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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3
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0answers
35 views

Check if a point is inside a rotated 2D NACA 0012 airfoil

I've already checked the rotated rectangle problem but this is (I think!) a little more complicated. I have a CFD calculation of a 2D NACA 0012 airfoil and I need to test if a point is inside the ...
1
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1answer
32 views

Equation of one of the two lines whose angle bisector is given.

A ray of light falling along the line $ lx+my+n=0 $ strikes a plane mirror at point $P$. Find the equation of reflected ray if $px+qy+r=0$ is the equation of normal to the plane at point $P$.
1
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1answer
278 views

How do you convert velocity given a heading and speed to ECEF coordinates?

If you are given positional data in latitude, longitude, altitude along with a given velocity and heading, how do you convert the velocity into Earth Centered Earth Fixed (ECEF) based values? In this ...
0
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0answers
28 views

Change of coordinate matrix for polynomials

So I am trying to do a simple change of basis. I have my 2 basis for $p_1$: $\beta={(1,x)}, \beta'=(1+x,1-x)$ The matrix for coordinate change from $\beta$ to $\beta'$ is $$Q = \begin{bmatrix} ...
1
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0answers
39 views

Spherical coordinates what are the coordinates of a vector?

I am getting a bit confused about the language for coordinate systems. I have chosen spherical coordinates as an example, but my confusion is general. Let's say we have a vector $$\vec A=A_r ...
1
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2answers
74 views

How to know if two points are diagonally aligned?

If I have two points at different X/Y coordinates, I know that: They are vertically aligned if both are at the same X coordinate; They are horizontally aligned if both are at the same Y coordinate. ...
1
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1answer
105 views

Manifolds, coordinate systems, books

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a ...
3
votes
2answers
119 views

Scalar product invariance

The question is somehow silly, but I can't seem to find a way out right now. Consider the vector $v\in\mathbb R^2$, expressed in Cartesian coordinates as $(1,1)$ and in polar coordinates as $(\sqrt ...
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0answers
40 views

Mapping cartesian coordinates to circular coordinates

I have some "rectangular" x,y coordinates which I like to convert to "circular" coordinates. Notice I'm not sure if I'm using the correct terms here, so bear with me - I drew an example to show how ...
1
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0answers
23 views

Parabolic Coordinates and the Normal Derivative

I would like some help with the following problem. Thanks for any help in advance. Determine the largest region Ω’ in R^2 spanned by (u,v) such that the transformation T that defines parabolic ...
1
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0answers
99 views

Mapping A point from one 3D Coordinate System to Another 3D coordinate System with Euler Angles between the two systems given

Suppose I have a point in the green coordinate system, and I wish to describe it in reference to the orange coordinate system. I know the roll, pitch, and yaw of the green system with respect to the ...
1
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0answers
75 views

distance in n-dimensional space

According to answer of this question : Distance between 2 points in 3D space (in spherical polar coordinates) The distance between 2 points in 3 dimensional space is : $$ ...
1
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1answer
19 views

Coordinate transformation (or conversion) into yards

Following is a soccer field with its dimensions. There is a similar field, but I am capturing coordinates via mouse-movement. So, what (115,75) shows here, is ...
1
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1answer
40 views

Square labeled with same number.

Recently I met this combinatorics problem: "Let all points with integer coordinates in a plane be labeled with one of the numbers $1,2,3,...,n$. Prove that there is a rectangle whose vertices are ...
0
votes
1answer
67 views

How to rotate an orientation (Euler angles)

If I have an orientation defined by Euler angles and I want to simulate a rotation of the coordinate system about the origin (doesn't matter to me how the rotation is specified), how would I get the ...
0
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0answers
39 views

How to express standard deviations and correlations of 6 points as one measure of uncertainty?

Algorithm that I use, uses a weight matrix P of size n x n to compute $XPY^T$ where $X$ and $Y$ are 3 x n matrices - representing X,Y,Z coordinates for n points. This means that points are weighted ...
0
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1answer
44 views

coordinates of 3rd point (vertex) of a right triangle knowing lengths and direction

In a last post I wanted to know the 3rd point of vertex, actually I have some similar problem .... I think I have all data.. for example...: 3 vertex cordinates in order to have the direction(gray ...
0
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0answers
28 views

Making Homogenous Parabola Equation

Find the locus of the mid-points of the chords of the parabola $y^2=4ax$ which subtend a right angle at the vertex of the parabola. Now we say $y^2=\frac{4ax(yk-2ax)}{k^2-2ah}$ coefficient of ...
4
votes
1answer
114 views

How to get coordinates of a point after an image is rotated? (with images)

I have a problem that involves a rotating image and finding a previously known point. Firstly, there is a sequence with the rotation. We start with an empty image. A line is drawn vertically, ...
0
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0answers
14 views

linear functional basis and affine map

Suppose that the domain under consideration is the right triangle with coordinates (0,0), (1,0), and (0,1). At each vertex, let $\sigma_k(x,y)$ be such that $\sigma_k(x_k,y_k) = 1$ and ...
0
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1answer
28 views

How do I find the symmetrical point B given the centre of symmetry C and another point A?

I have a point $A (-2k; 3)$ and a point $B$ that is symmetrical to the point A given the centre of symmetry $C (-1; 0)$. I tried applying the following formula, where $x_o$ and $y_0$ are the ...
-3
votes
1answer
78 views

Need Help in Coordinate Geometry (Straight Lines) [closed]

A line $L$  is drawn from the point $P\equiv(4,3)$  to meet the parallel lines $L_1: 3x + 4y + 15 =0$  and $L_2 = 3x + 4y + 5 =0$ at points $A$  and $B$ respectively. From $A$, a line perpendicular ...
0
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2answers
40 views

Calculate new(x,y) for a line

Suppose if tanks has to rotate its main gun by $30^\circ$ to hit the target, what will be its new $(x,y)$ coordinate or a formula to calculate it as shown in image? If bullet is fired from the main ...
0
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0answers
36 views

Cylindrical coordinates - Orthonormal system

Using cylindrical coordinates and the orthonormal system of vectors $\overrightarrow{e}_r, \overrightarrow{e}_{\theta}, \overrightarrow{e}_z$ describe each of the $\overrightarrow{e}_r$, ...
0
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3answers
68 views

The geometric meaning of certain mappings written in cylindrical or spherical coordinates

What is the geometric meaning of the following mappings, that are written in cylindrical coordinates? The mappings are: $$(r, \theta, z) \rightarrow(r, \theta , -z) \\ (r, \theta , z) \rightarrow (r, ...
0
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3answers
36 views

Cylindrical - Spherical coordinates

We are given a point in cylindrical coordinates $(r, \theta , z)$ and we want to write it into spherical coordinates $(\rho , \theta , \phi)$. To do that do we have to write them first into ...
1
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1answer
24 views

Cylindrical coordinates - Surfaces

I found the following: Cylindrical coordinates $(\rho , \theta , z)$. This system consists of the following coordinate surfaces: Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ ...
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0answers
40 views

Show that the point satisfies the conditions

A round membrane in space, is over the space $x^2+y^2 \leq a^2$. The maximum coordinate $z$ of a point of the membrane is $b$. We suppose that $(x, y, z)$ is a point of the inclined membrane. ...
0
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0answers
70 views

Volume between a paraboloid and a parabolic cylinder

Find the volume of the region bounded by the paraboloid x=y^2 + z^2 and the parabolic cylinder x=2-y^2. I set up the integral as the integral of theta from 0 to 2pi, integral of (2/(sin^2theta + ...
0
votes
1answer
30 views

Multiplication order of rotation matrices

I have three 3D coordinate frames: O, A and B, as shown below. I want to know the rotation matrix RAB between A and B, that is the rotation that is required, with respect to the frame A, to move ...
0
votes
1answer
87 views

Transformations between two coordinate systems on a rigid body

I have two coordinate frames, A and B, which are rigidly attached to each other on a body. This body then translates and rotates, such that A starts at A1, and moves to A2, and B starts at B1, and ...
0
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2answers
62 views

Rotation matrix between right- and left-handed systems, with an additional rotation

I have two coordinate frames, A and B. I want to create the rotation matrix RAB which takes you from A to B. A is a right-handed system, and B is a left-handed system. Furthermore, after moving from a ...
6
votes
3answers
88 views

Why is $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$?

I have to show the identity I wrote in the title: it should be $\partial_z\partial_{\bar z}=\frac14\left(\partial_r^2+\frac1r\partial_r+\frac1{r^2}\partial_{\theta}^2\right)$ but some computation ...
0
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0answers
41 views

Integrals For AMC 10B #25?

Could an answer be found to this AMC 10B #25, using integration? A rectangular box measures abc, where a, b, and c are integers and 1<=a<=b<=c. The volume and the surface area of the box are ...
0
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0answers
26 views

Multiplication order for coordinate frame transformations

Suppose I have three coordinate frames: $A$, $B$ and $C$. If $T_{AB}$ is the transformation from $A$ to $B$, then which of the following is correct? $T_{AC} = T_{AB} \cdot T_{BC}$ $T_{AC} = T_{BC} ...
15
votes
1answer
798 views

Abscissa, Ordinate and ?? for z-axis?

Like x-axis is abscissa, y-axis is ordinate what is z-axis called? It is one of basic doubts from my childhood.
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0answers
25 views

Moving die on cartesian plane so as to minimize sum of facing face

I have a problem that I have been working on for which I cannot find a solution. Problem: Assume you are on a cartesian plane, and you want to move a die to a specific point. You can move the die ...
0
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0answers
35 views

Tangent undefined for polar curves ($r^2=a^2sin(s\theta)$)?

I am considering the function $r^2=a^2\sin(2\theta)$ and am trying to find tangents perpendicular to the initial line, so $\frac{dx}{d\theta}=0.$ However, when I take the derivative by implicit ...
0
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0answers
67 views

Tangents perpendicular to the initial line for cardioid? Polar coordinates…

For the polar curve $r=a(1+cos\theta)$, I am trying to find the equations of the tangents perpendicular to the initial line by setting $\frac{dx}{d\theta}$ equal to zero. I am able to factorise a sine ...
0
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1answer
38 views

How can I prove non-geometrically that there is a bijective correspondence between polar and cartesian representations of coordinates?

We have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ as $f(x,y) = (\sqrt{x^2 + y^2}$, $\tan^{-1}\left(\frac{y}{x}\right))$ which takes a Cartesian pair $(x,y)$ to its polar form, and a ...
0
votes
1answer
24 views

Minimum points to choose from a set such that midpoint is also in the set

Consider the set $S =\{ (x, y)$ : $x$ and $y $ are integers $\}$. The midpoint of a pair of points $ P_1$ = ($ x_1 $, $y_1 $) and $P_2 $= ($x_2$ , $y_2$ ) is $ ( \frac{x_1 + x_2}{2}, ...
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0answers
19 views

Confused by Parameterizations and Coordinate Conversions

So I have a few questions regarding parameterizations and coordinate conversion. Ever since dealing with parametric equations last semester I have felt like I have never truly understood ...
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0answers
104 views

Parametric integration negative area?

I know there is a question very similar to mine already here Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation? , but I am still a bit ...
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0answers
22 views

Polar coordinates for vector difference in $\mathbb{R}^2$

I have a function $F(\boldsymbol X)=\tilde F(x,y)$ of $x$ and $y$ in the plane, and I can transform it in a function of $r$ and $\theta$, say $f=f(r,\theta)$, through the change of coordinate $$x=r ...
0
votes
1answer
70 views

Using Barycentric coordinates to check whether a point lies within a Degenerate triangle

http://www.blackpawn.com/texts/pointinpoly/ I used this site to learn how to determine whether a point lies within a triangle. However, the site does not say whether or not this method can handle ...
0
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0answers
48 views

Coordinate Systems on the Complex Plane: Rectangular, Polar, Exponential, … Imaginary?

In the complex plane there is a nice relationship between rectangular, polar, and exponential coordinates: $$(x+iy) = r(\cos\theta + i~\sin\theta) = re^{i\theta} $$ $$where~~x ,y ,\theta, r \in ...
0
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0answers
24 views

Cylindrical coordinates: Paradox?

Let's say the vector field $\vec A_1 = \begin{pmatrix} r\\ 0 \\ z\end{pmatrix}$ in cylindrical coordinates is given and I want to calculate $\vec A_1 \cdot \vec e_r$, where $\vec e_r = \begin{pmatrix} ...
0
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0answers
22 views

Transforming a vector field in cylindrical coordinates - simple

I want to transform $\vec F = \alpha \vec r$ in cylindrical coordinates, i.e. $x=\rho \cos(\phi), y=\rho \sin(\phi), z=z$ Now since $\vec r = \begin{pmatrix} x \\ y \\ z\end{pmatrix}$ I receive $ ...
0
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0answers
24 views

How to formulate a coordinate transformation

Thank you in advance for taking the time to consider this. I'm trying to figure out how to formulate a coordinate transformation problem (at least that is what I think it is). Background: I have an ...
0
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0answers
26 views

Transform to cylindrical coordinate system

I tried so many approaches , at least give me a hint on how to find The unit vectors $$ \vec{V} = y\vec{i} + x\vec{j} + \frac{x^2}{\sqrt[2]{x^2+y^2})} \vec{k}\ $$