Surf
This is a small library I wrote to visualize 2D manifolds in 3D space, along with associated quantities (like their curvature or the areas of their surface elements) -- partially because it's cool, and partially to check homework :)
For a visual demo, check out the notebook.
Usage
First, define a function that maps from 2D space to 3D space. It should take in a vector of two components and return a vector of three components. It should also support having a batch dimension (which should be the final index). For example:
def catenoid(x):
U, V = x
X = np.cosh(U)*np.cos(V)
Y = np.cosh(U)*np.sin(V)
return np.array([X,Y,U])Then you can turn this into a Surface and compute interesting quantities:
from surf import Surface
surface = Surface(catenoid)
inputs = np.array([[0,1,2,0,1,2],[0,0,0,1,1,1]]) # (U, V)
nu = surface.normal(inputs)
jac = surface.jacobian(inputs)
hes = surface.hessian(inputs)
gij = surface.metric_tensor(inputs)
hij = surface.shape_tensor(inputs)
K = surface.gauss_curvature(inputs)
H = surface.mean_curvature(inputs)
print(nu.shape) # (3, n)
print(jac.shape) # (3, 2, n)
print(hes.shape) # (3, 2, 2, n)
print(gij.shape) # (2, 2, n)
print(hij.shape) # (2, 2, n)
print(K.shape) # (n,)
print(H.shape) # (n,)There are also helpers for plotting, which are demonstrated in the notebook. Gradients are computed using the wonderful autograd library.