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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Euler Angle Transformation from righthanded to lefthanded cartesian coordinate system

I have a righthanded and a lefthanded cartesian coordinate system defined as follows: I have Euler angles (x, y, z) defined in the righthanded system and want to transform them to the lefthanded ...
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33 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 ...
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3answers
91 views
+50

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
327 views

Find parametric expression of an arc given its start point, end point and central angle in 3D cartesian coordinate system

In a 3D cartesian coordinate system, the coordinates of start point and end point have been given as $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$. If the central angle of the two points (the one smaller ...
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2answers
47 views

Fix the radius when drawing a circle.

I found this function and it draws an oval rather than a circle. What do I need to do to fix the calculations to make a circle? Thanks. ...
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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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1answer
6 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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2answers
26 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.)\ ...
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1answer
52 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
31 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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1answer
10 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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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
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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
21 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
169 views

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 ...
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4answers
21k 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 given ...
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2answers
20 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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1answer
23 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
18 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
12 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
652 views

coordinate geometry - find point in right-angled triangle

I'm making a map. I've come across a geometry problem, and I'm not so knowledgeable about maths! Let me illustrate with pictures. I am trying to plot flightpaths with a curved line, using a ...
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0answers
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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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0answers
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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
327 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
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22 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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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
48 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 ...
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0answers
27 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
12 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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17 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
28 views

find which two points an arbitrary point is nearest to

I would like to solve for a point $P$ regarding its proximity to the line segment it resides within. I can make a guarantee that the point will be placed along a line. In the included example, we ...
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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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1answer
387 views

How can I transform coordinate systems with quaternions?

I have a coordinate system 0 which I'd first like to rotate about its z-Axis which gives me system 1, and afterwards rotate system 1 about its y-axis which gives me system 2. See picture: Both ...
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2answers
303 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
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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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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
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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 ...
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1answer
20 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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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 ...
2
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2answers
49 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
16 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 ...
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0answers
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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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1answer
564 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 ...
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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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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 ...
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2answers
59 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
16 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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1answer
47 views

How to obtain a rectangle's side's positions if its origin isn't in its middle?

Basically I have an algorithm which generates rooms and corridors randomly and each time a room is made, a new corridor is placed on $1/4$ of the room's sides and its origin point is set to that ...
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3answers
19 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
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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 ...