Questions tagged [coordinate-systems]

This tag involves questions on various coordinate systems. The usual Cartesian coordinate system can be quite difficult to use in certain situations. Some of the most common situations when Cartesian coordinates are difficult to employ involve those in which circular, cylindrical, or spherical symmetry is present. For these situations it is often more convenient to use a different coordinate system.

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1answer
51 views

Why it is easier to draw on some grids?

I would like to know is there any way to prove that drawing in a grid on the left would be more precise than in a grid on the right. For me, that make sense since we have more points on the left. But, ...
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1answer
32 views

Equation of a hyperplane in n- dimensions.

I know that the equation of a hyperplane in n-dimensions is given by: $$w^Tx+w_0=0$$ Where $w$ is a vector that is perpendicular to the surface of the hyperplane and $w_0$ is a constant. I also know ...
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2answers
19 views

How to transform these coordinates by substitution in classical mechanics?

The goal is to transform the following coordinates: $$x(t)= R(\Phi-\sin\Phi)$$ and $$z(t)=R(2 +\cos\Phi)$$ with the substitution: $u=\cos\left(\Phi/2\right)$ in order to get: $$x(t)=2R(\arccos(u)-u\...
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Why do we sum the projections to get the new coordinate point when doing coordinate transformations?

I'm going through Classical Dynamics of Particles and Systems from Thorton and Marion, and I'm the section 1.3 is showing how to deal with coordinate transformations. Using the drawing below, the ...
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8 views

Coordinate system transformations using a rotation angle

I am working with an application used on CNC machines and coding an alternative means of calculating a point on a grinding wheel from a point on a component (workpiece) - the current method is based ...
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26 views

Normalizing directed graphs from image space to compute joint differences

My question is how to transform a 'skeleton' (directed graph) from image space so that I can compare it joint-wise against another 'template' or reference skeleton that may be differently sized/...
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surface of revolution and coordinate change. Does order matter?

The construction: Consider $xy=1$ revolved about $y=x$ to bring the surface, $S$ into $\Bbb R^3.$ Then rotate $S$ to get: $$ x^2+y^2+z^2-4xy-4yz-4zx=-1 $$ Now change the coordinates of $xy=1$ and, ...
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1answer
35 views

JEE Circles Doubt

Can you derive a general formula for length of intercept made by a circle $x^2+y^2+2gx+2fy+c=0$ on the line $ax+by+c=0$? If no, then is there any other method to solve these type of questions?
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1answer
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How to find a value of a point in a new coordinate system?

question refers to this image The known data: length of $AO$, $OB$, angles $\alpha$, $\beta$, point value of $B= \left(b_1,b_2\right)$ in the Cartesian coordinate system where $O$ is the origin. It is ...
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19 views

Convert from Cylindrical coordinate to Cylindrical vector

Let's assume there is a function of curve represented with Cylindrical Coordinate system as $r(\phi, z)$, each point can be represented as $P(r(\phi,z),\phi,z)$. I want to convert each point into a ...
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9 views

solve coordinate and quadratic system

Refer to the diagram below, AB and CD have slopes of -1 and BC has slope of 2. Given that BE: CE = 2:3, find the area of triangle ACF: area of quadrilateral AFDB. I need help with how to find it. many ...
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1answer
50 views

Prove that the normals at A, B, C incribed in $xy=c^2$ will meet in a point if $cot2\alpha + cot2\beta + cot 2\gamma = 0$

The sides of a triangle ABC, inscribed in a hyperbola $xy = c^2$, makes angles $\alpha, \beta, \gamma $ with an asymptote. Prove that the normals at A, B, C will meet in a point if $cot2\alpha + cot2\...
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Partial derivatives in a skew coordinate system

I am a math hobbyist. I'm learning partial derivatives. All the examples I've come across deal with orthogonal coordinates. I am wondering what, if any, adjustments need to be made when taking partial ...
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1answer
37 views

Focus of rectangular hyperbola touching a given ellipse $\frac{x^2}{36}+\frac{y^2}{18}=1$

Consider an ellipse $\frac{x^2}{36}+\frac{y^2}{18}=1$ There is a hyperbola whose one asymptote is the major axis of a given ellipse. If the eccentricity of the given ellipse and hyperbola are ...
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2answers
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Unit vector in $ds$ substitution (while changing coordinate systems) [closed]

I had to calculate the line integral of a homogenous vector field $E=E.e_y$ from an angle $\phi=o$ to $\phi=\phi_{e}$ having a radius $\rho$. the formula normally is $\varphi=\int E \space ds$ but ...
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9 views

Dot product of a cartesian vector with a spherical vector

I have a vector of the form $P\hat{z}$ in the Cartesian system. I need to perform a dot product with another vector which is the spherical form. I figured it would be easier to convert $P\hat{z}$ into ...
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43 views

Change of coordinates and diffeomorphisms

In physics, we often use diffeomorphisms to change coordinates on a smooth manifold $(M,A)$. But, from what I've seen, "changing coordinate" simply corresponds to give our self another atlas ...
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3answers
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How does $axy+bx+cy+d=0$ represent a rectangular hyperbola?

I'm told a curve of this form $$axy+bx+cy+d=0$$ is a rectangular hyperbola. But I don't seem to understand where are it's $x^2$ and $y^2$ terms since $x^2-y^2=a$ is also a rectangular hyperbola. Sorry ...
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1answer
36 views

How do find if (1,2) lies in between the acute or obtuse region of two lines .

The two lines are $\sqrt{3}x-y+5=0$ and $\sqrt{3}x+y-1=0$. How will I find in which region the point lies: the acute region or the obtuse region? I feel this can't be done if anyone could suggest a ...
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On the parabola y^2=4ax, 3 points E,F,G are taken so that their ordinates are in G.P.

Prove that the tangents at E and G intersect on the ordinate of F. This was my approach: Taking the three points as: E$(at_1^2,2at_1)$ , F$(at_2^2,2at_2)$, G$(at_3^2,2at_3)$ Using the G.P. condition ...
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9 views

Transform measurements in spherical coordinates to new basis with an arbitrary axis

I'll use the Earth to illustrate my question: There's a point $P$ on the surface of the Earth with coordinates $P(r, \theta, \phi)$ where $r\geq0, \quad 0\leq\theta\lt180, \quad 0\leq\phi\lt360$. $r$ ...
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How to calculate start angle and end angle in 3D ARC for DXF file?

Not sure if this is the right place to ask to this question, but I can see so many DXF queries are resolved here. I have three 3D points (P1 <x1,y1,z1> ; P2 <x2,y2,z2> ; P3 <x3,y3,z3>...
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Having problems creating an original coordinate system. Can it be done?

I have a problem that someone with more experiece and knowledge of differential geometry can hopfully solve. While studying a problem I was working on, I encountered a specific "manifold" in ...
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8 views

How to define a modified coordinate space given an arbitrary starting function?

CLARIFYING IMAGE How would I go about transforming a rectangular space (with some simply connected, arbitrary function f(x,y)) into an undefined, unique space where f(x,y) is transformed from an ...
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Finding the Projection Plane of a Dimensionality Reduction that uses an unknown Transformation Matrix.

Say I have a set of data points, and applied some form of linear dimensionality reduction. I know that this projection is linear, so there must be a projection matrix unknown to me. Question : How do ...
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85 views

Number of non-intersecting paths for two bodies on a Cartesian grid

Each body is allowed a jump of $+1 \uparrow$ or $+1 \rightarrow$ in one step. There are $n$ steps available to each body. Find the number of non-intersecting path pairs between a fixed source and a ...
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1answer
33 views

How to find the value of L in the figure? [closed]

How do I find the value of L in the problem below?
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28 views

How do you find the intersection of these two trig curves?

How do you find the intersection of these two trig curves? $ \cos(x_1) + \cos(x_2) + \cos(x_3) = 1 \\ \sin(x_1) + \sin(x_2) + \sin(x_3) = 0 $ Alternatively, how would you find a general solution for $...
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1answer
32 views

Difference in values of centroid for a Kite with different approaches

Suppose a Kite with coordinates as (0.5,0.5), (1/3,1/3), (0.5, 0) and (1,0) Centroid abscissa=1/4*(0.5+1/3+0.5+1)=7/12 Consider 2 triangles with coordinates ...
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21 views

Meaningful or coincidental distance formula relationship to intersections of lines in circles?

Consider two lines, written as $y=m_1x+b_1$ and $y=m_2x+b_2$; their $y$-intercepts must be between $-1$ and 1, inclusive ($-1\le b_1 \le 1$ and $-1\le b_2 \le 1$). Thus, their intersection may fall ...
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1answer
26 views

How to find the coordinates with the given data

In the following figure, Is there a way to calculate the coordinates of $X$ if we know the coordinates of $A$, $B$ and $C$ and the distances $AX$, $AB$ and $AC$?
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2answers
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prove : The tangent to the curve ${x}^3+{y}^3=3axy$ at $(\frac{3a}{2},\frac{3a}{2})$ makes obtuse angle with the positive direction of $x$ axis?

I have differentiated the equation from both side and put the value of coordinates given. Is there any better way? Can anyone help me to solve the problem?
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Been trying to convert absolute co-ordinate alterations to relative ones based on current Euler angles. Can anyone give me a good method? Thanks

I've spent weeks searching the internet and experimenting to find a method for changing the position of XYZ co-ordinates that are relative to an object's Euler angles (Yaw, Pitch, Roll), but in an ...
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2answers
43 views

Find location of a point on an oval shape

I have an oval shape (let's assume the center is (0,0)), and I have two laser pointers (at known locations) pointing to the two different objects (marked as X). One object is on the perimeter, the ...
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8 views

Solving the Kepler Problem With Cartesian Coordinates?

if we look at the Lagrangian for the Kepler problem in cartesian coordinates, we obtain sth like $$\mathcal L\left( \vec X, \vec x, \dot{\vec{X}}, \dot{\vec{x}} \right) = \frac{M}{2} \dot{\vec{X}}^2 + ...
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0answers
45 views

Real world coordinate tracking problem between positive/negative infinity.

I am an engineer working on a steering system with many wheels. In order to steer correctly, the wheels must all be oriented perpendicular to a single point. So if the wheels were in a tight turn, ...
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1answer
31 views

Prove the formula for Jacobian determinant for this coordinate transform for general $n$

Let $n$ be some integer $n \geq 2$, $(x_1, \cdots, x_n)$ are the old coordinates and $(r, \phi_1, \phi_2, \cdots, \phi_{n-1})$ are some new coordinates. I want to transform $(x_1, \cdots, x_n)$ to $(r,...
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1answer
18 views

Get Point on the Perimeter of a Rectangle from Topleft Corner

I stumbled upon the need to get a random point on the perimeter of a rectangle while making a game, the solution programmatically is easy enough, tho i was curious of a mathematical solution too. Say ...
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1answer
41 views

Map a 3D point into its corresponding 2D coordinates on a plane

My problem is the following: I have three 3D points, having coordinates $P_0 = (x_{p0}, y_{p0}, z_{p0})$, $P_1 = (x_{p1}, y_{p1}, z_{p1})$, $P_2 = (x_{p2}, y_{p2}, z_{p2})$. I know the 2D coordinates ...
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29 views

Is there a coordinate geometry calculator

I am looking for a calculator where I can input the coordinates of some points, connect some points, and find the intersection of lines (as coordinates). Does something like this exist? Ideally, it ...
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1answer
36 views

Is this definition of distance between two points correct?

I came across this definition but it's confusing to me. Shouldn't $A$ be $(x_1,y_1)$ instead of $(x_1,x_2)$? Same for $Y$. And even if $A$ was correct like that then shouldn't the formula for the ...
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1answer
35 views

Explicitly solving for inverse function

Consider a mapping $f:\mathbb{R}^2\to \mathbb{R}^2$ given by $$ f(x,y) = \left(-x+\sqrt{x^2+y^2},-x-\sqrt{x^2+y^2}\right). $$ I want to find an explicit inverse mapping, on the neighborhoods for which ...
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9 views

Reforming coordinate data from axis histograms?

So, I have a set of ~2 million datapoints from a microscopy image that I have plotted in R, and then used histogram analysis on the x- and y-axes in order to detect bands of datapoints (it's a very ...
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1answer
27 views

Finding coordinate expressions for the Euclidean metric in $\mathbb{R}^3 $ and spherical coordinates

I am trying to get a better sense of pullbacks/tensors geometrically by looking at the problem of finding coordinate expressions for the Euclidean metric in $\mathbb{R}^3 $ and spherical coordinates. ...
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19 views

Tensor identity in orthogonal coordinate systems

In Riley, Hobson & Bence's "Mathematical methods for physics and engineering" third edition, in the chapter about tensors, one of the exercises involves finding the expression for the ...
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1answer
40 views

Is it possible to draw an ellipse given only the perimeter point tangent, and the distance & angles between that and one of the foci?

Full disclosure, I am a complete novice when it comes to this so if my terminology is off or I don't know some relatively basic things, that will be why. I am currently trying to determine the rest of ...
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1answer
32 views

Find Reflection of A point with respect to a line mirror in 3D

I need to find the reflection of point $P(1,2,3)$ w.r.t line mirror $(x-1)/2 =(y-1)/3 = (z+1)/1$ I know one method to do it i.e by first finding the foot of perpendicular of P on the line by using ...
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0answers
30 views

Intuition vs calculus: average of angles in range.

Let's say I want to find some kind of average of angles $OPR$, where $O$ and $P$ are constant points and $R$ linearly changes in range $<O, E>$ ($E$ is also constant). For simplicity let's start ...
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2answers
15 views

Partial derivatives and Galilean transforms

I have trouble understanding the following: \begin{equation} \frac{\partial}{\partial x'} = \frac {\partial x}{\partial x'} \frac{\partial}{\partial x} + \frac{\partial t}{\partial x'} \frac{\partial}...
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7 views

Decomposing Circumferential strain to Radial and Tangential coordinates

I am currently working on a problem where a cylindrical sample is being compressed (say in the z direction, vertically) causing to increase in circumference (in x-y plane). I have a gauge which ...

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