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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5
votes
3answers
249 views

Alternative form of equation of circle?

In a problem set I was solving, one of the solutions used the equation of a circle in the form $$(x-h)^2 + (y-k)^2 + \lambda(ax + by +c) = 0$$ where, $(h,k)$ is any point on the circle $ax+by+c ...
4
votes
1answer
103 views

Find the maximum length of a line segment enclosed in a given area

$A = \{ (x, y) : x = u + v , y = v , (u^2) + (v^2) \le 1 \}$ . Then what is the maximum length of a line segment enclosed in this area? My friend suggested the answer $\sqrt{5}$, but I think it ...
0
votes
1answer
50 views

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 ...
1
vote
1answer
682 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 ...
0
votes
2answers
45 views

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 ...
2
votes
1answer
367 views

Problem of a circle tangent to three other circles [on hold]

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 ...
1
vote
1answer
30 views

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 $h,k$?
0
votes
1answer
53 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 $OA\cdot OB\cdot OC$?
0
votes
1answer
42 views

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 ...
0
votes
1answer
23 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 ...
0
votes
1answer
253 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 ...
0
votes
1answer
266 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 ...
1
vote
1answer
1k 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 ...
0
votes
2answers
245 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 ...
0
votes
2answers
41 views

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 ...
0
votes
1answer
58 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 ...
0
votes
2answers
2k 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 $P_1\equiv (a=33855.05, b=21129.55)$ $P_2\equiv (c=33745.04, ...
0
votes
1answer
58 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 ...
0
votes
2answers
143 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 ...
0
votes
1answer
562 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 ...
0
votes
1answer
86 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 ...
0
votes
1answer
51 views

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 ...
0
votes
1answer
42 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, ...
-1
votes
1answer
64 views

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: ...
1
vote
1answer
303 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 ...
1
vote
0answers
50 views

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 ...
1
vote
0answers
123 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 ...
1
vote
2answers
63 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 ...
0
votes
1answer
159 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 ...
3
votes
3answers
102 views

“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 ...
1
vote
1answer
68 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, ...
0
votes
1answer
56 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 = ...
2
votes
1answer
232 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) & ...
2
votes
1answer
53 views

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 ...
1
vote
1answer
248 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: ...
0
votes
1answer
105 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. $x=0.4,y=22.0$ ...
1
vote
1answer
85 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 ...
1
vote
0answers
47 views

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 ...
0
votes
0answers
39 views

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 ...
1
vote
1answer
952 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: ...
0
votes
1answer
31 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!
0
votes
2answers
111 views

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: ...
1
vote
1answer
139 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 ...
-1
votes
1answer
1k 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 ...
-1
votes
1answer
87 views

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 ...
0
votes
3answers
57 views

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 :'( ...
1
vote
1answer
237 views

Mapping points on a plane in space onto a coordinate plane

Is there a way to isometrically map points on a plane in space onto a coordinate plane? So for example, given the three points $\left( 1,1,0 \right)\;, \left( -1,0,2 \right),\; \text{and }\left( ...
0
votes
3answers
61 views

Determining if a point is inside two planes

I have two planes(Plane 1 and Plane 2) the normals ($n_1$ and $n_2$) of which are known to me. How do I determine if a point is inside the two planes? By inside I mean the 3d space between Planes 1 ...
0
votes
2answers
140 views

How to find the perpendicular distance from B to AC?

"Find the area of triangle $ABC$ with vertices $A(2, 1), B(12, 2) $and $C(12, 8)$. Hence or otherwise, find the perpendicular distance from $B$ to $AC$." I found the area, which is 30 units by find ...
1
vote
3answers
2k views

Find the point on the y-axis which is equidistant from the points $(6, 2)$ and $ (2, 10)$.

Find the point on the y-axis which is equidistant from the points $(6, 2) $ and $ (2, 10)$. Please help, there are no examples of this kind of sum in my book! I don't know how to solve it.