Questions tagged [coordinate-systems]
This tag involves questions on various coordinate systems. The usual Cartesian coordinate system can be quite difficult to use in certain situations. Some of the most common situations when Cartesian coordinates are difficult to employ involve those in which circular, cylindrical, or spherical symmetry is present. For these situations it is often more convenient to use a different coordinate system.
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Finding Cartesian unit vectors in terms of cylindrical unit vectors
When we want to find what the unit vectors of a cylindrical coordinate system would be in Cartesian we can use the following strategy. First we define a position vector in Cartesian:
\begin{align}\vec{...
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How to Convert Equations of Motion from Cylindrical to Custom Coordinates for a Cylindrical Shell?
I have derived the equations of motion for a cylindrical shell using cylindrical coordinates. However, I would like to switch to a specific coordinate system that is commonly used in mechanical ...
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Find the volume and total surface area of the solid formed
Let the planes be defined by $$|x|+|y|+|z|=1$$
Find the volume of the solid enclosed and the total surface area of the solid thus generated.
I am not able visualise the solid. What will it be$?$ I ...
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Show that |E×F|=EFsinα
Given are the two vectors E=5.2ar + 6.5 az and F=8.3ar + 12.8aφ - 3az
The question is
Show that |E×F|=EFsinα
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Is it appropriate to add vectors from different coordinates?
Assume a $X_{W}Y_{W}Z_{W}$ orthogonal coordinate system is transformed to an arbitrary
orthogonal system $X_{C}Y_{C}Z_{C}$ by a 3D transformation matrix M, where M is represented by
4x4 matrix:
$$
\...
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Can a hexagonal grid embed rectangular coordinates?
I'm trying to figure out if a hexagonal grid can embed rectangular coordinates in whole numbers of "Y-steps". In the image below, one "Y-step" is the spacing between red hexagon ...
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Conversion from machine-local polar coordinates (yaw, pitch, roll) to global (azimuth, elevation)
I'm writing a 3D LOGO module in Python.
I recently realized that my approach is flawed. What I implemented is: the turtle besides right/left commands which change its azimuth in the xy plane, received ...
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Straight Lines, Co-ordinate geometry
If the pair of lines $ax^2+2(a+b)xy+by^2=0$ lie along the diameters of a circle and divide it into four sectors such that the area of one is thrice the others. Then:
$3a^2 - 10ab + 3b^2 = 0$
$3a^2 - ...
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Hyperbolic model on $US=\big \lbrace (r,u): r,u \in (0,1) \big \rbrace.$
I'm referencing some material from Hyperbolic coordinates.
Consider the Half Plane (HP)
$$HP=\big\lbrace(p,q):p\in \Bbb R, q>0\big \rbrace$$
and the Quadrant (Q)
$$Q= \big\lbrace(x,y):x<0, y<...
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How would Cartesian Plane look at infinity?
I’m relatively new to mathematics, and I admit I’m still trying to grasp the concept of infinity, so there might be some errors in my thinking.
I understand that $\infty$ is not a real number, which ...
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For what value of "a" the line and the plane are parallel in real coordinate space? [closed]
The question that I am facing is the following:
"For which choice of a real number $a$ are the line $x = y-1 = \frac{(z+1)}{2}$ and the plane $ax+y+2z=3$ in $\mathbb{R}^{3}$ parallel?
a) $a=-5$
b)...
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Find the points at which the diameter intersects the circumference of the circle.
I am working on an optic simulation program. I reached a stage where I am unable to find an algorithm for the desired points.
Here is the required problem,
I figured out how to find the equation of ...
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Is a polynomial having $d+1$ coefficients related to how it uniquely satisfies $d+1$ points? [closed]
A degree $d$ polynomial $p(x)=a_d x^d+a_{d-1} x^{d-1}+\cdots+a_1 x+a_0$ has $d+1$ coefficients. Is this fact related in any meaningful way to how a polynomial uniquely satisfies $d+1$ points with ...
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How to convert Cartesian coordinate to polar coordinate in $-180°$ to $+180°$
I wanted to plot temperature at circumference of a circle. I extracted the data by converting cartesian coordinates into polar coordinates by using ATAN2 function, but the plot is having negative and ...
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Position of a point with respect to parallel lines. [closed]
Prove the fact that if a point in the X-Y plane $(x_1,y_1)$ lies between two parallel lines $ax+by+r=0$ and $ax+by+s=0$ then:
$(ax_1+by_1+r)(ax_1+by_1+s)<0$
I tried to assert the above by using the ...
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Could Hilbert Spaces offer the vehicle to make Synergetic's IVM System Functional?
This one is a bit out there in left field, and hope it doesn't draw too much flak.
Intended more for the "Mathematically Adept" out there ... which I am not.
Please understand that I am not ...
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Parametric form of Parabola [closed]
Recently I was going through co-ordinate geometry. While studying about parabola I encountered something called "parameteric coordinates". In my textbook some parameteric coordinates were ...
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Conformal mapping from quadrics to the plane
I am writing a raytracer (a type of 3D engine) to render quadrics, and I am working on rendering a one-sheeted hyperboloid defined by the zero set of $x^2-y^2+z^2=1$. I need 2D coordinates $a \in [0, ...
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Number of ways to reach a certain point on an hexagonal grid by taking halving steps.
Imagine you begin at the center of a hexagon, with center-to-vertices distance ( $=$ radius) $1$, and can step in any of the directions of the vertices of the hexagon. Every time you take a step, the ...
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Prove that incentre and excenter of a triangle are harmonic conjugates.
Given the coordinates of a triangle $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, prove that the incentre and excenter of the triangle $\triangle ABC$ are harmonic conjugates with respect to the $A$ and $...
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A Problem on Family of circles and family of corresponding orthogonal circles
Problem Statement:
Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two fixed points on xy plane and $R(\alpha , \beta)$ is a point such that $PR:QR=k , \ (k≠1)$ and locus of R for different values of k be ...
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Is it possible to construct a coordinate system which gets "pulled in"
I really hope i can explain our issue adequately and keep this purely math bound, even if it is technically a graphics programming related question. But over on their exchange ill never get an answer. ...
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What's the equivalent for spheres to homogeneous coordinates for projective spaces?
Projective or homogeneous coördinates are an extremely useful parameterization of projective spaces (indeed often used to define them!), but they are redundant -- a projective space of dimension $n$ ...
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Coordinate system for the region between a circle and a square
Consider the region outside a circle of diameter $a$ and inside the square of side length $b$ where $b>a$. I would like to represent this region using a coordinate system $(u,v)$ where the circle ...
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About the definition of a vector in "Mechanics I" by Akira Harashima. Do I understand this definition?
I am reading "Mechanics I" (in Japanese) by Akira Harashima.
A quantity is given by three real numbers $(A_x, A_y, A_z)$ in an orthogonal coordinate system $S$, and for another orthogonal ...
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Nonlinear Coordinate Transform via Intersections?
I've defined a sort of 'warp' procedure for 2d shapes, and I'm curious whether it's familiar within the math canon or even has a name. Given its simplicity I'm sure that there's a formal definition ...
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Are Non-Rectangular Coordinate Systems Ever Used?
In my study of coordinate geometry, I am using the textbook provided by S.L. Loney. In the textbook, he derives various formulas for rectangular coordinates as well as coordinates inclined at a ...
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Coordinates equipped with equivalences in other conventional (or unconventional) ways
I know coordinates with equivalences of the type :
(a,b,c, ...) ~ (a+k, b+k, c+k, ...)
(a,b,c, ...) ~ (ak, bk, ck, ...)
Does exist some other type of coordinates (and its geometry associated) that
...
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Understanding The Parabola $x^2 + 2y = 8x –7$
Now, I am reading S.L. Loney's book on coordinate geometry, and am having a hard time figuring the way he examines parabolas. The standard equation to the parabola is introduced to be $y^2 = 4ax$ ...
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If length AB, $(x_1,y_1)$ angle POQ are given. Is it possible to get point B $(x_2,y_2)$ ? Yes or No ? Or is the information insufficient?
This is a modification of my previous question here:
Is it possible to get the value of angle BAC in this case?
If length AB (say $100$m), $(x_1,y_1)$ say $(-50,0)$ and angle POQ ($30$ degrees) are ...
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curvilinear coordinates - meaning of a function
I am taking a course in mathematics for physicists and in one of the lectures my professor has written the following paragraph:
"given the transformation $q1 = q1(x,y,z)$, $q2 = q2(x,y,z)$, $q3 = ...
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Unit Vectors and Coordinate Systems
Are $x$, $y,$ and $z$ unit vectors coordinate system independent like any true vector? The same question can be asked of cylindrical and spherical unit vectors. I know that one may perform the same ...
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Standard and Polar Coordinate Conversions on $S^1$
I'm having a little trouble formalizing the idea of using "ambient standard coordinates" on level sets of $\mathbb{R}^n$. For example, we typically write $S^1=\{(x,y)\in\mathbb{R}^2:x^2+y^2=...
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Intersecting lines could have same direction cosines?
When direction cosines of a line are calculated as (per their definition states) the 'cosines' of the angles made by the line with positive X, Y, Z axis (angles measured counterclockwise), we get a ...
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Value of $m$ for which points are collinear.
Find value of $m$ for which $(1,4),(4,5)$ and $(m,m)$ are collinear.
What I try : If points are collinear , Then all points lie on same line
So equation of line through $(1,4)$ and $(4,5)$ is
$\...
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Find the shortest distance of the curve $\frac{(x-3)^2}{4}+\frac{(y-1)^2}{9}=1$ from the origin
https://math.stackexchange.com/a/185627
Referring to the methods mentioned in this answer, after trying the first method I got a term that looked like this:
$$\sqrt{14 + 5\sin^2{\theta} + 6\sin{\theta}...
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Consider $A(3, 1)$ and $C(4, 3)$. There is a point $B(x, y)$ on the curve $y = x^2$. Minimise $|AB|^2 + |BC|^2$.
I took B as $B(x, x^2)$. So far, I tried using the distance formula to get an expression for $|AB|^2 + |BC|^2$. I got $(3 - x)^2 + (1 - x^2)^2 + (x - 4)^2 + (x^2 - 3)^2$. Expanding this out gives $2x^...
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How can I find how far above the ground is point $A$ with gradients?
This is the problem. I feel like there is something missing I can't relate how we use gradients to find the point
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Euler angles to ENU coordinates
Given the LLA coordinates and Euler angles (orientation) of a phone, where alpha = beta = gamma = 0 when the top of the phone is pointing north and facing up, I would like to find the unit ENU ...
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What is Meant by "Change of Origin" in Coordinate Geometry?
I don't think I understand what is meant by "to shift the origin of coordinates to the point $(h,k)$ in coordinate geometry. I've read Loney's book on coordinate geometry in which he says that to ...
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Problem With Transformation of Coordinates
In S.L. Loney's book on Coordinate Geometry, he introduces the reader a way to switch from one origin of coordinates to another. The method he shows is: to change the origin to the point $(h,k)$ you ...
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Homogeneous coordinates in projective geometry
I am studying Projective Geometry in 3D for Computer Vision. I am confused on the high-level rationale behind our need to map from heterogeneous to homogeneous coordinates, and I would like to confirm ...
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When are spherical harmonic expansions valid?
It is known that a square integrable function on the sphere can be expanded in a basis of spherical harmonics,
$$
f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l c_l^m Y_l^m(\theta,\phi)
$$
where $\...
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Doubts Regarding the group structure on Lie groups.
I have some conceptual doubts on the notion of a lie group, or perhaps on the notion of topological group. A lie group is manifold equipped with a continuous group structure of multiplication and ...
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How to relate a derivative in tortoise coordinates with the radial coordinate?
Since the radial coordinate and the tortoise coordinate are related by
r* = r + 2M Log (r/2M-1)
How can I write the derivative ...
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Is It using basis change? Which is the way?
Let be $F : P^4(R)\rightarrow P^4(R)$ an homography with the matrix A. (P is the projective space)
$A = [[1,1,0,0,0], [0,1,1,0,0], [0,0,1,0,0], [0,0,1,2,1],[1,0,0,0,1]]$
I have the plane $Z : x_2 = ...
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$100$ random points and a line to divide them into half
Once in a math interview in college my friend was asked the question,
"If there are $100$ points that are chosen on a plane randomly such that no three points are collinear. Can we always draw ...
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Suppose there exist a point $(x_{0}, y_{0})$ on the circle $(x-r-1)^2+y^2=r^2$ satisfying $4x_{0}-y_{0}^2\leq0$. Then minimum value of $|r|$ is
Suppose there exist a point $(x_{0}, y_{0})$ on the circle $(x-r-1)^2+y^2=r^2$ satisfying $4x_{0}-y_{0}^2\leq0$. Then minimum value of $|r|$ is
My Approach:
Put given point $(x_{0},y_{0})$ in the ...
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0
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Can a Quadratic have a straight line graph or can a linear equation have a parabola graph
While reading a Olympiad book about quadratic equations, I came across a question in my mind that I will express below.
Can a linear equation like ab+by=c have a parabola graph or Can a quadratic ...
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How to prove that shortest distance between any two points is always a straight line? [duplicate]
While solving a book there was a statement that prove that shortest distance between two points is always a straight line by
Geometry
Coordinate geometry
I was unable to prove it.
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