# 3D Graphing - from: 
#        http://doc.sagemath.org/html/en/tutorial/tour_plotting.html


# Use plot3d to graph a function of the form z=f(x,y)

x, y = var('x,y')
plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))

# Alternatively, you can use parametric_plot3d to graph a parametric surface where each of x,y,z is determined # by a function of one or two variables (the parameters, typically u and v). The previous plot can be expressed # parametrically as follows:

u, v = var('u, v')
f_x(u, v) = u
f_y(u, v) = v
f_z(u, v) = u^2 + v^2
parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))

# The third way to plot a 3D surface in Sage is implicit_plot3d, which graphs a contour of a function like 
# f(x,y,z)=0 (this defines a set of points). We graph a sphere using the classical formula:

x, y, z = var('x, y, z')
implicit_plot3d(x^2 + y^2 + z^2 - 4, (x,-2, 2), (y,-2, 2), (z,-2, 2))

# Here are some more examples ... 
# (see Wikipedia for additional explanations of Whitney's umbrella, Cross cap, etc.)

# Yellow Whitney’s umbrella:

u, v = var('u,v')
fx = u*v
fy = u
fz = v^2
parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1), frame=False, color="yellow")

# Cross cap:

u, v = var('u,v')
fx = (1+cos(v))*cos(u)
fy = (1+cos(v))*sin(u)
fz = -tanh((2/3)*(u-pi))*sin(v)
parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red")

# EXERCISE: plot your favorite surface!
# The graph of a function is a parametric plot: f_x=x, f_y=y, f_z=z(x,y)
# ...