Polar Coordinates
Description
To translate our standard Cartesian coordinates into polar coordinates we can use the functions
(1)\begin{align} r & = \sqrt{x^2 + y^2} \\ \theta & = \tan^{-1} \left(\frac{y}{x}\right). \end{align}
The following Sage interact converts Cartesian coordinates into polar coordinates. The default values are $(x, y) = (1,1)$, provided $x > 0$.
Code
@interact
def _(x = input_box(default=1), y=input_box(default=1)):
r = sqrt( x^2 + y^2 )
t = arctan(y/x)
pretty_print(html(r"$x = %s$" %latex(x)))
pretty_print(html(r"$y = %s$" %latex(y)))
pretty_print(html(r"$r = %s$" %latex(r)))
pretty_print(html(r"$t = %s$" %latex(t)))Sage Cell
Option
We can also convert polar coordinates back into Cartesian coordinates using the formulas
(2)\begin{align} x = r \cos \theta \\ y = r \sin \theta. \end{align}
The Sage interact below converts polar coordinates into Cartesian coordinates. The default values are $(r,\theta) = (2, \frac{ \pi}{2})$.
Code
@interact
def _(r = input_box(default=2), t=input_box(default= pi/2 )):
x = r*cos(t)
y = r*sin(t)
pretty_print(html(r"$r = %s$" %latex(r)))
pretty_print(html(r"$t = %s$" %latex(t)))
pretty_print(html(r"$x = %s$" %latex(x)))
pretty_print(html(r"$y = %s$" %latex(y)))Options
Primary Tags—Precalculus: Polar coordinates & vectors
Secondary Tags—Polar coordinates & vectors: Polar and rectangular coordinates
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Author: J. A. Phillips
Attribute
Date: 30 Oct 2018 17:31
Submitted by: James A Phillips