point$(x_1,y_1)$on its circumference is $xx_1+yy_1–a^2+d^2/2=0$. I tried to solve ...
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Prove function in two variables is convex

Prove that $$f:(x,y)\to\log(\exp(x)+\exp(y))$$ on $\mathbb{R}^2$ is convex. By direct computation, the Hessian matrix is $H\begin{bmatrix}a&-a\\-a&a\end{bmatrix}$ where $a=\tfrac{e^{x+y}}{(e^...
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Why are Cartesian coordinates ordered x, y but matrix coordinates are ordered r, c?

Lets say I have the following plane: 4 3 2 1 0 1 2 3 4 -1 -2 -3 -4 and a matrix like this: ...
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2answers
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Measuring the angle between skew coordinate axes using the skew coordinates

My application is that I have a cheap Chinese CNC machine where the X and Y axes are not quite orthogonal. They are close, but not quite there, and I'd like to measure the angle between the axes so as ...
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1answer
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Finding Coordinates of a Triangle using Azimuth?

Good Evening, I am just starting out in GIS and it's been a long time since I took math in university. I am teaching myself through a textbook but there's a few questions I am stumped on. If anyone ...
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Finding coordinates of points using distance between points.

I got a question in my homework which I can't solve. Here is the question: (I am not a native speaker so please explain step by step and clearly.) Point $C$ internally divides the segment ...
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Calculate row (y) based on given height

I am trying to solve mathematical problem. I want to find a pattern for getting correct row based on current height of mouse position (height in local system of coordinates). I have written on paper ...
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How to trace a spline in the middle of an irregular curved street?

I'm creating a program that should drive a car through the middle of a street. The street, as shown below, is a bitmap with irregular curves. I would like my program to calculate a central route, ...
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1answer
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Equation of circle touching 2 circles and x-axis [closed]

You are given 2 circles with center (2,2) and (8,4) respectively. Both the circles touch x-axis. You need to find the equation of a circle which touches these ...
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Specify a vector in circular coordinates

If circular coordinates are understood as: Then a point is indicated by ($r$, $\theta$). But how should a vector be specified? ($r-r_0$, $\theta$-$\theta_0$) ? $\theta - \theta_0$ seem a wrong step,...
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Converting the coordinates of a point to cylindrical coordinates with positive values.

I'm trying to convert the coordinates of point $(x,y,z) = (-2,-1,0)$ to cylindrical coordinates, with positive values for $\theta$ and $r$. I know that: $r^2 = x^2 + y^2$ so... $r = \sqrt(5)$ ...
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Aligning point cloud using camera local coordinates only [closed]

I have the stanford bunny point clouds taken from different views and the config file containing all the local camera poses for each of these clouds. ...
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How do transfer from a xy coordinate system to r and θ

Let S $\subset$ $R^2$ and let S be formed by : the region inside the circle $x^2$+$y^2$=9 below the line y=x above the x axis laying to the right of x=1 Evaluate $\int xydA$. I know how the ...
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Translating a 3D point from one frame to another

Me and my friend have ran into trubbels with translating a point in a 3D-space from one cartesian coordinate system to another. In this case we have 3 coordinate systems called the Base-frame, h-...
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1answer
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Relation between transformation matrices and conversion formulas between coordinate systems?

We are learning how to work with different coordinate systems in my Mechanics class (spherical and cylindrical mainly), and about form factors, general formulas for the gradient, the curl, the ...
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If ABC is an isosceles right angled ∆ at B and coordinates of A and C are (1,2) and (4,-2) then coordinates of Vertex B can be

My approach : I have tried it by different methods but find them too lengthy like , by using Basic Distance formula . I am finding a short solution for it . A Hint will help.
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Graphically representing vectors with polar unit vectors without converting to Cartesian coordinates

Short version : How do you graphically represent a vector(without converting to Cartesian) given components in direction of $\hat r$ and $\hat \theta$ (unit vectors in polar coordinates)? and what is ...
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How to get coordinates of point knowing distance from x,y without know angle?

Starting from this question How to get coordinates of point knowing distance from x,y and angle? I would to know if there is a method to get the same result but without the knowing of the $\alpha$ ...
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Coorddinate system described by following 2D change of variables

I found following change of variables that can be also seen as a coordinate system $(y_1, y_2)$ described according to cartesian coordinates $(x_1, x_2)$: $$y_1 = x_1 · (1+(x_1-1)\sin(\frac{\pi x_2}{...
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Transformation between two coordinate systems using inverse rotation matrix

Let's say I have data in one coordinate system. It is somehow rotated against the real world coordinate system. I have a tracker device and its rotation matrix in the mentioned coordinate system. What ...
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Can we draw a graphic function in a different $x-y$-axis?

Why $y$ and $x$ axis must be perpendicular? For example if we had a function $y=f(x)$ could we draw it on a coordinate system where $y$ and $x$ axis would have an angle of $30$ degrees?
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1answer
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A coordinate geometry problem from Hungarian olympics

Here is the 2nd problem from Hungarian Kurshak maths competition I can’t solve. Please help! Let $v_1,v_2,\dots,v_n$ be different vectors in the 3D space in the Cartesian coordinate system, such that ...
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Why is the homogeneous line through all points at infinity (1:0:0) and not (0:0:0)?

So I just had a geometry lecture that introduced me to homogenous coordinates. To be clear with notation let me recap: Homogenous coordinates in $\mathbb R^n$ space are described as $$(x_0:x_1: ... : ...
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How do I revolve a general 2D coordinate system?

$\newcommand{\dd}{\partial}$ Question I wish to construct a general 3D revolved, orthogonal, curvilinear co-ordinate system, where the axis of revolution is coincident with the Cartesian $z$-axis. ...
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1answer
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Homogeneous second degree equation problem

We are given a curve having the equation: $ax^2+2hxy+by^2+2gx+2fy+c=0$ and a line having the equation: $lx+my=n$ $\frac{lx+my}{n} = 1$ While making the equation of the curve "homogeneous", we ...
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Radius and angular derivatives expressed in Cartesian coordinates

Given a point with Cartesian coordinates $(x,y)$ and with Cartesian velocity $(\dot{x},\dot{y})$, I would like to express its radius $r$, its angle $\theta$, its radius velocity $\dot{r}$, and its ...
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how to get the equation of a surface equation like $F(x,y,z)=0$ by parallel projecting onto a plane?

taking example 1: we want to project the standard ellipsoid equation: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 $$ onto the plane equation $$z=0$$ here we can just substitute $z$ with ...
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Area of region bounded by locus of a point P

The area of the region bounded by the locus of point P satisfying d(P,A)=4, where A is (1,2) is ____. Where we define the distance between two points P(x,y) and Q(a,b) as $$d(P,Q)=max(|a-x|,|b-y|)$$. ...
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The number of points on given line which are at some distance from a fixed given point.

Question The number of points on the line $3x+4y=5$, which are at a distance $$(sec(\theta))^2 + 2 (cosec(\theta))^2$$, where $\theta$ ∈ $R$ , from the point (1,3) is___ . My attempt $$d= \frac{|3(...
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Locus problem for vertex of equilateral triangle

Question An equilateral triangle PQR is formed where P(1,3) is a fixed point and Q is moving point on line x=3. Find the locus of R. My attempt Took Q as (3,p) and R(h, k). Then substituted to ...
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1answer
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How to draw lines and circles on cylindrical projection map?

I am trying to draw a circle with known radius around a coordinate on a cylindrical projection map. Which is a circle around equator and egg shaped closer to the poles. And also trying to draw a line ...
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Count how many different final position are there?

A frog started from the origin of the coordinate plane and made three jumps. Each time the frog jumped a distance of 5 units and landed at a point with integer coordinate. How many different ...
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1answer
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Trouble understanding coordinate systems as charts on differentiable manifolds.

I had a question involving the notion of coordinates. Here, I will use polar coordinates on $\mathbb{R}^2$ as an example. From my admittedly lacking understanding of differential geometry, polar ...
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Hexagonal Grid from Cube

It's fairly well-known that an intersection with the integer cube grid (i.e., the vertices of the grid are in $\mathbb{Z}^3$) with the plane $u-v+w=0$ creates a hexagonal grid (usually people use $u+v+...
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Transforming the coefficients of polynomial

Im having a polynomial in one co-ordinate frame. Lets say it with respect to (0,0). I calculated the coefficients too. Now I'm transforming the points to different frame. Make it (10,10). How will I ...
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Finding normal vector to polar equation

It’s been a long time since I’ve looked at coordinate systems and I’m struggling on a derivation in my research. Suppose I have an equation of the form $r=r(\phi)$ which is just an azimuthal orbit. ...
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How do I derive the volume element $ dV = \sqrt{g} du^1 du^2 du^3 $ in a 3D curvilinear coordinate system?

I am trying to derive $ dV = \sqrt{g} du^1 du^2 du^3 $ for some general curvilinear coordinate $(u^1,u^2,u^3)$ system in $\mathbb{R}^3$ where $g = \mathrm{det}[g_{ij}]$. I am using the following facts:...
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“Abscissa”, “Ordinate”, and “Applicate” — Origins.

How did the terms "abscissa", "ordinate", and "applicate" (for the $x$-axis, $y$-axis, and $z$-axis, respectively) originate? Note: I feel the need to explain this question before someone says that ...
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How to define n-dimensional 'planes'?

A plane/surface is the 2-dimensional analog of a point (zero dimensions) a line (one dimension) and three-dimensional space. Is 'coordinate system' the unifying term for any n-dimensional model? ...
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How to check if point is in elliptical sector without float-point arithmetic?

How can I check whether point lies in elliptical sector without float-point arithmetic if I know a and b from the ellipse ...
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1answer
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Distance of the point from the curve

Original question The solution of the equation $$ \frac{dy}{dx}= \frac{3x-4y-1}{3x-4y-3}$$ passes through the origin. The distance of it from the point (-1,1) is ___. *My attempt * Found the ...
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1answer
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Rotation matrix along a custom axis

For a certain software that I'm developing, I need to create a rotation matrix for a custom axis, and being almost completely self-taught in math, I am trying to wrap my mind around it, yet failing ...
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1answer
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Coordinates of a segment formed by the sum of two angles

If I add two arbitrary angles A + D, how do I find the coordinates of the arc formed by GH? Corners C and B can be swapped. Corners E and F can be swapped. As long as A and D get added together. ...
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Vector Spaces are Free Objects

Warning: I know little linear algebra and my assertions below may all be incorrect. I am interested in lists --i.e., free monoids-- and my interest has led me to [finite-dimensional] ...
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Find spherical polar components of $\frac{\partial \textbf{F}}{\partial \phi}$

I have to find the partial derivatives in spherical form of $\textbf{F}=r[\textbf{e}_{\theta}+\textbf{e}_{\phi}],$ and I managed to find all the others except for the one over $\phi$. I got this far: ...
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$TS^2$ coordinate expression and coordinate change.

I want to express coordinate the (co)tangent bundle of sphere. I think $TS^2$ (or $T^*S^2$) $=\{(x_1,x_2,x_3,v_1, v_2 , v_3 | x_1^2+x_2^2+x_3^2=1, <(x_1,x_2,x_3),(v_1,v_2,v_3)>=0 \}$ and using ...