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.

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, circles or spheres. Most coordinate systems use two numbers, a coordinate, to identify the location of a point. Each of these numbers indicates the distance between the point and some fixed reference point, called the origin.

A coordinate system consists of four basic elements:

$(1)$ Choice of origin

$(2)$ Choice of axes

$(3)$ Choice of positive direction for each axis

$(4)$ Choice of unit vectors for each axis

Some well-known coordinate system:

  • Cartesian coordinate system
  • Polar coordinate system
  • Spherical coordinate system
  • Cylindrical coordinate system

There are also some other common coordinate systems, for more details you please find https://en.wikipedia.org/wiki/Coordinate_system

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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 given and then apply Heron's formula. Is this the…
TSP1993
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Are rational points dense on every circle in the coordinate plane?

Are rational points dense on every circle in the coordinate plane? First thing first I know that rational points are dense on the unit circle. However, I am not so sure how to show that rational points are not dense on every circle. How would one…
Hidaw
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What is the analogue of spherical coordinates in $n$-dimensions?

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by $$x_1=r \cos \theta$$ $$x_2=r \sin \theta$$ For…
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Painting the plane, and finding points one unit apart

An old (rather easy) contest problem reads as follows: Each point in a plane is painted one of two colors. Prove that there exist two points exactly one unit apart that are the same color. This proof can be easily written by constructing an…
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Hexagon grid coordinate system

If I had a grid of squares, they can be labeled with Cartesian coordinates such that given square $(x,y)$, you know it shares a boundary with squares $(x+1,y),(x-1,y),(x,y+1),(x,y-1)$. Is there a way of labeling a tessellated hexagon grid, so that…
Joshua Kidd
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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 represented by the equations…
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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.
Pervez Alam
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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 spherical coordinate system so that the unit vectors…
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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 +/-100,000:+/-100,000) So if this is my grid: [-1, 1][ 0,…
Jane Panda
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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. What is your limiting position $(x,y)$? This is…
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How to "rotate" points through 90 degree?

I am trying to do some intersection tests and so the math gets weird if two certain points have the same $x$ coordinate and so infinite slope. The points can be anywhere in any quadrant. I want to "rotate" all my points through $90^o$ which will…
KaliMa
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Check if line intersects with circles perimeter

Say I have this circle here. What would be an algorithm to check if lines like the green or blue lines intersect the edge of the circle, but not the red line. So lets say if(amountOfPointsHit(line, circle) > 1) then return true else return…
FabianCook
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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 will also give me some more intuition about the…
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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 polar-coordinate. If I use simple area rule or the…
Sijo Joseph
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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?
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