How to Create 3D Torus from Circle Revolved about x=2r, r is the radius of circle (Python or JULIA)

Question

I need help to create a torus out of a circle by revolving it about x=2r, r is the radius of the circle.

I am open to either JULIA code or Python code. Whichever that can solve my problem the most efficient.

I have Julia code to plot circle and the x=2r as the axis of revolution.

using Plots, LaTeXStrings, Plots.PlotMeasures
gr()

θ = 0:0.1:2.1π
x = 0 .+ 2cos.(θ)
y = 0 .+ 2sin.(θ)
plot(x, y, label=L"x^{2} + y^{2} = a^{2}",
    framestyle=:zerolines, legend=:outertop)

plot!([4], seriestype="vline", color=:green, label="x=2a")

I want to create a torus out of it, but unable, meanwhile I have solid of revolution Python code like this:

# Calculate the surface area of y = sqrt(r^2 - x^2)
# revolved about the x-axis

import matplotlib.pyplot as plt
import numpy as np
import sympy as sy

x = sy.Symbol("x", nonnegative=True)
r = sy.Symbol("r", nonnegative=True)

def f(x):
    return sy.sqrt(r**2 - x**2)

def fd(x):
    return sy.simplify(sy.diff(f(x), x))

def f2(x):
    return sy.sqrt((1 + (fd(x)**2)))

def vx(x):
    return 2*sy.pi*(f(x)*sy.sqrt(1 + (fd(x) ** 2)))
  
vxi = sy.Integral(vx(x), (x, -r, r))
vxf = vxi.simplify().doit()
vxn = vxf.evalf()

n = 100

fig = plt.figure(figsize=(14, 7))
ax1 = fig.add_subplot(221)
ax2 = fig.add_subplot(222, projection='3d')
ax3 = fig.add_subplot(223)
ax4 = fig.add_subplot(224, projection='3d')
# 1 is the starting point. The first 3 is the end point. 
# The last 200 is the number of discretization points. 
# help(np.linspace) to read its documentation. 
x = np.linspace(1, 3, 200)
# Plot the circle
y = np.sqrt(2 ** 2 - x ** 2)
t = np.linspace(0, np.pi * 2, n)

xn = np.outer(x, np.cos(t))
yn = np.outer(x, np.sin(t))
zn = np.zeros_like(xn)
for i in range(len(x)):
    zn[i:i + 1, :] = np.full_like(zn[0, :], y[i])

ax1.plot(x, y)
ax1.set_title("$f(x)$")
ax2.plot_surface(xn, yn, zn)
ax2.set_title("$f(x)$: Revolution around $y$")

# find the inverse of the function
y_inverse = x
x_inverse = np.power(2 ** 2 - y_inverse ** 2, 1 / 2)
xn_inverse = np.outer(x_inverse, np.cos(t))
yn_inverse = np.outer(x_inverse, np.sin(t))
zn_inverse = np.zeros_like(xn_inverse)
for i in range(len(x_inverse)):
    zn_inverse[i:i + 1, :] = np.full_like(zn_inverse[0, :], y_inverse[i])

ax3.plot(x_inverse, y_inverse)
ax3.set_title("Inverse of $f(x)$")
ax4.plot_surface(xn_inverse, yn_inverse, zn_inverse)
ax4.set_title("$f(x)$: Revolution around $x$ \n Surface Area = {}".format(vxn))

plt.tight_layout()
plt.show()

Answer 1

Here is a way that actually allows rotating any figure in the XY plane around the Y axis.

"""
Rotation of a figure in the XY plane about the Y axis:
    ϕ = angle of rotation
    z' = z * cos(ϕ) - x * sin(ϕ)
    x' = z * sin(ϕ) + x * cos(ϕ)
    y' = y
"""

using Plots

# OP definition of the circle, but we put center at x, y of 4, 0
#    for the torus, otherwise we get a bit of a sphere

θ = 0:0.1:2.1π 
x = 4 .+ 2cos.(θ)  # center at (s, 0, 0)
y = 0 .+ 2sin.(θ)
# add the original z values as 0
z = zeros(length(x))
plot(x, y, z, color=:red)

# add the rotation axis
ϕ = 0:0.1:π/2 # for full torus use 2π at stop of range
xprime, yprime, zprime = Float64[], Float64[], Float64[]

for a in ϕ, i in eachindex(θ)
    push!(zprime, z[i] + z[i] * cos(a) - x[i] * sin(a))
    push!(xprime, z[i] * sin(a) + x[i] * cos(a))
    push!(yprime, y[i])
end

plot!(xprime, yprime, zprime, alpha=0.3, color=:green)