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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Transformation matrix in polar coordinates

I'm trying to write a software widget that allows the user to resize the component, so I can write a transformation matrix $\mathbf T_\text{xy}$ that will map $(x,y)$ to a transformed $(x',y')$, that ...
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15 views

Diagonal of parallelepiped circumscribed around ellipsoid is constant

There are many rectangular parallelepipeds that can be circumscribed around a given ellipsoid in $\mathbb R^n$. Prove that the length of the main diagonal does not depend on the choice of such ...
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2answers
58 views

Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix.

Parabola having focus $(1,2)$ touches both axes. Find the equation of directrix. As perpendicular tangents meet at directrix, the directrix passes through origin. So the directrix has equation of the ...
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1answer
60 views

Contravariant vector example with polar coordinates

My book gives me this definition for contravariant vector: Let an n-tuple of real numbers $a^1,a^2, \dots, a^n$ be associated with a point P of an n-dimensional Riemannian space with coordinates $u^1,...
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3answers
23 views

What is the general equation of lines going through 'a' particular point?

I want to know the general equation of lines going through a single point where there will be arbitary constants which will change and cause the line to rotate in a circle and consequently the center ...
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1answer
22 views

Why aren't the derivatives of this transformation and its inverse reciprocals of each other?

Consider the coordinate transformation under rotation that \begin{align} y' & = y\cos\phi + z\sin\phi \\ z' & = -y\sin\phi + z\cos\phi , \end{align} we can get \begin{equation} \frac{\partial ...
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1answer
42 views

Changing variables: partial derivatives of a tensor

Given is the tensor $T$ in Cartesian coordinates $T=\operatorname{diag}\{T_{xx},T_{yy},T_{zz}\}$ in cylindrical coordinates $T=\operatorname{diag}\{T_{rr},T_{\theta\theta},T_{zz}\}$ How does one ...
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81 views

Finding the number of normals to a parabola

Find the number of normals to the parabola $y^2=8x$ through (2,1) $$$$ I tried as follows: Any normal to the parabola will be of the form $$y=mx-am^3-2am$$ Since the point (2,1) lies on the ...
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47 views

Prolate spheroidal coordinates and spheroidal harmonics expansion

I recently started to study problems with prolate spheroidal geometries, for which prolate spheroidal coordinates are most suited. In particular I have the advantage that the problem is axisymmetric ...
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22 views

Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$ Taking its derivative ...
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1answer
52 views

Where can I learn about other types of coordinate systems?

As a junior math major the only coordinate systems I've thus far been exposed to are Cartesian and polar (including cylindrical and spherical) coordinates. But of course these aren't the only ones. ...
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41 views

World To Screen Game math

I have a Coordinate system, I have my XYZ, pX,pY,pZ and the other player, eX, eY, eZ and I want the Pitch and YAW First the YAW: I first take VectorX = eX - pX VectorZ = eZ - pZ then I ...
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12 views

Trying to get 2D X angle through 3D

Im trying to make an aimbot for Call of duty 4 for educational purpose I tried this: I declared a few variables enemy coords = eX, eY, eZ Player = pX, pY, pZ Vector length for X, Z I have 4 ...
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1answer
47 views

The arctangent is a strange floating point number

I have 2 players in a game (Cod 4). I read X, Y, Z and store them in: ...
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2answers
27 views

coordinates format to GPS

I was wondering if someone could help me figure out how to go about converting the coordinates on a map I was sent from abroad from the format I see to the usual latitude/longitude I'm used to. (I'm ...
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1answer
27 views

Create unique identifier from close coordinates

Suppose that I have the following list of coordinates: ...
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1answer
34 views

Projection of a plane on coordinate planes

$$\left|\begin{matrix} x & y & z & 1 \\ x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \end{matrix}\right|=0$$ This is the equation ...
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1answer
24 views

How can vector functions define coordinate systems?

I was watching a lecture in which the professor states that by taking the derivative of a vector function, one can find the basis of the coordinate system being studied. My question is, how can a ...
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1answer
73 views

Spherical distance between two points in terms of latitude and longitude

I have seen the answer to this question - Great arc distance between two points on a unit sphere However in a fortran program that I have this is the code to calculate spherical distance between two ...
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0answers
30 views

Equal areas of segments in the lazy caterer problem?

In the book "Build Your Brain Power" by Wootton and Horne, they mention the lazy caterer's problem, asking for a way to cut a circular cake into 8 equally sized pieces with 3 cuts. Clearly since the ...
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1answer
39 views

Actual convention for the term Sinusoidal Phase “Shift” or “Offset”

I have seen the convention for a sinusoid appear as: $x(\theta) = A \cdot \sin( B \cdot (\theta - \phi) \ ) + D$ $y(\theta) = A \cdot \sin( B \cdot (\theta + \phi) \ ) + D$ Is "offset": $\phi$ ...
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1answer
33 views

Forming a line from two points

In the book Multiple View Geometry in Computer Vision, it says "a line is defined by the join of two points" in section 3.2.2 Later on it goes to say there is a point $P^+x$ (the point that projects ...
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1answer
33 views

Do you know the formula?

I have three data: coordinates $(x,y)$ angle in degrees ($\gamma^\circ$) distance in meters (m) How should I calculate in general a new position $(a_1,b_1)$? For example, assume the following ...
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146 views

Wind vector transformation from Gaussian grid to displaced pole grid

I have been given the "u" and "v" component with respect to an earth coordinate reference system(Gaussian grid - https://en.wikipedia.org/wiki/Gaussian_grid http://www.nco.ncep.noaa.gov/pmb/docs/on388/...
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18 views

symetrical coordinates of algebraic variety

Let $$M = \{ \mathbf{x} \in \mathbb{R}^n : x_i \geq 0 \}$$ and $c \in \mathbb{R}$, $\alpha_i \in \mathbb{Z} \setminus \{0\} $ for $i \in \{1,\dots,n\}$ $$ \mathcal{S} = \{ \mathbf{x} \in M : c = \...
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1answer
33 views

Wave equation with angular variable

Suppose that the variable of a mono-dimensional wave equation is an angle: $$\frac{\partial^2 f(\phi)}{\partial \phi^2} + k_{\phi}^2 f(\phi) = 0$$ This equation is derived from a more complex (and ...
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1answer
41 views

Orthogonal Basis for a family of Curves

Given a family of curves $\mathcal{F}_1$ and an orthogonal family $\mathcal{F_2}$ in rectangular coordinates, how does one create a coordinate system and basis based on these two families. For ...
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1answer
66 views

A better way to solve this question on Parabola?

Q: Let $P$ be a point on Parabola $y^2= 4x$ and $Q$ be a point on the line $L_1: 2x+y+4=0$. If the line $L: x-y+1=0$ is the perpendicular bisector of $PQ$ ; then the coordinates of $P$ can be ? My ...
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1answer
34 views

Choosing the best coordinate system for triple integrals?

I am working with double and triple integral in multi-variable calculus and have found that it is extremely useful to convert between different coordinate systems including: Spherical: Cylindrical: ...
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2answers
95 views

Why are homogenous coordinates needed in image projection?

The above image shows how a 3D object is projected onto a 2D image by a camera. Which makes perfect sense to me. However it's then said that division by z is non linear (why?), so homogenous ...
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31 views

Family of circles, maybe?

Circles are drawn passing through the origin O to intersect the coordinate axes at points P and Q such that $(m)(OP)+(n)(OQ)=k$, then the fixed point satisfying all such circles is? (A) $\left(m,n\...
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1answer
19 views

Calculating surface area of a sphere with cylindrical coordinates

Calculating the surface area of a sphere of radius $a$ with spherical coordinates is quite easy ($4\pi a^2$). I'm trying to do the same with cylindrical coordinates, $\rho$, $\theta$, $z$, (just for ...
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2answers
113 views

Formula that describes the movement of a bishop in chess

I'm programming a chess game and I'm trying to validate the movements every player tries to make. Obviously, every piece can move differently and I've had no trouble validating their moves up until ...
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2answers
31 views

How do I scale a triangle given its cartesian cooordinates?

Given the cartesian $(x,y)$ coordinates of three points $a, b$ and $c$ that form an equilateral triangle $ABC$, how do I scale them using its center point so that its position on the cartesian plane ...
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1answer
160 views

Finding the third vertex of a isosceles triangle given the vertices of the base and knowing that the medians are perpendicular

Full problem: The points A(0, c) and B(c, 0) are the vetrices of the base of an isosceles triangle. In addition, the medians AD and BE are perpendicular to each other. What are the coordinates of the ...
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3answers
36 views

Finding center of circle from 3 coordinates

How would I solve this question (from the SAT): In the coordinate plane,the points $F (-2,1)$, $G (1,4)$, and $H (4,1)$ lie on a circle with center P. What are the coordinates of point P ? $a)...
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1answer
69 views

circle inscribed in a rectangle

I am given a lot of points and the points are suppose to be from a rectangle. I'm required to calculate the boundary of the rectangle. Moreover, I have to figure out radius of an empty candle(...
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1answer
43 views

Maximum value of $n$ such that the distances are equal

For a natural number $j$, let $Q_j$ denote the point $(0,j)$ in the coordinate plane. Find the maximum value of $n$ such that there are $n$ points $P_1,P_2 \ldots P_n$(not necessarily different) with ...
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1answer
29 views

Finding the area under a quadrilateral whose sides are given by four linear equations.

Let there be $4$ equations, namely: $y=m_1x+c_1$ $y=m_2x+c_2$ $y=m_3x+c_3$ $y=m_4x+c_4$ Assuming that these $4$ lines form a quadrilateral, how do I calculate the area of the quadrilateral? One ...
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6 views

Calculate rotation of object related to edge of other object

Given: A triangle in 2D-space, the coordinates of all corners are known. Solution: When clicking on 1 of the blue "+3-signs, a new triangle will appear, where one of the edges will be connected to ...
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1answer
63 views

Conversion from Cartesian to spherical coordinate for vectors - ray tracing application

I'm implementing a ray tracer that support physically based rendering, so is based on various BRDF models. At the moment I'm focused on Oren-Nayar and Torrance-Sparrow model. Each one of these is ...
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1answer
27 views

Calculate vector from two points and angle

I have three points: P1 (5065, 423) P2 (4935, 281) P3 (0, 0) I calculated the angle between P3,P2 and ...
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1answer
147 views

MATLAB fsolve giving wrong solution

I tried to solve equations of 3 spheres using MATLAB's fsolve function, but it is giving the wrong solution. here is my MATLAB function ...
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0answers
23 views

Maximum points on boundaries

Given $N$ points on a $2D$-plane of type , each coordinate of type $(x,y)$. We need to make a rectangle on this plane in such a way that maximum number of points lie on the boundary of this rectangle. ...
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1answer
53 views

Tangent spaces at different points on manifolds

Why are tangent spaces on a general manifold associated to single points on the manifold? I've heard that it has to do with not being able to subtract/ add one point from/to another on a manifold (...
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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 <...
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3answers
99 views

Show that a parabola passes through the point (0,1) [closed]

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$.
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2answers
77 views

Circular Arc Prametrization not Using Radius

In an optimization problem I have to parametrize a circular arc. Thus far, I have reduced a more general problem to the figure below: The figure shows a symmetrical circular arc, with chord length ...
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2answers
67 views

Find angles of triangle formed by images of vertices about opposite sides of an isosceles triangle

In $\triangle ABC$, $AB=AC$ and $\measuredangle BAC=30^\circ$. If $A^\prime$, $B^\prime$ and $C^\prime$ are the reflections of $A$, $B$, and $C$ about $BC$, $CA$ and $AB$, How to find $\measuredangle ...
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1answer
32 views

Cylindrical coordinates: why require $0≤\varphi≤\pi$? [closed]

In cylindrical coordinates: Why is there a requirement that $0≤\varphi≤\pi$?