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### How can I get an image in 3D using the axial symmetry of the graph in polar coordinates.

I have equations and the 2D curve drawing in polar coordinates:

Equations and 2D curve drawing in polar coordinates.

var('beta, phi, R, d') a(phi,beta)=(cos(phi)+beta)/(1+betacos(phi)) # Transformation function
r1(phi,beta,R,d) = d
a + sqrt(d^2(a^2-1)+R^2) # First equation in polar coordinates r2(phi,beta,R,d) = da - sqrt(d^2*(a^2-1)+R^2) # second equation submission of both charts for parameters: beta=0.8, R=1, d=2 polar_plot(r1(phi,0.8,1,2),(-pi/2+0.16,-0.16+pi/2),color='red') + polar_plot(r2(phi,0.8,1,2),(-pi/2+0.18, pi/2-0.18))

### How can I get an image in 3D using the axial symmetry of the graph in polar coordinates.

I have equations and the 2D curve drawing in polar coordinates:

Equations and 2D curve drawing in polar coordinates.

var('beta, phi, R, d') a(phi,beta)=(cos(phi)+beta)/(1+betacos(phi)) d')

a(phi,beta)=(cos(phi)+beta)/(1+beta*cos(phi)) # Transformation function

r1(phi,beta,R,d) = da + sqrt(d^2(a^2-1)+R^2) # First equation in polar coordinates coordinates

r2(phi,beta,R,d) = da - sqrt(d^2*(a^2-1)+R^2) sqrt(d^2(a^2-1)+R^2) # second equation equation

submission of both charts for parameters: beta=0.8, R=1, d=2 d=2

polar_plot(r1(phi,0.8,1,2),(-pi/2+0.16,-0.16+pi/2),color='red') + polar_plot(r2(phi,0.8,1,2),(-pi/2+0.18, pi/2-0.18)) pi/2-0.18)) 3 None

### How can I get an image in 3D using the axial symmetry of the graph in polar coordinates.

I have equations and the 2D curve drawing in polar coordinates:coordinates.

Equations and 2D curve drawing in polar coordinates.

var('beta, phi, R, d')d')

# Transformation function
a(phi, beta) = (cos(phi) + beta)/(1 + beta*cos(phi))

# Two equations in polar coordinates
r1(phi, beta, R, d) = d*a + sqrt(d^2*(a^2 - 1) + R^2)
r2(phi, beta, R, d) = d*a - sqrt(d^2*(a^2 - 1) + R^2)


Plot both charts for parameters beta=0.8, R=1, d=2.

a(phi,beta)=(cos(phi)+beta)/(1+beta*cos(phi)) # Transformation function

r1(phi,beta,R,d)

p = da polar_plot(r1(phi, 0.8, 1, 2), (-pi/2 + sqrt(d^2(a^2-1)+R^2) # First equation in polar coordinates r2(phi,beta,R,d) = da 0.16, -0.16 + pi/2), color='red')
p += polar_plot(r2(phi, 0.8, 1, 2), (-pi/2 + 0.18, pi/2 - sqrt(d^2(a^2-1)+R^2) # second equation submission of both charts for parameters: beta=0.8, R=1, d=2 polar_plot(r1(phi,0.8,1,2),(-pi/2+0.16,-0.16+pi/2),color='red') + polar_plot(r2(phi,0.8,1,2),(-pi/2+0.18, pi/2-0.18))0.18))
p