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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14
votes
2answers
193 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 ...
10
votes
1answer
756 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 ...
9
votes
2answers
164 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
423 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
2answers
2k 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?
8
votes
3answers
118 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 ...
7
votes
0answers
426 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 ...
6
votes
4answers
4k 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 ...
6
votes
1answer
278 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
1answer
131 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
2answers
282 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
2answers
121 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
132 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
936 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
1answer
119 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
172 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
131 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
0answers
170 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 ...
4
votes
1answer
178 views

Number of integer solutions of $xy - 6 (x+y)=0$

What are the number of integer solutions of $xy - 6 (x+y)=0$ with $x\leq y$ is ? Equation $xy - 6 (x+y)=0$ can also be written as $1/x + 1/y = 1/6$
4
votes
1answer
178 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 ...
4
votes
2answers
327 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 ...
4
votes
1answer
49 views

What is the name of two points that share one coordinate?

Is there an adjective to characterize two points in $\mathbb R^2$ that have the same value for one of the coordinates?
4
votes
1answer
3k 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 ...
4
votes
1answer
98 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
4
votes
1answer
30 views

Family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$

Problem : If $\sin(\alpha + \beta)\sin(\alpha -\beta) =\sin\gamma (2\sin\beta +\sin\gamma), 0 < \alpha , \beta ,\gamma <\pi$ then the family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$ ...
4
votes
2answers
143 views

Coordinates of parallel triangle with a distance of 'd' between the parallel edges?

I have a triangle with Co-ordinates $\{(x_1,y_1),(x_2,y_2),(x_3,y_3)\}$. I need to find co-ordinates of a triangle,whose edges are exactly $\alpha$ distance from previous triangle. Below is the figure ...
4
votes
3answers
277 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 ...
4
votes
2answers
358 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 ...
4
votes
1answer
54 views

Find the maximum length of a line segment enclosed in a given area

$A = \{ (x, y) : x = u + v , y = v , (u^2) + (v^2) \le 1 \}$ . Then what is the maximum length of a line segment enclosed in this area? My friend suggested the answer $\sqrt{5}$, but I think it ...
4
votes
2answers
842 views

derivatives transformation

I'm currently doing a calculation for the connection coefficients using the standard space-time coordinates, namely x[0],x[1],x[2],x[3]. The setup is a spherically symmetric problem. In my ...
4
votes
2answers
358 views

How to calculate x,y position of 3D points?

I have points in 3D system like this $$p1=(2,3,4)$$ $$p2=(3,5,5)$$ Here I would like draw point $p1$ and $p2$ in $2D$ view. Project type = orthographic. Coordinate system = Cartesian X- axis, ...
4
votes
1answer
85 views

Calculate the X,Y values of an ellipse

I guess am confused somewhere. I have the length(l) and breadth(b) of an ellipse enclosing rectangle. I know the center point and the angle(x) that the line makes with the center. I want to know the ...
4
votes
0answers
41 views

Solving a system of equations

I'm trying to prove the existence of a solution to the system of equations $$c_i = \gamma x_i + (1-\gamma) \frac{x_i^2}{\sum_{j=1}^\infty x_j}$$ for $i\in\{1,2,....\}$ where $\sum c_i=1$. I am also ...
4
votes
0answers
186 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
3
votes
3answers
255 views

What is wrong with this method for a rotated and shifted parabola?

$(x+2y)^2=4(x-y)$ Disecting the above parabola is the question. (vertex, axis,tangent at vertex,etc). So at first what I thought of was making its equations at LHS and RHS perpendicular. I thought ...
3
votes
3answers
468 views

Tetrahedron problem (proving)

Prove that if $P$ is the intersection of the altitudes of a tetrahedron $ABCD$ and $r$ is the circumradius then $PA^2+PB^2+PC^2+PD^2=4\cdot r^2$.
3
votes
4answers
1k views

Why do we draw the $xyz$ coordinate system like this?

Usually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: Imagine I actually got three ...
3
votes
2answers
387 views

Quick question regarding coordinate geometry

Note: My exam is in about 1 hour and i just realized that i have a unsolved paper, this is one of the questions that i wasn't able to answer from it. I would highly appreciate it if a full explanation ...
3
votes
3answers
5k 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?
3
votes
3answers
182 views

Two questions for coordinate geometry

Note: I am burning through dozens of questions from sample papers and these i couldnt understand, these are not homework and i would appreciate it if the full answer could be provided. The first ...
3
votes
4answers
759 views

Position of a point with fixed distance between other two points

I have two points, $p_1$ and $p_2$, in a cartesian plane, and a fixed radius, $r$. I want to find the coordinates of another point, $p_3$, that is in the same line of the $p_1$ and $p_2$, and always ...
3
votes
1answer
74 views

Given a set of 2D points (x,y) (cloud of points), find the points that, when connected, will contain all other points

Given a set of 2D points I have to find the points that when connected will form a polygon that contains all the points in the set. A quick example: imagine you have a set ...
3
votes
2answers
123 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
3
votes
2answers
393 views

Coordinates and distance in higher dimensional spherical and hyperbolic space

For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
3
votes
2answers
147 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
3
votes
1answer
1k views

Calculate Camera Pitch & Yaw To Face Point

How do you calculate pitch & yaw for a camera so that it faces a certain 3D point? Variables Camera X, Y, Z Point X, Y, Z Current Half Solution Currently I know how to calculate the pitch, ...
3
votes
3answers
3k views

Finding the coordinates of an unknown point.

I have 10 points on a 2D plane where I know the $(x,y)$ coordinates of 9 of the points. For 1 point, $p$, I do not know its location. Additionally, I have the distances from each of the known 9 points ...
3
votes
1answer
2k views

rotating 2D coordinates

I've tried googling this, but I always end up somewhere that just says it's easy. Anyhow, I have a coordinate system, where I need to rotate a bunch of points. It's all 2D. Coordinates varies and so ...
3
votes
2answers
55 views

Can a straight line be drawn from origin to co-ordinate X,Y?

Given a co-ordinate P(X,Y), can a straight line be drawn from origin to P, if there is wall existing with end points A(X1,Y1) and B(X2,Y2)? My Approach: I first of all wrote the equations from origin ...
3
votes
1answer
47 views

Cauchy-Riemann equations in arbitrary coordinates?

The CR equations in rectangular coordinates can be written as one equation in the following way: $$\frac{\partial f}{\partial x} = \frac{1}{i} \frac{\partial f}{\partial y}$$ Likewise, in polar ...