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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1answer
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Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ($\vec{e_1'},\vec{e_2'},\vec{...
9
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3answers
4k 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 ...
3
votes
2answers
318 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 ...
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vote
2answers
100 views

Calculate Point Coordinates

As you can see, In the image a rectangle gets translated to another position in the coordinates System. The origin Coordinates are A1(8,2) B1(9,3) from the length <...
2
votes
1answer
180 views

Example of a Problem Made Easier with Skew Coordinates

Skew or oblique coordinate systems are coordinate systems where the angle between the axes is not 90 degrees. The second answer to this question has formulas to convert between these systems with an ...
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7answers
61k 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 ...
2
votes
1answer
4k views

Transforming from one spherical coordinate system to another

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
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1answer
343 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
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votes
3answers
77 views

There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$

There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$. A point $P$ satisfies following condition: The straight line passing through $P$ and dividing the area of square $Q$ in the ...
4
votes
1answer
231 views

Transformations from n-sphere coordinates to cartesian coordinates.

I was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $. For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can ...
3
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1answer
5k 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, ...
5
votes
3answers
10k 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
4answers
1k 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
2answers
6k views

How do we prove the rotation matrix in two dimensions not by casework?

I was trying to prove: To carry out a rotation using matrices the point $(x, y)$ to be rotated from the angle, $θ$, where $(x′, y′)$ are the co-ordinates of the point after rotation, and the formulae ...
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2answers
4k views

Number of ways of reaching a point from origin [duplicate]

Possible Duplicate: How can I find the number of the shortest paths between two points on a 2D lattice grid? If we have a point p(x,y) in coordinate system [x>=0, y>=0; i.e 1st quadrant] How ...
3
votes
3answers
508 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...
2
votes
2answers
562 views

How to use a Rhumb Line?

I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
2
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0answers
90 views

Scale factors in cylindrical coordinates - geometrical meaning

I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can ...
1
vote
1answer
134 views

What is the maximum point for which number of way to reach is given

Previous question: link Say there are two points $P_1(a_1,b_1)$ and $P_2(a_2,b_2)$, the number of ways of reaching $P_1$ from the origin is $w_1$ and $P_2$ from $P_1$ is $w_2$. (Here $a_1<a_2$ and ...
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2answers
5k views

How to calculate the middle of a line?

My question is following. I have a line with a given (X1, Y1) and (X2, Y2) coordinates (see figure below). I need to calculate ...
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vote
4answers
187 views

How can I index 2 dimensional points, starting at the origin and going outwards?

Is there any way that I can mathematically determine a unique index number for 2D points that increases the further away I get from the origin? I do not know how far out that this coordinate system ...
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1answer
2k views

Determine whether or not a point lies within a rotated rectangle

I need some maths help for a 2D game I am programming. In this game I have a rectangle, specified by its centers' X and Y coordinates, and its width and height. I then rotate this rectangle via ...
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 ...
5
votes
1answer
190 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
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3answers
188 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}+ h\sqrt{h^2+8-...
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votes
2answers
75 views

Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the given three lines

Problem : Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the lines $11x+6y+14=0$, $9x+y-12=0$, $2x+5y-17=0$ (a) $0$ (b) $2$ (c) $3$ ...
3
votes
1answer
4k 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
436 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 ...
2
votes
1answer
2k views

invariance of cross product under coordinates rotation

Question goes as If $\vec A$ and $\vec B$ are invariant under rotation, the prove that $ \vec A \times \vec B $ is also invariant. However solution of on the other page is not given. Says ...
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1answer
72 views

Book on coordinate transformations

I am looking for a book that covers various coordinate systems in 3 dimensions, various methods of representing rotations and other transformations like rotation matrices and quarternions, including ...
6
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2answers
180 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
0answers
145 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 \left((x_k)^...
5
votes
2answers
566 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
3answers
866 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 +/-100,...
4
votes
5answers
1k views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
3
votes
5answers
3k views

How do I determine if a point is within a rhombus?

I know the coordinates of the 4 rhombus' vertices. I also have the coordinates of another arbitrary point (the result of a click on the screen). How do I determine if that point is within the ...
3
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1answer
1k views

Computing gradient in cylindrical polar coordinates using metric?

I am trying to understand coordinate transformations properly (having studied some general relativity in the past). Let us consider the transformation from cartesian to cylindrical coordinates, $x=\...
2
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1answer
99 views

Pullback metric, coordinate vector fields..

I'm doing this computation on $\mathbb{R}^3$ with cylindrical coordinates $(r, \theta, z)$, (which aren't defined on the whole of $\mathbb{R}^3$, but I don't care about that) and I seem to get a ...
2
votes
1answer
2k views

Find coordinates of equidistant points in Bezier curve

I have to find points (say 10 points) in Bezier curve with 2 control points such that they are at equidistant positions in the curve. Currently I am using the following formula which gives me points ...
2
votes
3answers
59 views

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point $A$...
2
votes
1answer
488 views

conversion of laplacian from cartesian to spherical coordinates

In cartesian coordinates, the Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\qquad(1)$$ If it's converted to spherical ...
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1answer
6k views

How to find coordinates of 3rd vertex of a right angled triangle when everything else is known?

I want to locate precisely the 3rd coordinate of a right angled triangle. I have: the length of three sides The three angles The other two coordinates of the triangle The triangle can lie in any ...
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2answers
183 views

All Intersection points of two spheres having arbitary centres?

I have read much about intersection of two spheres from spheres-intersect , circlesphere and collision-points but all are based on the assumption of spheres located at origin or $x$-axis or some ...
1
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1answer
79 views

Center of Distance

I am given $N$ points in a 2D plane($x$ and $y$ coordinates). I have to find a point in this plane with coordinates $X$ and $Y$ such that: $$\sum_{i=1}^N \max\{|X - A_i|, |Y - B_i|\}\text{ is ...
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2answers
4k views

Is it possible to find function that contains every given point?

Let say we have a arbitrary number of given points and there is at least one function, for which every point lies on its graph. Is it possible to find that function using only X and Y coordinates of ...
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2answers
302 views

In my textbook,the coordinate (x,y) by sine and cosine addition formula seems to form a circle,is that a coincidence?

In my textbook,the coordinate (x,y) by sine and cosine addition formula seems to form a circle,is that a coincidence ?Since the addition formula once was defined by acute case and to prove by geometry ...
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4answers
80 views

A vector should more be thought an identity of an entity in space rathar than magnitude + direction?

Can I say that vector is more like a "unique identity" of an entity in space rather than calling it an entity with magnitude and direction ? For example a line. A vector $(10,10,0)$ is the identity ...
0
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1answer
52 views

Find the circle circumscribing a triangle related to a parabola [duplicate]

Consider the following lines $x-y-1=0$ $x+y-5=0$ $y=4$ The line 1 is the axis of the parabola, the line 2 is the tangent at the vertex to the same parabola, and the line 3 is another tangent to ...
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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 ...
4
votes
1answer
49 views

Formula for the angle of a line $y = mx$ as a function of $m$.

I was wondering if there was a way to calculate the angle made by a line $(\space y=mx)$ in the Cartesian plane using only $m$. I used the Pythagorean theorem in this figure: $$AO= \sqrt{AB^2+OB^2}=\...