Parametric Curves and Polar Coordinates

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
from sympy import plot_parametric, symbols
from IPython.display import display, Markdown
from functools import partial

## This is a 'partial function' so that we don't have to set figure size and aspect ratio every time.
splot = partial(plot_parametric, aspect_ratio = (1,1),size=(4,4),axis_center = (0,0))


def _displayM(text, expr):
  """
  Display text and expression inline. 
  """
  display(Markdown('{} {}'.format(
            text,
            sp.latex(expr, mode='inline')
          ))
  )

Example (r = 1 - cos(t))

# x, y = sp.symbols('x y')
r, t = sp.symbols('r t')

r = 1 - sp.cos(t)

x = r * sp.cos(t)
y = r * sp.sin(t)
splot((x,y), (t, 0, 2 * sp.pi))

Example ($r^2 = 4 cos(t))

## NUMPY
## -----
# Define the parameter
t = np.linspace(0, 2 * np.pi, 1000)

# Compute r from the parametrization
r = np.sqrt(4* np.cos(t))

# Compute x and y for the two lobes
x1 = r * np.cos(t)
y1 = r * np.sin(t)

x2 = -r * np.cos(t)
y2 = -r * np.sin(t)

# Plot using Matplotlib
plt.figure(figsize=(6, 6))
plt.plot(x1, y1)
plt.plot(x2, y2)

plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(True, which='both')
plt.gca().set_aspect('equal', adjustable='box')
plt.show()

C:\Users\Shehry\AppData\Local\Temp\ipykernel_24584\4250302634.py:7: RuntimeWarning: invalid value encountered in sqrt
  r = np.sqrt(4* np.cos(t))

## SYMPY
##-------
r, t = sp.symbols('r t', positive = False)

r_sq = 4 * sp.cos(t)
r_lobes = [sp.sqrt(r_sq), -sp.sqrt(r_sq)]

## Find x and y
x_lobes = [lobe * sp.cos(t) for lobe in r_lobes]
y_lobes = [lobe * sp.sin(t) for lobe in r_lobes]

splot((x_lobes[0],y_lobes[0], (t, - sp.pi/2, sp.pi/2)),
      (x_lobes[1],y_lobes[1], (t, - sp.pi/2, sp.pi/2))
)

Areas in Polar Coordinates

Of cardioid (r = 2(1+cos(\theta)))

# Total area

r, t = sp.symbols('r t')

r = 2 * (1 + sp.cos(t))

sp.integrate(1/2 * r**2, (t, 0, 2 * sp.pi))

\displaystyle 6.0 \pi

Area between curves

# Area between regions 

##-- Define curves
r1, r2, t = sp.symbols('r1 r2 t')

r1 = 1 - sp.cos(t)
r2 = 1

##-- Find Area

res = sp.integrate((r2**2 - r1**2)/2, (t, -sp.pi/2, sp.pi/2))
_displayM("Area between curves is:", res)


##-- Plot curves
x1 = r1 * sp.cos(t)
y1 = r1 * sp.sin(t)

x2 = r2 * sp.cos(t)
y2 = r2 * sp.sin(t)

splot((x1,y1, (t, -sp.pi/2, sp.pi/2)),
      (x2,y2, (t, -sp.pi/2, sp.pi/2))
)

Area between curves is: 2 - \pi / 4

Intersecting curves

r1, r2, t = sp.symbols('r1 r2 t')

r1 = 2 * sp.cos(t/3)
r2 = sp.sqrt(2)

## Find intersects to r1 == r2.
res = sp.solve(r1 - r2, t, dict = True)

res
## Display neatly
# [_displayM(f"Solution {idx}: ", i[t]) for idx, i in enumerate(res)];

[{t: 3*pi/4}, {t: 21*pi/4}]

x = [r * sp.cos(t) for r in (r1, r2)]
y = [r * sp.sin(t) for r in (r1, r2)]

splot(x[1], y[1], (t, 0, 2 * sp.pi))

splot((x[1], y[1], (t, 0, 2 * sp.pi)),
(x[0], y[0], (t, 0, 4 * sp.pi))

)

Lengths in Polar Coordinates

Length of cardioid (r= 1 - cos(\theta))

# Total area

from sympy.simplify.fu import TR11, TR0


r, t = sp.symbols('r t', real = True)

r = (1 - sp.cos(t))
drdt = sp.diff(r,t)


f = sp.sqrt(r**2 + drdt**2)
integrand = sp.simplify(f)
# integrand = integrand.subs(sp.sqrt(1-sp.cos(t)), sp.sin(t/2)*2)
integrand
# sp.integrate(integrand, (t, 0, 2 * sp.pi))

\displaystyle \sqrt{2 - 2 \cos{\left(t \right)}}

from sympy.simplify.fu import TR11, TR0
from sympy import cos, sin, pi
from sympy.abc import x
TR0(TR11(integrand, base = t*2))

\displaystyle \sqrt{2} \sqrt{1 - \cos{\left(t \right)}}