3d plot with multiple lines showing the projection on the xy plane - matplotlib

Question

I was wondering how to have a 3d plot with multiple lines showing the projection on the xy plane by means of something like fill_between but in 3D. I have here a sample code.
fig, ax = plt.subplots(figsize=(12,8),subplot_kw={'projection': '3d'})
for i in np.arange(0.0,1,0.1):
x1=np.arange(0,1-i+0.01,0.01)
y1=1.0-i-x1
def z_func(x,y,z):
return x+y**2+0.5*z #can be any fn
coordinates3= [[i,j,1-i-j] for j in np.arange(0,1-i+0.01,0.01)]
z1=np.array([z_func(*k) for k in coordinates3])
ax.plot(x1,y1,z1)
ax.view_init(azim=10,elev=20)
plt.show()
I'd like to have each line 'projected' on the xy plane, with a shaded filling in between the curve and its projection. Anybody knows a quick way?

Answer 1

After the suggestion in the comments of #ImportanceOfBeingErnest, I was able to write a solution. I came up with this:
fig, ax = plt.subplots(figsize=(12,8),subplot_kw={'projection': '3d'})
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
for color,i in enumerate(np.arange(0.0,1,0.1)):
x1=np.arange(0,1-i+0.01,0.01)
y1=1.0-i-x1
def z_func(x,y,z):
return x+y**2+0.5*z #can be any fn
coordinates3= [[i,j,1-i-j] for j in np.arange(0,1-i+0.01,0.01)]
z1=np.array([z_func(*k) for k in coordinates3])
verts=[[(k[1],k[2],z_func(*k)) for k in coordinates3]]
verts[0].insert(0,(coordinates3[0][1],coordinates3[0][2],0))
verts[0].insert(0,(coordinates3[-1][1],coordinates3[-1][2],0))
poly = Poly3DCollection(verts,color=colors[color])
poly.set_alpha(0.2)
ax.add_collection(poly)
ax.plot(x1,y1,z1,linewidth=10)
ax.view_init(azim=10,elev=20)
plt.show()
One thing that puzzles me is that the shade doesn't get the color of the line and that I had to supply it myself. If you remove the color=colors[color] in the Poly3DCollection you always get blue shades, whereas the lines automatically get the different colors, as one can see in the question. Anybody knows a reason for this?