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Questions tagged [polar-coordinates]

Questions on polar coordinates, a coordinate system where points are represented by their distance from the origin ($r$) and the angle the line joining the point and the origin makes with the positive horizontal axis ($\theta$).

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Parametric equation of a rose with $5$ petals, orthogonal to $x+y+z=3$, with radius $4$, center $(1,1,1)$, and a petal in the direction of $(0,0,3)$

I am kind of stuck in this exercise. Write the parametric equation of a rose-shaped curve with 5 petals, radius 4, centered on $(1,1,1)$, orthogonal to the plane $x+y+z=3$, such ...
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2answers
54 views

Convert $𝑥^2+6𝑦−9=0 $to polar.

I need to convert this rectangular equation to polar. I already took a look at some solutions but I have a problem understanding this part: how was the square root simplified to three?
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2answers
50 views

Find all solutions in $\mathbb C$ for $z^4 = 1$

To start, I write the equation in polar form: $$|z|^4(cos^4\theta + isin^4\theta) = 1(1 + 0i)$$ Next, I want to solve for $\theta$: $$cos4\theta = 1 \textrm{ and } sin4\theta = 0$$ $$4\theta = cos^{-...
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0answers
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Appropriate boundary conditions of the wave equation

Here, I have some acoustic waves generated in a compressible fluid by small oscillations of a cylinder with boundary at $r=a$. In this problem, I am only interested in the solution outside of the ...
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2answers
26 views

Do we use the x-axis for defining polar coordinates?

"To represent any point in a two dimensional space, you need two variables" right? For example (using our good old friend, The cartesian coordinates) we can talk about any point in two dimensional ...
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1answer
37 views

Domain range of a straight line y = 2x + 3 for polar coordinate θ

A straight line is defined by equation $y=2x+3$ in Cartesian coordinate system XY. I need to specify the domain range for the polar coordinate u which is valid for this straight line. Firstly, to ...
2
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1answer
26 views

Is the derivative of the radius function of polar coordinates well-defined?

In polar coordinates there is the following definition: $ x = \rho \cos \phi $, from which it follows that $ \frac{ \mathrm d x}{\mathrm d \rho} = \cos \phi $ and that $ \rho = x \cos^{-1} \phi $. ...
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generic parabola in polar coordinates

Starting from the equation $y=ax^2+bx+c$ substituting I get the next equation in polar coordinates: $$a\cos^2 \theta\ \rho^ 2 + (b \cos \theta - \sin \theta)\ \rho + c = 0$$ in case C was $0$ we could ...
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Need help understanding polar coordinates transformations on higher dimensions

I'm trying to solve a supposedly simple problem in probability where $x$ and $y$ are vectors in $\mathbb R^n$ and $$f(x,y) = c \exp\left[-\frac{1}{2} \left(||x||^2 + ||y||^2\right)\right],$$ where $||...
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1answer
15 views

Archimedean spiral - symmetry test

It is usually stated in Precalculus textbooks that in polar coordinates when a relation between $r$ and $\theta$ passes a symmetry test then the curve described by that relation has that symmetry. ...
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40 views

Find the polar form of $12 + 5i$

Polar form: $\vert z \vert \big(\cos\theta + i\sin\theta \big)$ $$\begin{aligned}z^2 &= \vert 12^2 + 5^2 \vert\\ z &= \vert 13 \vert \\ \arctan \frac{5}{12} &= 22.61^\circ\\ z &= |13| ...
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4answers
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Writing a number in polar form (help converting $\theta$ to $\pi$)

Write $w = \sqrt{3} - i$ in polar form. How is $\theta = \frac{-1}{\sqrt3} \textrm{ converted into } \frac{-\pi}{6}$? I understand that $w$ lies in the fourth quadrant of the unit circle, but ...
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0answers
24 views

Finding the whole triangle information by one point.

I wonder if there's any way to blend two (or more) RGB colors in a reversible way? I mean, imagine we have an RGB pixel (R: 55, G:35, B:255), and we need to extract the two other RGB pixels that ...
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2answers
25 views

Integration in polar coordinates

I was wondering if this property of integration is held true in the polar coordinates $$ \int_a^b f(\theta)\,\mathrm{d}\theta = -\int_b^a f(\theta)\,\mathrm{d}\theta $$
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Integrating $|x|^{-p} \chi_{\{|x|<1 \}}(x)$ two different ways

This question was adopted from the Leib and Loss textbook, my professor hasn't covered polar coordinates in class, and the first mention of polar coordinates in the textbook is this problem. I plan on ...
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40 views

Verify my work : Polar equation to Cartesian Equation

If we are given Polar equation $r = 2a\sin(\theta-\theta_{0})$ where $a,\theta_{0}$ are constants, we are asked to transform this equation to Cartesian equation. My attempt : $\sin(\theta-\theta_{0})...
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1answer
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Encoding numbers from 0 to 255 using Huffman coding.

How can I encode numbers from 0 to 255 using Huffman coding (or any other code), so that each number (especially the largest numbers such as 255) wouldn't take 8 bits of binary space? In other words, ...
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2answers
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Stuck in something: How to write coordinates in one number?

I have a X, Y coordinate system which starts from 0 and ends in 255 on each axis. Thus, I can fit 65,025 numbers in it. Imagine each number as a pixel, so I have 65,0250 pixels in my coordinate system....
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1answer
25 views

Complex number (finding range of values)

Let $z_1 = 2 e^{i\pi/6}$ and $z_2 = re^{i\theta}$, where $r>0$ and $0\le\theta<2\pi$. Find the range of values of $r$ and $\theta$ for which $z_1z_2$ is: a) a real number greater than $5$ ...
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3answers
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How do I convert $(x-1)^2 + (y-\sqrt 3)^2 ≤ 4$ into polar form?

I need to draw the region underneath: $y ≤ x/\sqrt 3$ and the the circle: $(x-1)^2 + (y-\sqrt 3)^2 ≤ 4$ My guess would be: $x = 2cos(\theta) +1$ and $y = 2cos(\theta) +\sqrt 3$. But the ...
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1answer
39 views

Does a change of variable affect the function

If I have a function $f: \mathbb{R}^2 \to \mathbb{R}$ then I can get $f$ in polar coordinates by doing : $f \circ g(r, \theta)$ where $g(r, \theta) = (r \cos \theta, r \sin \theta)$. Now my question ...
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1answer
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You are given the polar curve r=cos(2θ). Find the points where the tangent line is horizontal and where the tangent line is vertical.

Some answers are listed below that I have gotten right. Unfortunately I am not getting the right answers for the majority of them a. (a) List all of the points $(r,\theta)$ where the tangent line is ...
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3answers
48 views

How do I prove that two equations in Cartesian and Polar coordinates are equivalent?

I'm asked to verify that the set of points described by the cartesian equation $$(x-1)^2 + y^2 = 1$$ is equal to the set of points described by the polar equation $$r = 2 \cos{\theta}, \cos{\theta} &...
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2answers
46 views

Polar coordinates to find area

Is there a way to use polar coordinates to find the area enclosed by the loop of the curve $y^2=(x+1)^2(3-x)$ instead of using cartesian? Wondering if there is a neater method. Thanks!
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1answer
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Is there a formula for the silver spiral?

I can construct a golden spiral by rotating segments 90 degrees and scaling their segments by the golden ratio, and then use the polar formula on wikipedia to interpolate points at other angles to ...
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0answers
55 views

Find the extremals of the functional $\int\sqrt{x^2+y^2}\sqrt{1+(y'(x))^2}dx$?

I want to use the polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$. After transformation, I get $$\int r\sqrt{r^2+(r')^2}d\theta.$$ Then, I derived the Euler-Lagrange equation $$2r^2-rr''+3(r')^2=...
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0answers
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How do I graph a 30-petal rhodonea curve (rose)?

A rose of the form $r = \cos(c\theta)$ normally has $2c$ petals when $c$ is even and $c$ petals when $c$ is odd. A rose may also have $4c$ petals when $c$ can be expressed as $Z + \frac{1}{2}$ where $...
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2answers
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Polar coordinates transformation of $\dot{x} = -y + x(1-x^2-y^2)$ and $\dot{y} = x + y(1-x^2-y^2)$

We have $$\begin{cases}\dot{x} = -y + x(1-x^2-y^2)\\ \dot{y} = x + y(1-x^2-y^2) \end{cases}$$ By taking $x=r\cos\theta$, $y=r\sin\theta$, I am able to get to the expression $$\dot{r} = r(1-r^2) + \...
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How/Why do I find the Hyperplane to Calculate the Volume of $A$

Let $A:=\{(x,y,z) \in \mathbb R^{3}:x^2+y^2+z^2\leq 2\} \cap \{(x,y,z)\in \mathbb R^{3}:x^2+y^2\leq z\}$ I know that $\{(x,y,z) \in \mathbb R^{3}:x^2+y^2+z^2\leq 2\}$ is a sphere with radius $\sqrt{...
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1answer
40 views

polar curves - Why do we reflect when $r$ is negative

I've been looking for a answer to why we reflect the angle whenever the length "$r$" is negative. Why does this change the sign for $r$, when adding $180$ degrees to the angle $\theta$? It makes ...
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0answers
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How to integrate $ y\cos^4x$ over a disk?

I'm trying to integrate $$\int_D y\cos^4x \, \mathrm dx \mathrm dy$$ where $D=\{(x,y)\in R^2;x^2+y^2<\pi\}.$ I'm thinking using the polar coordinates but doing so the integral would become, $$\...
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3answers
67 views

How to determine the area of a rotated ellipse?

The ellipse $6x^2+4xy+5y^2+8x+8y+1=0$ is neither expressed in terms of $x$; like $y=\pm\sqrt{a^2-x^2}$, nor in terms of $y$; like $x=\pm\sqrt{a^2-y^2}$. Separation of $x$ (or $y%$) may be impossible. ...
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1answer
50 views

Differentiate polar curve

I have attempted this question but have not found a solution. I am currently stuck. Hints on how I may go further would be helpful. Thank You in advance. The Question: What I have done so far:
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1answer
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Computing the Laplacian in Polar Coordinates [duplicate]

Similar questions have been asked on this site but none of them seemed to help me. I'm asked to compute the Laplacian $$\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ in terms of ...
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1answer
21 views

$r = ycos(\theta) - xsin(\theta) $ derivation for Hough Transform

I am trying to see how $y = xtan(\theta) + \dfrac{r}{cos(\theta)}$ is made from the graph. Also how does the derivation work if $(x, y)$ is in the different quadrant? i.e. the $\theta$ location stays ...
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2answers
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Grad in polar coordinates

As part of my lectures, it is noted that $\nabla \sin \theta \propto \frac{1}{r^2}$ and $\nabla \phi \propto \frac{1}{r^2}$ where we are working in spherical polar coordinates with $\theta$ as the ...
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1answer
49 views

Absolute value of a complex expression

I've been stuck on this problem for a while: $$\vert 1-e^{-i2\pi f}+.5e^{-i2\pi f\cdot 2}\vert^2,$$ where $i =$ the imaginary unit, $(2\pi f) =$ a real value, and $(2\pi f\cdot 2)=$ a real value. I ...
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2answers
52 views

Using polar coordinates to find the critical points

I have written the following DE system $$\dot{x} = x+y-x(x^2+y^2)\\ \dot{y} = -x+3y-y(x^2+y^2) $$ as follows in Polar form: $$\dot{r} = -2r\cos^2\theta+r(3-r^2) $$ $$\dot{\theta} = 2\sin\theta\cos\...
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1answer
44 views

covariant metric tensor in case of polar transformation

Assume usual polar coordinates equations: $$ x=r\cos(\theta) $$ $$ y=r\sin(\theta) $$ In what I think is called covariant we have a Jacobian of: $$ J=\begin{pmatrix} \frac{\partial{x}}{\partial{r}} ...
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1answer
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Calculating the inverse of a system of 4 simultaneous equations that deal with image conversion from image space to cortical space.

I am completely stuck in calculating the inverse of these equations. These equations show how to convert a image coordinate to a cortical coordinate (x,y being the image coordinate and alpha being a ...
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1answer
123 views

Problems with putting $r=e^{-\theta \cot(\theta)}$ into rectangular form

Recently, I found a polar curve which contains all(?) complex numbers $z$, which satisfy $\Im(z^z)=0$; or in other words, all complex numbers which raised to itself give a real number. The curve is ...
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1answer
84 views

How can I convert $(a + bi)^{c+di}$ to polar form?

Specifically, I need to convert $(-8i)^{2+πi/3}$ to polar form. I understand I need to use Euler's formula, $e^{\theta i} =\cos \theta + i \sin \theta$, but I'm not sure about the full process. ...
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1answer
23 views

Quick ways of finding principle curvature in the Cartesian plane

Say I have some polar curve $r=f(\theta)$ in the Cartesian plane (smooth and twice-differential) and I've found a formula for the curvature, $k$. Does a clever trick exist (that isn't merely finding ...
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1answer
31 views

Finding polar coordinates angle for complex numbers given cartesian form

I have the following formula for finding $\theta$ given cartesian form of complex numbers. $$\theta = \begin{cases} \tan^{-1}(\frac{y}{x}) & x \leq 0 \\ \tan^{-1}(\frac{y}{x}) & ...
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4answers
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What is $(xy)^7=3$ in polar coordinates, in the form ____$=r^{14}$? [closed]

This isn't a question of converting coordinates, and I've tried every version of $x=r\cos\theta$, $y=r\sin\theta$ I can think of. So, what is $(xy)^7=3$ converted to an equation in polar coordinates. ...
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2answers
30 views

Differentiating polar functions using complex numbers

I was wondering, given some polar function $r(\theta)$ is it possible to convert it into a complex number in exponential form, then differentiate that and then convert it back and have the appropriate ...
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1answer
38 views

Mass transport equation Cartesian to polar coordinates

Can someone please advise on how to transform the following equation to polar coordinates? $$\frac{\partial \rho(x,t)}{\partial t}=v\frac{\partial \left(\rho(x,t) L(x)\right)}{\partial x}+D\frac{\...
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3answers
72 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
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0answers
24 views

Integral of function undefined at one point

Let us consider a plane with polar coordinates. Let us also consider the following integral over any area $A$ on the plane: $$\iint_A f(r,\theta)\ \hat{r}\ dr\ d\theta\ $$ Here the function is $\...
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1answer
43 views

Converting the polar equation $r=12-\sin\theta+2\sin3\theta+2\sin5\theta-\sin7\theta+3\cos2\theta-2\cos4\theta$ to rectangular form

How do I convert the following polar equation to rectangular equation? $$r = 12 - \sin(θ) + 2\sin(3θ) + 2\sin(5θ) - \sin(7θ) +3\cos(2θ) - 2\cos(4θ)$$