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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2
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1answer
23 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 ...
2
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1answer
67 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 ...
2
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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": $\phi$ ...
1
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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 ...
2
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0answers
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 ...
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0answers
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 = ...
3
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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 ...
3
votes
1answer
65 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
31 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
74 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 ...
0
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1answer
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) ...
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1answer
17 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
87 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
29 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
136 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
33 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 ? ...
1
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1answer
60 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 ...
2
votes
1answer
41 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 ...
0
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1answer
26 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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0answers
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 ...
3
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1answer
62 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 ...
0
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1answer
25 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
108 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. ...
1
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1answer
47 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
92 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
85 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
72 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
65 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 $0≤\varphi≤\pi$?
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0answers
40 views

Derivation of Equation of common chord of two circles(S1-S2) in Cartesion form

I come to know the equation of common chord of circle and it helped me a lot in finding common tangents(when circles touch each other). I want to know it's derivation and I seek help for this!
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0answers
15 views

Get the positions of a periodic system of labeled points in reference to another coordinate system

First of all, I'd like to apologize because I'm not familiar with the conventions (names and formats) used in the math community. The problem is the following, I have information about a Face ...
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0answers
17 views

Laplace's equation in cylindrical coordinates for a function that only depends on the angle

I need to solve Laplace's equation: $$\nabla^2\Phi = 0$$ with the boundary conditions: $$\Phi(\theta=0)=0$$ $$\Phi(\theta=\pi)=a_1$$ In cylindrical coordinates ($r,\theta,z$), for $\Phi ...
0
votes
2answers
62 views

Determine if u and v are parallel

Determine if u and v are parallel $u = <3, -6, 3>$, $v = <-1, 2, -1>$ So i seen a formula that says there is a$ "k"$ such that it if one is a multiple of the other. Is there a number$ ...
0
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1answer
23 views

Elliptic Coordinates - Inverting the transformation

The standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$: $x = a \cosh(\mu) \cos(\nu)$ $y = a \sinh(\mu) \sin(\nu)$ Is there any way to get the ...
0
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1answer
42 views

How to get a point coordinates in a 2-dimensional coordinate system?

For the famous 2 dimensional coordinate system in which the x-axis is orthogonal to the y-axis, to get a point, say P, coordinates it is enough to get its orthogonal projection on the x and y axis and ...
3
votes
1answer
23 views

$\Psi$ is a morphism of ringed spaces iff it is smooth in local coordinates

Here is what I have to show: Let $(M,\mathcal{F})$ and $(N,\mathcal{F}')$ be smooth manifolds of class $C^{\infty}$ and let $\Psi:M\to N$ be a continuous map. Show that the following ...
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0answers
15 views

Is the $r$-axis in spherical coordinates the same as the $z$-axis?

If Cartesian coordinates have an $x$-axis, $y$-axis, and $z$-axis, do spherical coordinates have an $r$-axis, a $\theta$-axis, and a $\phi$-axis? Since the Cartesian $z$-axis is just the set: ...
0
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1answer
139 views

Determine Direction of an Arc (CW/CCW)

I'm trying to write a function that will determine whether a circular arc travels clockwise or counter-clockwise. Given the X,Y coordinates for the start point, center point, and end point I first ...
3
votes
1answer
78 views

Surface integral of a partially constant Dirac delta

I am trying to integrate the product of a function and a partially constant delta function over a sphere of constant radius $r$. The integral is of the form $\int^{2\pi}_0 \int^{\pi}_0 ...
0
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2answers
41 views

Finding points in circles

so I have two questions Im stuck on and I really do not know what to do at all. Thank you. 1) 2)
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0answers
29 views

Coordinates of a plane with a handle

I am trying to find the appropriate coordinates for a plane with a handle (of topology $\mathbb{R}^2 \# \mathbb{T}^2$), without having to use several coordinate patches. My current intuition is to ...
0
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1answer
30 views

An Expression for Nabla in Cylindrical Coordinates

I'm looking for an expression for $$\big(\,u\cdot\nabla\,\big)\,\psi$$ in cylindrical coordinates where $\psi$ is a scalar field and $u$ is a vector field. The Wikipedia page given here ...
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1answer
38 views

A question concerning Jacobians of coordinate transformation

Apologies for perhaps a very trivial question, but I'm slightly doubting my understanding of Jacobians after explaining the concept of coordinate transformations to a colleague. Basically, as I ...
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2answers
27 views

What is the area bounded in the graph?

I was given a question which I think was incomplete.I want your opinion- Find the area of the figure bounded by $x-3=0,y-5=0,x+3=0,y+5=0$. My attempt- $x=3,x=-3,y=5,y=-5$ represent parallel ...
0
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1answer
42 views

Differentials squared - Divergence in general orthogonal curvilinear coordinates.

I was reading this document on how to get some common operators when dealing with general orthogonal curvilinear coordinates. I am interested in particular in equation (12). It basically defines three ...
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0answers
28 views

Coordinate change expansion

Suppose there is a function $f(\vec{x})$ which can be given as a perturbative expansion of the form $$f(x) = f_0 (x) + f_1 (x) + f_2 (x)+\cdots$$ where $f_n$ represents a function of order some ...