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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0
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0answers
11 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
2
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1answer
524 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
0
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1answer
52 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
3
votes
2answers
360 views

Find parametric expression of an arc given its start point, end point and central angle in 3D cartesian coordinate system

In a 3D cartesian coordinate system, the coordinates of start point and end point have been given as $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$. If the central angle of the two points (the one smaller ...
1
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0answers
969 views

Radians : negative and positive values

Recently I have been reading books on DSP where I came across Polar co-ordinates. I understand that on Polar graph (4 quadrants) we have $0, \frac{\pi}{2}, \pi, \frac{3}{2\pi}$ and $2 \pi$ radians as ...
2
votes
3answers
30 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
0
votes
1answer
560 views

Inverse rotation euler angles

I have three angles representing a rotation (Pitch, roll and yaw). I need the inverse rotation (working on coordinate system transforms). What I do now is transforming these angle to a rotation matrix ...
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0answers
19 views

Coordinates change movement problem

yesterday I asked a related question that was a specific case of my problem. It was solved here: Geometry/ Triangles problem but I would want to know if its possible to get a generic universal ...
2
votes
1answer
368 views

Problem of a circle tangent to three other circles [closed]

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 ...
0
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0answers
30 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
0
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0answers
18 views

How to perform statistical test for two sets of points?

(I have asked this question originally on Cross Validated; however, no good answer and someone suggested me to ask the question here). Thanks a lot in advance if anyone can help. We know that we can ...
0
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0answers
17 views

$A: \mathbb R^3 \to \mathbb R^3$ what is the representation of $A$ under coordinate change? (Tensor)

For our initial homework on Tensor Calculus we have to do the following: Consider a map $A: \mathbb R^3 \to \mathbb R^3$, where we use local coordinates $\mathbf{x}$ given by $x^1,x^2$ and $x^3$. ...
1
vote
0answers
10 views

How to detect how clustered coordinates are in a graph? [closed]

Assuming we have a 2D graph with n points, I want to detect how clustered are the points. By clustered I don't mean only how close they are in general, but a very clustered graph can be one with m ...
1
vote
0answers
14 views

How to determine changing scale factors when performing coordinate transfomations?

To explain: I have two coordinate systems. One (x,y) and the other (x',y') as seen in this photo. Coordinate systems I am trying to convert the coordinate in the (x,y) system to the rotated red ...
1
vote
2answers
49 views

Finding the equation of a circle from the equation of its tangents

Given the equation of a pair of lines : $36x² - 63xy + 20y² + 54x - 17y - 10 =0.$ If the circle touches one of the lines at (-3,-1) and the other at some point then find the equation of the ...
1
vote
1answer
16 views

Get coordinates in a scaled and translated canvas

I'm drawing an image to a scaled and translated HTML5 canvas. It's zoomed to a certain point on the image to be more specific, so only a smaller partition is visible. In this smaller part, I want to ...
1
vote
1answer
6k views

How to find equation of parabola when we only know the equation of latus rectum and coordinates of vertex?

Suppose the equation of latus rectum is x=4 and the vertex is (2,3). I am confused wouldn't there be many parabola with this same vertex and latus rectum.If not how to find the equation? The answer ...
0
votes
4answers
24 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
0
votes
5answers
49 views

Two points are given (-2,0) and (0,4/3) we can easily find the equation of …

Two points are given (-2,0) and (0,4/3) we can easily find the equation of straight line joining these two points Let A(-2,0) and B(0,4/3) Equation of straight line joining these two points ...
2
votes
2answers
224 views

How to find the eccentricity of this conic?

How to find the eccentricity of this conic? $$4(2y-x-3)^2-9(2x+y-1)^2=80$$ My approach : I rearranged the terms and by comparing it with general equation of 2nd degree, I found that its a ...
0
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0answers
20 views

Calculating the sun position fails

could you help me find the mistake(s) in my calculation of the sun position today on hawaii at 16:00? I'm following this Wikipedia article. Number of days since 2000/01/01 (2016/01/29): $$n ...
3
votes
2answers
4k views

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin ...
6
votes
1answer
3k views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...
0
votes
0answers
13 views

Relations between unit vectors in polar and curvilinear coordinates

I'm a bit confused about the relations between unit vectors in curvilinear and polar coordinates in a plane. Let $u_{R}$ be the radial unit vector in polar coordinates and $u_{N}$ the normal unit ...
2
votes
1answer
33 views

Rational points on a line

This question is quite unique. Does there exist some point in the coordinate system such that any line passing through it has at most 2 rational points lying on it?
0
votes
1answer
372 views

maximum radius of a circle inscribed in an ellipse

Consider an ellipse with major and minor axes of length 10 and 8 resp. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus ...
0
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0answers
24 views

Trying to develop a formula for relative angle relationships in fixed space

Fixed space is defined relative to origin $(0,0,0)$. Fixed space has an origin with angles $(0,0,0)$ as well. A particle $P$ has $(x,y,z)$ coordinates relative to the origin: $P_x$, $P_y$, and ...
0
votes
1answer
11 views

Rectangular to polar conversion angle error

I am trying to determine the polar form of the following rectangular vector: -105 + 140j The polar form is $\sqrt(-105)^2+140^2$ = 175 and the angle is ...
0
votes
2answers
24 views

If $x=r\cos(\theta)$ then in $?=r\cos(\theta+a)$ what is $?$ equal to?

What I mean by $r\cos(\theta+a)$ is that it's the same function $r\cos(\theta)$ but it's translated by $a$ units, if this makes any sense. I just want to know what it means in terms of $x$.
19
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7answers
48k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
3
votes
1answer
68 views

How to change between these two coordinate systems?

I have a problem with the coordinate system change between two 3D rotation sensors A and B. The coordinate systems have the same origin and are (mostly) perpendicular. I tested a somewhat pure ...
2
votes
0answers
25 views

What does it mean for a polar coordinate system to have basis vectors?

So I understand that every element of a vector space can be represented uniquely by a linear combination of the basis vectors: $v=\alpha_1v_1+\cdots+\alpha_nv_n$ Then coordinates to those basis ...
1
vote
1answer
27 views

Change of Variables in Laplacian

I know how to prove $$\nabla^2(\frac{1}{r}) = -4\pi\delta^3(\textbf{r}),$$ where in Spherical coordinates, $$ \nabla^2 = {\frac{1}{r^2} \, \frac{\partial}{\partial r} \left( r^2 \, ...
1
vote
1answer
907 views

Extrinsic and intrinsic Euler angles to rotation matrix and back

currently I'm working on the visualization of coordinate systems in space to understand rotation matrices better. Until now I thought everything would be ok, but there is a thing that does not get ...
0
votes
1answer
25 views

Rotating a 3 coordinate point

Im having trouble figuring out how to rotate the z axis 90 degrees. For a programming im making example new_x = y; // For a 90 degree clockwise rotation, the new x is the original y, new_y = ...
0
votes
2answers
35 views

How to get coordinates with a new basis?

I have this exercise. Consider the following vectors of ℝ²: $$ u_1 = \begin{pmatrix}1\\2\end{pmatrix}, \space\space\space u_2 = \begin{pmatrix}2\\1\end{pmatrix} $$ Determine the ...
6
votes
5answers
750 views

Basic understanding of a metric.

What is a metric ? Do a metric depend on the system of coordinates I use ? Does it depend on surfaces (or higher dimensional manifolds. Correct me if I'm wrong using the word) the coordinate frames ...
3
votes
2answers
3k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
2
votes
1answer
742 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
0
votes
1answer
21 views

Position of a point$\ \ C(x|y)\ \ $ on a line between $\ \ A(1|1)\ \ $ and $\ \ B(5|2)\ \ $ when $x (i.e. 1)$ away from $A$

How to calculate the position of a point on a line when the start end endpoint of the line and the distance of the point from the start are given$?$ Example: $A(1|1)$ $B(5|2)$ Distance of point ...
1
vote
1answer
38 views

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 ...
0
votes
1answer
11 views

Finding the second point of intersection from a normal on a hyperbola

The question is phrased as follows: "A rectangular hyperbola, W, has equation xy = 12" a) Show that the gradient of the normal, N, to W at the point P(2,6), is 1/3. b) Hence find an equation for ...
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0answers
10 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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vote
2answers
400 views

New x coordinate of a rotated line

I need help finding the equation to find $x$ I work in GIS and I'm working on a script that uses the new x coordinate of a rotated line. I havent work with trigonometry in a long long time so I ...
1
vote
2answers
42 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 ...
0
votes
1answer
48 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 ...
1
vote
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 ...
0
votes
2answers
649 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
0
votes
1answer
21 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 ...
0
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1answer
30 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 ...