# Klein Bottle, Spherical to XYZΒΆ

The main point of this example is that

The development of the functional definition of a surface is the hard part, visualizing the three-dimensional surface is fairly easy using S3Dlib with Matplotlib.

As seen in the below script, this surface was constructed using a SphericalSurface object. Alternatively, a PlanarSurface object could be used as demonstrated in the Mapping guide plot.

A detailed description of a Klein Bottle is found in Wikipedia where the functional definition is located.

```import numpy as np
from matplotlib import pyplot as plt
import s3dlib.surface as s3d
import s3dlib.cmap_utilities as cmu

#.. Klein Bottle, Spherical to XYZ

# 1. Define function to examine ....................................

def klein(rtp) :
r,t,p = rtp
u = p
v = t
cU, sU = np.cos(u), np.sin(u)
cV, sV = np.cos(v), np.sin(v)
x = -(2/15)*cU* \
(  ( 3 )*cV + \
( -30 + 90*np.power(cU,4) - 60*np.power(cU,6) + 5*cU*cV )*sU \
)
y = -(1/15)*sU* \
(  ( 3 - 3*np.power(cU,2) -48*np.power(cU,4) +48*np.power(cU,6) )*cV + \
(-60 + ( 5*cU - 5*np.power(cU,3) - 80*np.power(cU,5) + 80*np.power(cU,7) )*cV  )*sU \
)
z = (2/15)*( 3 + 5*cU*sU )*sV
return x,y,z

# 2. Setup and map surface .........................................
rez=6
cmap = cmu.mirrored_cmap('viridis',rev=True)
cmap = cmu.alpha_cmap(cmap,0.7)

surface = s3d.SphericalSurface(rez,basetype='octa_c', linewidth=0 )
surface.map_geom_from_op( klein, returnxyz=True )
surface.map_cmap_from_normals(cmap=cmap, direction=[1,1,1])
surface.transform(s3d.eulerRot(0,-90),translate=[0,0,2])

# 3. Construct figure, add surface plot ............................

fig = plt.figure(figsize=plt.figaspect(1))
fig.text(0.975,0.975, "Klein Bottle", \
ha='right', va='top', fontsize='larger', multialignment='right')
ax = plt.axes(projection='3d', aspect='equal')
minmax = (-1.5,1.5)
ax.set(xlim=minmax, ylim=minmax, zlim=minmax)
ax.set_axis_off()
ax.view_init(elev=20, azim=-125)