Plotting surface and curve in 3D and a curve in xy-plane, all in the same plot

To illustrate an optimization problem, I want all of this in the same 3D plot:

• A surface.
• A curve in the xy-plane.
• A curve/path on the surface which marks out the points on the surface that lies directly above the curve in the xy-plane.

This is my code so far:

import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator
import numpy as np
from mpl_toolkits import mplot3d

fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
X = np.linspace(-5,5,100)
Y = X
X, Y = np.meshgrid(X, Y)
Z = 50 - X**2 - Y**2

#Plotting curve on the surface
ax = plt.axes(projection='3d')

yline = np.linspace(-5,5,100)
xline = -np.sqrt(4/(2+yline**2)) #the x-values of the curve in the xy-plane
zline = 50 - xline**2 - yline**2 

ax.plot3D(xline, yline, zline, "black")

surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm)
ax.set_zlim(0, 50)

#Plotting curve in xy-plane
a = 5
g = 1 - 2*X - X*Y**2
plt.contour(X,Y,g, [a], offset=0)

plt.show()

Here is the plot from two different angles:

Some problems:

• First of all, it seems like the axes have been numbered twice. Is that because I make a meshgrid, and later on use ax.plot3D? That I use two different ways of plotting something, and as a consequence make the 3D space twice?
• The path on the surface appears weakly. Is there a way to make the path more visible?
• From the picture in bird perspective, we see that the path does not lie directly above the curve in the xy-plane. What would be easier, was if Python had a built-in function who could project the curve in the xy-plane directly onto the surface. Am I missing something here? Does anyone know of such a function?

These questions might be dummy questions, but answers and tips are highly appreciated!