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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20
votes
1answer
469 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
19
votes
7answers
48k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
16
votes
1answer
1k 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.
14
votes
2answers
226 views

Was there some prior idea that inspired both Fermat & Descartes to invent coordinates?

It seems incredible to me that both Descartes & Fermat could have both simultaneously discovered such a novel & significant idea, without there being some single prior idea that they both ...
13
votes
1answer
1k views

Conditions for two straight lines to intersect: is this exam question wrong?

I am pretty sure this question (from a university admission test exam) is wrong. Two lines: $a_1x+b_1y+c_1=0$, $a_2x+b_2y+c_2=0$, intersect only if (a) $a_1a_2-b_1b_2=0\;\;\;$ (b) ...
11
votes
4answers
5k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
11
votes
2answers
6k views

Plotting in the Complex Plane

I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$ What is the difference plotting this function in the complex plane or real plane?
11
votes
1answer
1k views

Can someone please explain the cube to sphere mapping formula to me?

I am wondering if anyone could explain how the following formula works, it is supposed to take the input as a point on a cube then map that to points on a sphere, please go gentle on me, I'm in 9th ...
10
votes
5answers
2k views

Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...
9
votes
4answers
691 views

Check if a point is inside a rectangle (not knowing the coordinates, but knowing distances to vertices)

I have to solve the following problem: I have 4 points (A, B, C, D) which form a rectangle, but I do not know their coordinates. I have another point (X), I do not know its coordinates either, but I ...
9
votes
2answers
240 views

Is there any way to find the equation for this situation?

$\mathbb N = \{1,2,3,4\dots\}$ Let us suppose we are starting at a point with coordinates $(0,0)$. Now draw a line from $(0,0)$ to $(1,0)$ and from $(1,0)$ to $(1,2)$. Now by the Pythagorean theorem, ...
8
votes
4answers
810 views

Is there any good reason why a protractor starts from right to left, unlike a scale, which starts from left to right?

While studying through the number system, i notice that positive side is from 0 to +ve infinity. The direction is left to right. However, this is opposite in case of angles. The sort of curved number ...
8
votes
3answers
3k views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
8
votes
1answer
641 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
7
votes
2answers
589 views

How do I convert a vector field in Cartesian coordinates to spherical coordinates?

I have a vector field in terms of $\mathbf{\hat i}$, $\mathbf{\hat j}$, and $\mathbf{\hat k}$, $$\mathbf{F} = x\mathbf{\hat i} + y\mathbf{\hat j} + z\mathbf{\hat k}$$ How do I convert it to the ...
7
votes
1answer
419 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
6
votes
5answers
750 views

Basic understanding of a metric.

What is a metric ? Do a metric depend on the system of coordinates I use ? Does it depend on surfaces (or higher dimensional manifolds. Correct me if I'm wrong using the word) the coordinate frames ...
6
votes
1answer
1k views

What is the name of this paradox?

What is the name of the mathematical paradox which is arises from the following? If we imagine a point on a two-dimensional coordinate system (line graph), which moves from the positive part of the ...
6
votes
1answer
2k views

Jacobian for a Cartesian to Polar-Coordinate Transformation

I have a simple doubt about the Jacobian and substitutions of the variables in the integral. suppose I have substituted $x=r \cos\theta$ and $y=r \sin\theta$ in an integral to go from cartesian to ...
6
votes
2answers
166 views

Number of solutions of a simple equation

Problem How to count the number of distinct integer solutions $(x_1,x_2,\dots,x_n)$ of the equations like : $$|x_1| + |x_2| + \cdots + |x_n| = d $$ the count gives the number of coordinate points ...
6
votes
3answers
92 views

Why is $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$?

I have to show the identity I wrote in the title: it should be $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$ but some computation ...
6
votes
1answer
101 views

Is this GRE math problem wrong?

I'm working out of the Manhattan GRE test prep book and I've come across a question that I can't figure out why they chose the answer they did. "Perpendicular lines m and n intersect at point (a,b), ...
6
votes
1answer
85 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
6
votes
1answer
134 views

Transforming a PDE given basis vectors

I have a non-orthogonal coordinate system defined by $\mathbf x=\mathbf x(r,\beta,z)$, and so I can find the basis vectors as $$ \mathbf g_r=\frac{\partial \mathbf x}{\partial r},\quad\mathbf ...
6
votes
1answer
230 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
6
votes
1answer
642 views

Minimum number of lines covering n points

Let there be n points in the plane. I want to know the minimum number of horizontal and vertical lines covering all the points in the plane. My initial approach started like this, 1) for each point I ...
6
votes
0answers
205 views

Physical components of a third-order tensor

Aris' book Vectors, Tensors, and the Basic Equations of Fluid Mechanics describes how to convert between covariant, contravariant, and physical components of vectors and tensors. For example, in ...
6
votes
1answer
3k views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...
6
votes
1answer
200 views

What do we know about non linear coordinate systems?

I first learned about coordinate systems by Gelfand and I knew that we basically have two axis x and y with origin O and some unit vectors $\hat i$ and $\hat j$ and if $\vec{OA}=x\hat i+y\hat j$ then ...
5
votes
3answers
9k views

Convert coordinates from Cartesian system to non-orthogonal axes

I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?
5
votes
2answers
392 views

Question about the nature of coordinate systems

I couldn't really think of a good one line title for my question, so I will try to elaborate. From what I have sort of gathered over my years, if you want to locate an arbitrary point in an ...
5
votes
1answer
295 views

Are spherical coordinates unique orthogonal coordinates on sphere?

Spherical coordinates on unit sphere are defined by the following transformation: $$\begin{cases}x=\sin\theta\cos\varphi\\ y=\sin\theta\sin\varphi\\ z=\cos\theta\end{cases}$$ Are these coordinates ...
5
votes
3answers
176 views

If ABCD is a square with A (0,0), C (2,2). If M is the mid point of AB and P is a variable point of CB, find the smallest value of DP+PM.

I assumed the coordinates of P = (h,2) to get the value of DP+PM= $\sqrt { (h-2)^2 +4}+\sqrt{h^2+1}$. Then I differentiated the equation wrt to h to get: $h(\sqrt{h^2+1}) -2\sqrt{h^2+1}+ ...
5
votes
3answers
629 views

On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?

I don't know exactly how to describe it, but in a programmatic way I want to spiral outward from coordinates 0,0 to infinity (for practical purposes, though really I only need to go to about ...
5
votes
2answers
130 views

Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?

I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are ...
5
votes
3answers
251 views

Alternative form of equation of circle?

In a problem set I was solving, one of the solutions used the equation of a circle in the form $$(x-h)^2 + (y-k)^2 + \lambda(ax + by +c) = 0$$ where, $(h,k)$ is any point on the circle $ax+by+c ...
5
votes
1answer
6k views

How to find an end point of an arc given another end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I calculate the other ...
5
votes
2answers
79 views

Finding equation of chord of hyperbola.

Equation of chord of hyperbola joining points $(a\sec\phi,b\tan\phi)$ and $(a\sec\phi_1,b\tan\phi_1) $ $$y-b\tan\phi=\frac{b\tan\phi-b\tan\phi_1}{a\sec\phi-asec \phi_1}(x-a\sec\phi) $$ This reduces ...
5
votes
3answers
683 views

move a point up and down along a sphere

I have a problem where i have a sphere and 1 point that can be anywhere on that sphere's surface. The Sphere is at the center point (0,0,0). I now need to get 2 new points, 1 just a little below the ...
5
votes
1answer
1k views

Calculate average latitude / longitude

I have the array of geographical coordinates (latitude & longitude). What is the best way to calculate average latitude and longitude? Thanks!
5
votes
2answers
149 views

under what conditions can orthogonal vector fields make curvilinear coordinate system?

I am considering n-dimensional Euclidean space $\mathbb{R}^n$. For any $x\in\mathbb{R}^n$, $v_1(x), \cdots, v_n(x)$ are orthogonal vectors. As functions of $x$, $v_i$'s are differentiable and non-zero ...
5
votes
2answers
526 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
5
votes
1answer
241 views

Using paraboloidal coordinates

I have the 3-dimensional paraboloidal coordinates $$s_{\pm}=\sqrt{x^2+y^2+z^2}\pm z$$ $$\phi=ArcTan(y/x)$$ with the inverse transformation $$x=\sqrt{s_+ \cdot s_-}\cdot cos(\phi)$$ $$y=\sqrt{s_+ ...
5
votes
1answer
508 views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
5
votes
1answer
175 views

Coordinate Transformations

I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds : Rotation of coordinate (and ...
5
votes
1answer
81 views

Evaluate integral with gaussian curvature

I thought evaluating it in the following way: $$\begin{align} \int_0^{2\pi}\int_0^{\pi}K(x,y)\sqrt{\det(g_{ij})} \, dy\,dx &= \int_0^{2\pi}\int_0^\pi \sqrt{\det L_{ij}}\cdot \sqrt{{\frac{\det ...
5
votes
2answers
628 views

Cylindrical coordinates on elliptic paraboloids.

I want to compute the volume bounded by: the cylinder $x^2+4y^2=4$. the $z=0$ plane. the elliptic paraboloid $z = x^2 + 6y^2$. I would like to use cylindrical coordinates. However I have never ...
5
votes
0answers
123 views

Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
5
votes
0answers
105 views

Elastic wave equation in curvilinear coordinates: how do you perform a coordinate change?

The essence of this question is that I don't know how to convert an equation from Cartesian coordinates into curvilinear coordinates, and would like to know how, preferably using the language of ...
4
votes
2answers
272 views

Why don't global coordinates always exist for a manifold?

Let $M$ be a manifold and $(\phi,U)$ a patch. Then $\phi(P)=\bar{x}=\begin{bmatrix} x^1\\ x^2\\ \vdots\\ x^n \end{bmatrix}$ for each $P$ in $U$. But each $P$ in $M$ is in some patch, so this ...