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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57 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
6
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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 ...
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4answers
592 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 ...
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2answers
30 views

Find the number of possible points $R$.

$P(3,1),Q(6,5)$ and $R(x,y)$ are three points such that the angle $\angle PRQ=90^{\circ}$ and the area of the triangle $\triangle PRQ=7$.The number of such points $R$ that are possible is . $a.)\ ...
6
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1answer
71 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 ...
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2answers
44 views

Find the equation of line in new co-ordinate system.

A line is represented by equation $4x+5y=6$ in the co-ordinate system with the origin $(0,0)$.You are required to find the equation of the straight line perpendicular to this line that passes ...
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0answers
20 views

Find the length of the intercept cut by the side $BC$ on the y-axis .

The equation of two equal sides $AB$ and $AC$ of isosceles triangle $\triangle ABC$ are $x+y=5$ and $7x-y=3$ respectively.What will be the length of the intercept cut by side $BC$ on the y-axis? ...
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2answers
33 views

Find the area of $\triangle POQ$ .

If $P$ and $Q$ are two points on the line $3x+4y=-15$ such that $OP=OQ=9$, then the area of $\triangle POQ$ will be ? $\color{green}{a.)18\sqrt2}\\ b.)3\sqrt2\\ c.)6\sqrt2\\ d.)15\sqrt2$ The ...
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3answers
31 views

Find the relationship between $a$ and $b$.

If the medians $PT$ and $RS$ of a triangle with vertices $P(0,b),Q(0,0)\ \text{and}\ R(a,0)$ are perpendicular to each other,which of the following satisfies the relationship between $a$ and $b$? ...
2
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1answer
9 views

Compound map in manifolds

In the description of a manifold, we often start with the mathematical definition that $M=\cup M_i$ and if $m\in M_i \subset M$, where m is a point on the manifold, then it is mapped by a one-to-one ...
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4answers
188 views

What will be the equation of side $BC$.

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively . What will be the equation of the side $BC$ if the area of the triangle ...
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2answers
21 views

Which of the following point is outside the triangle?

If $P(6,7),Q(2,3)\ \text{and}\ R(4,-2)$ be the vertices of the triangle , then which of the point is not contained in the triangle? $a.)(4,3)\quad \quad \quad \quad b.)(3,3)\\ c.)(4,2)\quad ...
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0answers
20 views

What's the relation between 2 points from 2 different planes?

I'm trying to find the relation between my "text" objects, and my "world" objects. This may be related to development, but I thought this question was better fit for this exchange. I have two ...
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0answers
15 views

Definition of curvilinear coordinates?

Please can someone give me a formal definition of curvilinear coordinates, preferably with as source. The once that I have found don't seem to be very formal.
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1answer
53 views

Conversion from 2-dimensional parabolic coordinates to cartesian and cylindrical

I have been looking at the Wolfram Mathworld page on parabolic coordinates here: http://mathworld.wolfram.com/ParabolicCoordinates.html and I'm having trouble grasping how to convert between parabolic ...
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0answers
39 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
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1answer
13 views

Scalar functions and manifolds

This paragraph is taken from Supergravity book by Freedman and Van Proeyen.he simplest objects to define on a manifold $M$ are scalar functions $f$ that map $M \rightarrow \mathbb{R}$. We say that ...
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0answers
21 views

Change of Basis Matrix: Cartesian to Spherical Laplacian

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of ...
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1answer
265 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
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0answers
38 views

Transforming 2d coordinate system while keeping points placement

I don't know if my question is too simple. Hopefully I can make it clear. I want to find a matrix that I could multiply point with in one coordinate system into another. The first coordinate system ...
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0answers
30 views

Finding bounds of integration

Let $S$ be the region in the first quadrant of the $xy$-plane bounded by the $x$-axis and the parabolas $$x=1-\dfrac{1}{4}y^2,$$ $$x=\dfrac{1}{4}y^2-1$$ and $$x=4-\dfrac{1}{16}y^2.$$ Use the ...
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1answer
13 views

Change of Coordinates matrix.

If Q is the change of coordinates matrix From some basis B to B', then Q inverse is the change of coordinates matrix from B' to B? Is this true? I think/ know it is the, but don't know how to prove ...
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0answers
20 views

Build an orthogonal coordinate transformation?

I am supposed to build an orthogonal coordinate transformation $Q:G\to \Omega$ whose inverse is not orthogonal $Q^{-1}:\Omega\to G$. I am allowed to make the coordinate transformation as simple as ...
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2answers
61 views

find which two points an arbitrary point is nearest to

I have a line segment of connected points (a path in 2D), and a point $P$ that is not calculated based on this segment, although I can guarantee that the point will be placed along the path. Based on ...
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0answers
15 views

How to find the value of standard coordinate frame in a new coordinate frame?

I have a custom coordinate frame which has T as a point and A, B, C are three orthogonally normalized vectors whose coordinates are T = [Xt Yt Zt], A = [Xa Ya Za], B = [Xb, Yb, Zb] and C = ...
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0answers
30 views

Realistic Bounce (Using Trig?)

background: I am making a graphics program where the major purpose of it is to have a ball (traveling on an arbitrary slope) to bounce realistically off of a line (which is also at a arbitrary slope). ...
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0answers
10 views

How can I find the domain of this diffeomorphism (coordinate transformation)?

I have been struggling with this coordinate transformation in $R^2$. $Q:\begin{bmatrix}\rho\\\phi\end{bmatrix}\to \begin{bmatrix}\cosh(\rho)cos(\phi)\\sinh(\rho)sin(\phi)\end{bmatrix}$ I am ...
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0answers
41 views

Problem in calculating angles of a tringle with co-ordinate geometry

There are three equations. $$(a+b)x+(a-b)y-2ab = 0 \tag1$$ $$(a-b)x+(a+b)y-2ab = 0 \tag2$$ $$x+y = 0 \tag3$$ The question is, So that the triangle formed by these equations is an isosceles ...
0
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1answer
29 views

Find the tangents to circle

Let $ \Gamma : x^{2} + y^{2} - 6x - 4y + 8 = 0 $ be a circle. Find the equations of the tangents to $ \Gamma $ which pass through $ D(8, 7) $.
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33 views

Question about the following coordinate transformation.

$Q:\begin{bmatrix}\rho\\\phi\end{bmatrix} $$\to$ $\begin{bmatrix}cosh(\rho)cos(\phi)\\ sinh(\rho)sin(\phi)\end{bmatrix}$ The task is to pick a domain as big as possible so that Q is a ...
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0answers
35 views

transform orthonormal coordinate system to another

I have one orthonormal coordinate system ABC that it's origin is the point p0. I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1. I know how to ...
2
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2answers
59 views

Check if point is within quadratic area.

I have two dimensional Cartesian system and I need to check if certain point is within the geometry formed by four other points. I have few thousands different geometries all with different points to ...
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0answers
150 views

Converting fractional coordinates to cartesian coordinates

I have a set of fractional coordinates with the following vectors: 2.950 -1.475 0.0000 0.000 2.5547 0.0000 0.000 0.0000 77.5379 with the ...
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0answers
16 views

Computing local coordinates

Let $p=[x_0,y_0,1]\in \mathbb P_2(k)$ (projective space). Determine a projective transformation $\phi\in GL(3,k)$ such that $\phi(p)=[0,0,1]$ and name the coordinates explicit. Its easy to see that ...
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0answers
57 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
5
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2answers
102 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 ...
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0answers
21 views

Education tool for learning 3D angles

I hope it is not an off-topic. I have started working on 3D frame transformation and transforming a vector such as acceleration or angular velocity from one coordination to earth coordination. My ...
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3answers
36 views

How to get the coordinates of the point of a line segment

I have the coordinates of the 2 end points a and b thanks to which I can draw a line. How can I get the coordinates of all the points of the segment of line starting from point a to point b ?
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1answer
78 views

Torus coordinates and triple integration

Torus coordinates are given by $$x=(R+r \cos\theta)\cos\phi,\space y=(R+r \cos\theta)\sin\phi,\space z=r\sin\theta,$$ with $R>0$. A given torus $T_b$ is defined by $$T_b=\{(r,\theta,\phi):0\leq ...
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0answers
40 views

Calculate point's coordinates relative to rotation in 3D-space

I have a point "A" in 3D-space, let's say in coordinates (2, 3, -1). Then there's a point "B" which position is A + (-1, 2, -1), so now it's ...
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0answers
36 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
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1answer
38 views

Transform vector from xy plane to “another vector's plane”

I've not had linear algebra yet, so bear with me if I write something weird. Given vector A and B as shown above, how do I transform vector B so that one of its components is parallel to A, and the ...
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0answers
13 views

Transforming coordinates according to Point of View.

Imagine I have two cameras (A and B) in the same axis and looking at the same white wall with a black dot. Both cameras see the black dot. I'd like to know how to proceed in order to derive the black ...
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1answer
60 views

Coordinate systems in physics problem (damped pendulum)

A simple pendulum of lenght $\ell$ is vibrating freely (with small oscillations). The pendulum is set into forced vibration by moving its point of suspension horizontally with a Simple Harmonic ...
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1answer
89 views

Using trigonometry to predict future position

Intro I'm currently creating an AI for a robot whose aim is to shoot another robot. All I want to do is to be able to calculate at what angle to shoot my bullet, so that it hits my enemy, with the ...
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2answers
26 views

How to calculate angles and areas (circles)- AS Maths

Hi here's the question: A(-1,-4) and B(6,-5) are points on the circumference of a circle, centre D(3,-1). The tangents at A and B intersect at C. How would I find the angle ACB and the area of ACBD? ...
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0answers
37 views

Using azimuthal and polar angles in ECEF coordinate system

I have a physical cone, which its vertex located in some point (x,y,z) in ECEF coordinates, and I want to check if another point is inside this cone. In order to do it, I have to take into ...
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1answer
37 views

Benefit from local coordinates

I am reading Elliptic Curves by Anthony Knapp. Its the first time that I am dealing with local coordinates. In page 21 he introduces them as follows: Let $[x_0,y_0,w_0]\in \mathbb P_2(k)$ where $k$ ...
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1answer
45 views

Clamp Distance Between Two Vectors

I was wondering if there is a formula to clamp distance between two Vectors. Let me elaborate. I have two Vectors, say, $V_1(x_1,y_1)$ and $V_2(x_2,y_2)$. I can find the distance '$d$' between them ...
3
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1answer
32 views

convert coordinates from non-orthogonal axes to cartesian in 3-dimensions

I'm having issues converting from a non-orthogonal coordinate space (abc) to a cartesian space (xyz). I think I may have just thought myself into a corner, and I can't get out. The systems have a ...