Catenoid to Helicoid

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Once Loop Reflect

Animation control:

Visualization	Frame Value
Surface geometry	functional parameter per frame
Surface position	fixed to the coordinate axis
Surface color	color per frame
Shading and highlighting	fixed to the coordinate axis
Axis coordinate	constant

Based on the static plots from the Surface Displacement Vector Field and Parametric Set of Surfaces 2 examples.

A parametrization of the deformation is given at Helicoid transformation

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

#.. Helicoid Transformation Animation

# 0. Define animation control parameters ............................

totalTime, f_domain, numFrames = 4, (-1.0,1.0), 80  # time in seconds
frames=np.linspace(*f_domain, numFrames, endpoint=False)
interval = int(1000.0*totalTime/numFrames)          # milliseconds

# 1. Define functions to examine ....................................

def catenoid_helicoid(rtz, A) :
    r,t,z = rtz
    theta = A*np.pi  #  -1 < A < 1
    cosT, sinT = np.cos(theta), np.sin(theta)
    U, V = t, z   
    x =  cosT * np.sinh(V) * np.sin(U) +   sinT * np.cosh(V) * np.cos(U)
    y = -cosT * np.sinh(V) * np.cos(U) +   sinT * np.cosh(V) * np.sin(U)
    Z = ( U/np.pi- 1.0 ) *cosT +  V * sinT
    return x,y,Z

def colormap_by_A(A) :
    hue = (A + 1.0) / 2.0
    return cmu.hsv_cmap_gradient( [hue,1.0,0.25], [hue,0.5,1],smooth=1.6 )

def indicator_by_A(fig, A, vOld=None) :
    symbol, blank = r'$\blacktriangleright$', r'$\blacksquare$'
    horz, vCen, vRng = 0.8, 0.5, 0.28
    vert = vCen + vRng*A
    if vOld is not None: 
        fig.text(horz,vOld,blank, ha='right', va='center', fontsize='x-large', color='w')
    fig.text(horz,vert,symbol, ha='right', va='center', fontsize='large')
    return vert

# 2. Setup and map surfaces .........................................
rez, start_F = 5, -1.0

surface = s3d.CylindricalSurface(rez, basetype='squ_s', cmap=cmu.hue_cmap())
surface.map_geom_from_op( lambda rtz : catenoid_helicoid(rtz,start_F), returnxyz=True )

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

fig = plt.figure(figsize=plt.figaspect(1))
ax = plt.axes(projection='3d', aspect='equal')
cbar = plt.colorbar(surface.cBar_ScalarMappable, ax=ax, ticks=np.linspace(-1,1,5), shrink=0.6 )
cbar.set_label(r'$\theta$ parameter', rotation=270, labelpad = 15)
cbar.ax.set_yticklabels([r'-$\pi$',r'-$\pi$/2','0',r'$\pi$/2',r'$\pi$'])
# surface mapping placed here to change colormap after colorbar defined.
surface.map_cmap_from_op( lambda rtz : rtz[0],colormap_by_A(start_F))
prevIndicator = indicator_by_A(fig, start_F)
# ....
minmax = (-1.2,1.2)
ax.set(xlim=minmax, ylim=minmax, zlim=minmax )
ax.view_init(25, -28)
ax.set_axis_off()

ax.add_collection3d(surface)

fig.tight_layout(pad=1)
#plt.show()

# 4. Animation ======================================================

def update_fig(frame):
    global surface
    global prevIndicator
    surface.remove()

    surface = s3d.CylindricalSurface(rez, basetype='squ_s', antialiased=True)
    surface.map_geom_from_op( lambda rtz : catenoid_helicoid(rtz,frame), returnxyz=True )
    surface.map_cmap_from_op( lambda rtz : rtz[0],colormap_by_A(frame))
    prevIndicator = indicator_by_A(fig, frame, prevIndicator)

    ax.add_collection3d(surface)
    return surface,
 
anim = FuncAnimation(fig, update_fig, frames, interval=interval, repeat=True)
anim.save('cat2heli.html',writer='html')

msg = "saved {} frames, values: [{:.3f} to {:.3f}] @ {} milliseconds/frane"
print(msg.format(numFrames,np.min(frames),np.max(frames),interval))