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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Change of Basis - Homework Question

Please help me understand what is being asked, I feel I am missing something. Compute the change of basis matrix for each of the bases, and use it to find the coordinate vector v with respect to B ...
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40 views

How to determine the direction of one point from another, given their coordinates?

If I have the coordinates of two points, how would I determine what direction the second point lies in, relative to the first point? Specifically, I'm writing an application that involves basically ...
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Region closer to one given point than to any other given point

(Q) *Consider 6 points located at P0=(0,0), P1=(0,4), P2=(4,0), P3=(-2,-2), P4=(3,3), P5=(5,5). Let R be the region consisting of all points in the plane whose distance from P0 is smaller than that ...
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70 views

Problem of a circle tangent to three other circles

Two circles with centres A and B and radii 14 and 7 units respectively touch each other externally. M is the mid point of segment DE and is the centre of the circle with radius 21 units. The two ...
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Finding conditions for a point residing in the interior of an ellipse

I have an ellipse with the equation $x^2+2y^2-2xy-1=0$. Suppose $(h,k)$ is a point residing in the interior region of the ellipse. Should this point satisfy any condition in terms of $h,k$?
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homogenous coordinate system vs cartesian coordinate

“The homogeneous coordinate system is used in projective geometry as much of the math ends up simpler in homogeneous coordinate space than it does in a regular Cartesian space.” Excerpt From: Haemel, ...
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22 views

3D cartesian cordinates to 3D isometric cordinates

I am working on Isomer opensource project https://github.com/jdan/isomer and I need to convert 3D cartesian cordinates to 3D isometric cordinates. We already have 3D iso to 2D cartesian ...
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35 views

Straight Lines and Curves

If the line $y=x\sqrt3$, intersects the curve $x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0$, in three points $A,B,C$. If $O$ is the origin, then what's the product $OA\cdot OB\cdot OC$?
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Rotation of axes help?

This is not a duplicate of my other question in regard to this. I really am not understanding this rotation of axes stuff. If we want to graph a 45 degree shifted ellipse for example, we can think of ...
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22 views

Y-Coordinate of a Point - Notation

Given point $P$ on curve $\omega$, what expression is generally use to denote the $y$-coordinate of point $P$, also, the $x$-coordinate. Would it be $P_y$? Also, let $\omega$ be in $\mathbb{R}^2$, not ...
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51 views

Find a point using multilateration or triangulation

Suppose we have 3 points in a 3d coordinate system with the following locations. A=(100,0,0) B=(0,100,100) C=(0,0,100) If there is a 4th point D, where the distances from A, B, and C are known, can ...
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192 views

Finding the expected value of the length of a minimal spanning tree of n randomly generated nodes bound in a box with edge length a.

Say we specify a number (n) of random points (x,y), bound within the axes and x=a, y=a. Given the number of points and the constraints on the boundaries, how can you calculate the expected value of ...
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1answer
42 views

Finding the locus of a mid-point

Let $A$ be the fixed point $(0, 4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $R$. Find the locus of the ...
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2answers
48 views

Calculating the expected value of n randomly generated numbers?

Say I have a random number generator that will generate x numbers - not necessarily integers - on the continuous range between a and b. How can I calculate the expected values for these numbers? My ...
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Rotation of axes give the same point in space?

I am playing around with rotation of axes formulas and not getting it. I don't understand how this rotates anything when it is just giving you different coordinates for the SAME point in space. How ...
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1answer
45 views

Cartesian Line to Projective Coordinates

I have an equation of a line written in slope intercept form $y = mx + b$ How would I translate it from the 2d space into the projective space? I have been reading ...
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2answers
79 views

How to find center of a circle given 2 points on the circle and the radius?

I would like to find an equation that I can put into excel to calculate the coordinates for the center of the circle. Given P1 (a) 33855.05, (b) 21129.55 P2 (c) 33745.04, (d) 21221.69 Radius 590 I ...
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45 views

Geometric interpretation of a complex solution

A straight line in 2-D $x+y=3$ and a circle in 2-D $x^2+y^2=4$ do not have a point of intersection in the plane containing the two. But on solving these equations analytically, on gets 2 complex ...
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53 views

Is the reference point (x, y) above or below the non-linear equation?

BACKGROUND In short, I have a series of 3 to 10 data points that will be used to represent a curve. For example: $X=0, Y=10$ $X=4, Y=7$ $X=9, Y=12$ $X=16, Y=10$ What I am trying to do is ...
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147 views

Help understanding polar coordinates and conversion between polar and rectangular

I'm not understand this. I understand that you can take normal functions with x's and y's and convert them into polar coordinates. I also understand that the polar form of that function will have the ...
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76 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...
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18 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
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An error in Wikipedia? (trigonometry)

https://en.wikipedia.org/wiki/N-sphere In "Spherical coordinates" section the hyperspherical coordinates are results of arccosinus function. In some other sources there is arccotangent instead: ...
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41 views

Find coordinates for points on circle given R, 2 Points, and angle or 2 points and center?

I would like to find coordinates for points on a circle given: Radius of circle Coordinates of 2 points on the circle Angle of point 1, center, and point 2. Ultimately, I would like to write a ...
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Parabolic Coordinates Radius

Given Cartesian $(x,y,z)$, Spherical $(r,\theta,\phi)$ and parabolic $(\varepsilon , \eta , \phi )$, where $$\varepsilon = r + z = r(1 + \cos(\theta)) \\\eta = r - z = r(1 - \cos( \theta ) ) \\ \phi ...
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38 views

Fit cartesian coordinate system to point cloud

I have a cloud of points that initially lie in a plane and have a coordinate system attached to them. I then displace the points slightly, and I want to find how a 'best fit' of the coordinate system ...
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32 views

Angle between planes , when only two points are given [closed]

Find the equation of plane passing through points $[1,0,0]$ and $[0,1,0]$, and making an angle of $45°$ degree with the plane $x + y - 3 = 0$.
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2answers
43 views

Tangent definition

As far as the definition of a tangent goes it is a line that touches a curve only at one point. Now let us consider the sine function .At (pi)/2 it attains its maximum value and so does it at ...
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56 views

rotation matrix to axis angle

from wikipedia the above rotation matrix has a rotation of -74 degrees. What does it mean "around the axis (−1⁄3,2⁄3,2⁄3)"? How can I determine how many degrees is rotated on X axis, Y axis and Z ...
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1answer
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“Rigorous” definition of Cartesian coordinates

I, like most, first learned about Cartesian coordinates very early on in my educational career, and so the most instructional way to think about them was that you place down some perpendicular lines ...
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1answer
40 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
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45 views

Geometric reasoning and triangular coordinates

The following is from a book: I do not understand the sentence "... the point $(t_1, t_2, t_3)$ can be plotted by plotting $(t_1 = t_3, t_2 = t_3)$...", what is meant by the point $(t_1 = t_3, t_2 = ...
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1answer
25 views

Rotation of velocity vectors in Cartesian Coordinates

I want to rotate a $(X,Y,Z)$ coordinate-system around it $Z$-axis. For the coordinates this can be done with the rotation matrix: $$ R_Z(\theta)= \begin{pmatrix} cos \theta & -\sin(\theta) & ...
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1answer
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Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
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1answer
65 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
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30 views

How to draw a straight line and determine the gradient from the graph?

The following gives experimental values of two variables $x$ and $y$ which are known to be connected by a relation of the form $xy=a+bx$. So, this information was given in a table. $x=0.4,y=22.0$ ...
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How to move coordinate systems using rotation matrices.

I am having some trouble with this question. I understand that the rotation matrix will be 4x4 and that the first 3 columns will just be $u$, $v$ and $n$ transposed but I dont know what I am ...
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1answer
56 views

Using Cylindrical Coordinates to Compute Curl

I am given a vector field $\vec{A} = A_\rho \space \hat{e_\rho} + A_\phi \space \hat{e_\phi} + A_z \space \hat{e_z}$, and I am supposed to use the unit vectors (provided below) in cylindrical ...
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Find the coordinates of P.

A trapezium ABCD, AB parallel to DC, BC is perpendicular to both AB and DC. $A (4, 2), B (16, 8), C (13,14), D(5, 10)$. Lines AD and BC produced to meet at P. Find the coordinates of P. I tried to ...
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Finding coordinates of nodes in a graph

I have a complete graph in which the edges represent the euclidean distance between the nodes which is known. Assuming a node to be (0,0), I want to find (approximately) the coordinates of other ...
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A sufficient condition for a convex body to lie completely inside another convex body?

Suppose we have two convex bodies in 3D space. Let’s call them $B_1$ and $B_2$. Let’s denote their projection curves on the xy plane by: $P_{1xy}$ and $P_{2xy}$, on the yz plane by: $P_{1yz}$ and ...
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Detect Regions Described By Lines in Rectangular Coordinates

Need some help from the superior math minds here. This problem is part of a software project. Essentially, I have a Cartesian grid. The user can create lines by plotting points (every 2 points ...
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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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1answer
27 views

Find the equation of the line.

Find the equation of the line passing through the point $(5,7)$ and parallel to the line $5x+4=0$ If I say $m=5$, how should I find $c$? There is no $y$ in the equation!
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Find the equation of the straight line.

Find the equation of the straight line that is parallel to $2y-x=7$ and bisects the line joining the points $(3,1)$ and $(1,-5)$. So, I found the gradient = $\frac{1}{2}$ And I solved the midpoint: ...
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1answer
61 views

Transfer Transformation from Physics to Vector Graphic

Upfront, I am not a professional in Maths and hope that the formulation of my question describes the problem well enough. I am creating a jump'n'run game, which uses a physics engine (Box2D) and SVG ...
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166 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
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How to find coordinates of a point on a 3D cylinder in Cartesian system if any one point on cylinder and dimensions of cylinder are known?

Consider a cylinder of known dimensions inserted in 3D cartesian space. I know the cartesian coordinate of one point located on the surface of cylinder. Using this information I want to find out the ...
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Find the coordinates of $P, Q, R$.

In triangle $PQR$, $A(-2,3), B(5,-1)$ and $C(-4,-7)$ are the midpoints of $PQ, QR$ and $PR$ respectively. Find the coordinates of P, Q and R. I really can't understand how to begin....Please hint :'( ...