<p>I have an approach that should be applicable to an arbitrary 3D surface, even when that surface has holes in it or is noisy.  It's pretty slow right now, but it seems to work and may give you some ideas for how to do this.</p>

<p>The basic premise is a differential geometric one and is to:</p>

<p>1.) Generate a pointset representing your surface</p>

<p>2.) Generate a k nearest neighbors proximity graph from this pointset (I also normalized distances across dimensions here as I felt it captured the notion of "neighbors" more accurately)</p>

<p>3.) Calculate the tangent spaces associated with each node in this proximity graph by using the point and its neighbors as columns of a matrix that I then perform SVD on.  After SVD, the left singular vectors give me a new basis for my tangent space (the first two column vectors are my plane vectors, and the third is normal to the plane)</p>

<p>4.) Use dijkstra's algorithm to move from a starting node to an ending node on this proximity graph, but instead of using euclidean distance as edge weights,  use the distance between vectors being parallel transported via tangent spaces.</p>

<p>It's inspired by this paper (minus all the unfolding): <a href="https://arxiv.org/pdf/1806.09039.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1806.09039.pdf</a></p>

<p>Note that I left a few helper functions I was using in that probably aren't relevant to you directly (the plane plotting stuff mostly).</p>

<p>The functions you'll want to look at are get_knn, build_proxy_graph, generate_tangent_spaces, and geodesic_single_path_dijkstra.</p>

<p>The implementation could also probably be improved.</p>

<p>Here's the code:</p>

<pre><code>import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from mayavi import mlab
from sklearn.neighbors import NearestNeighbors
from scipy.linalg import svd
import networkx as nx
import heapq
from collections import defaultdict


def surface_squares(x_min, x_max, y_min, y_max, steps):
    x = np.linspace(x_min, x_max, steps)
    y = np.linspace(y_min, y_max, steps)
    xx, yy = np.meshgrid(x, y)
    zz = xx**2 + yy**2
    return xx, yy, zz


def get_meshgrid_ax(x, y, z):
    # fig = plt.figure()
    # ax = fig.gca(projection='3d')
    # ax.plot_surface(X=x, Y=y, Z=z)
    # return ax
    fig = mlab.figure()
    su = mlab.surf(x.T, y.T, z.T, warp_scale=0.1)


def get_knn(flattened_points, num_neighbors):
    # need the +1 because each point is its own nearest neighbor
    knn = NearestNeighbors(num_neighbors+1)
    # normalize flattened points when finding neighbors
    neighbor_flattened = (flattened_points - np.min(flattened_points, axis=0)) / (np.max(flattened_points, axis=0) - np.min(flattened_points, axis=0))
    knn.fit(neighbor_flattened)
    dist, indices = knn.kneighbors(neighbor_flattened)
    return dist, indices


def rotmatrix(axis, costheta):
    """ Calculate rotation matrix

    Arguments:
    - `axis`     : Rotation axis
    - `costheta` : Rotation angle
    """
    x, y, z = axis
    c = costheta
    s = np.sqrt(1-c*c)
    C = 1-c
    return np.matrix([[x*x*C+c,    x*y*C-z*s,  x*z*C+y*s],
                      [y*x*C+z*s,  y*y*C+c,    y*z*C-x*s],
                      [z*x*C-y*s,  z*y*C+x*s,  z*z*C+c]])


def plane(Lx, Ly, Nx, Ny, n, d):
    """ Calculate points of a generic plane 

    Arguments:
    - `Lx` : Plane Length first direction
    - `Ly` : Plane Length second direction
    - `Nx` : Number of points, first direction
    - `Ny` : Number of points, second direction
    - `n`  : Plane orientation, normal vector
    - `d`  : distance from the origin
    """

    x = np.linspace(-Lx/2, Lx/2, Nx)
    y = np.linspace(-Ly/2, Ly/2, Ny)
    # Create the mesh grid, of a XY plane sitting on the orgin
    X, Y = np.meshgrid(x, y)
    Z = np.zeros([Nx, Ny])
    n0 = np.array([0, 0, 1])

    # Rotate plane to the given normal vector
    if any(n0 != n):
        costheta = np.dot(n0, n)/(np.linalg.norm(n0)*np.linalg.norm(n))
        axis = np.cross(n0, n)/np.linalg.norm(np.cross(n0, n))
        rotMatrix = rotmatrix(axis, costheta)
        XYZ = np.vstack([X.flatten(), Y.flatten(), Z.flatten()])
        X, Y, Z = np.array(rotMatrix*XYZ).reshape(3, Nx, Ny)

    eps = 0.000000001
    dVec = d #abs((n/np.linalg.norm(n)))*d#np.array([abs(n[i])/np.linalg.norm(n)*val if abs(n[i]) &gt; eps else val for i, val in enumerate(d)]) #
    X, Y, Z = X+dVec[0], Y+dVec[1], Z+dVec[2]
    return X, Y, Z


def build_proxy_graph(proxy_n_dist, proxy_n_indices):
    G = nx.Graph()

    for distance_list, neighbor_list in zip(proxy_n_dist, proxy_n_indices):
        # first element is always point
        current_node = neighbor_list[0]
        neighbor_list = neighbor_list[1:]
        distance_list = distance_list[1:]
        for neighbor, dist in zip(neighbor_list, distance_list):
            G.add_edge(current_node, neighbor, weight=dist)
    return G


def get_plane_points(normal_vec, initial_point, min_range=-10, max_range=10, steps=1000):
    steps_for_plane = np.linspace(min_range, max_range, steps)
    xx, yy = np.meshgrid(steps_for_plane, steps_for_plane)
    d = -initial_point.dot(normal_vec)
    eps = 0.000000001
    if abs(normal_vec[2]) &lt; eps and abs(normal_vec[1]) &gt; eps:
        zz = (-xx*normal_vec[2] - yy*normal_vec[0] - d)/normal_vec[1]
    else:
        zz = (-xx*normal_vec[0] - yy*normal_vec[1] - d)/normal_vec[2]
    return xx, yy, zz


# def plot_tangent_plane_at_point(pointset, flattened_points, node, normal_vec):
#     ax = get_meshgrid_ax(x=pointset[:, :, 0], y=pointset[:, :, 1], z=pointset[:, :, 2])
#     node_loc = flattened_points[node]
#     print("Node loc: {}".format(node_loc))
#     xx, yy, zz = plane(10, 10, 500, 500, normal_vec, node_loc)
#     # xx, yy, zz = get_plane_points(normal_vec, node_loc)
#     print("Normal Vec: {}".format(normal_vec))
#     ax.plot_surface(X=xx, Y=yy, Z=zz)
#     ax.plot([node_loc[0]], [node_loc[1]], [node_loc[2]], markerfacecolor='k', markeredgecolor='k', marker='o', markersize=10)
#     plt.show()


def generate_tangent_spaces(proxy_graph, flattened_points):
    # This depth should gaurantee at least 16 neighbors
    tangent_spaces = {}
    for node in proxy_graph.nodes():
        neighbors = list(nx.neighbors(proxy_graph, node))
        node_point = flattened_points[node]
        zero_mean_mat = np.zeros((len(neighbors)+1, len(node_point)))
        for i, neighbor in enumerate(neighbors):
            zero_mean_mat[i] = flattened_points[neighbor]
        zero_mean_mat[-1] = node_point

        zero_mean_mat = zero_mean_mat - np.mean(zero_mean_mat, axis=0)
        u, s, v = svd(zero_mean_mat.T)
        # smat = np.zeros(u.shape[0], v.shape[0])
        # smat[:s.shape[0], :s.shape[0]] = np.diag(s)
        tangent_spaces[node] = u
    return tangent_spaces


def geodesic_single_path_dijkstra(flattened_points, proximity_graph, tangent_frames, start, end):
    # short circuit
    if start == end:
        return []
    # Create min priority queue
    minheap = []
    pred = {}
    dist = defaultdict(lambda: 1.0e+100)
    # for i, point in enumerate(flattened_points):
    R = {}
    t_dist = {}
    geo_dist = {}
    R[start] = np.eye(3)
    t_dist[start] = np.ones((3,))
    dist[start] = 0
    start_vector = flattened_points[start]
    for neighbor in nx.neighbors(proxy_graph, start):
        pred[neighbor] = start
        dist[neighbor] = np.linalg.norm(start_vector - flattened_points[neighbor])
        heapq.heappush(minheap, (dist[neighbor], neighbor))
    while minheap:
        r_dist, r_ind = heapq.heappop(minheap)
        if r_ind == end:
            break
        q_ind = pred[r_ind]
        u, s, v = svd(tangent_frames[q_ind].T*tangent_frames[r_ind])
        R[r_ind] = np.dot(R[q_ind], u * v.T)
        t_dist[r_ind] = t_dist[q_ind]+np.dot(R[q_ind], tangent_frames[q_ind].T * (r_dist - dist[q_ind]))
        geo_dist[r_ind] = np.linalg.norm(t_dist[r_ind])
        for neighbor in nx.neighbors(proxy_graph, r_ind):
            temp_dist = dist[r_ind] + np.linalg.norm(flattened_points[neighbor] - flattened_points[r_ind])
            if temp_dist &lt; dist[neighbor]:
                dist[neighbor] = temp_dist
                pred[neighbor] = r_ind
                heapq.heappush(minheap, (dist[neighbor], neighbor))
    # found ending index, now loop through preds for path
    current_ind = end
    node_path = [end]
    while current_ind != start:
        node_path.append(pred[current_ind])
        current_ind = pred[current_ind]

    return node_path


def plot_path_on_surface(pointset, flattened_points, path):
    # ax = get_meshgrid_ax(x=pointset[:, :, 0], y=pointset[:, :, 1], z=pointset[:, :, 2])
    # ax.plot(points_in_path[:, 0], points_in_path[:, 1], points_in_path[:, 2], linewidth=10.0)
    # plt.show()
    get_meshgrid_ax(x=pointset[:, :, 0], y=pointset[:, :, 1], z=pointset[:, :, 2])
    points_in_path = flattened_points[path]
    mlab.plot3d(points_in_path[:, 0], points_in_path[:, 1], points_in_path[:, 2] *.1)
    mlab.show()


"""
    True geodesic of graph.
    Build proximity graph
    Find tangent space using geodisic neighborhood at each point in graph
    Parallel transport vectors between tangent space points
    Use this as your distance metric
    Dijkstra's Algorithm
"""
if __name__ == "__main__":
    x, y, z = surface_squares(-5, 5, -5, 5, 500)
    # plot_meshgrid(x, y, z)
    pointset = np.stack([x, y, z], axis=2)
    proxy_graph_num_neighbors = 16
    flattened_points = pointset.reshape(pointset.shape[0]*pointset.shape[1], pointset.shape[2])
    flattened_points = flattened_points
    proxy_n_dist, proxy_n_indices = get_knn(flattened_points, proxy_graph_num_neighbors)
    # Generate a proximity graph using proxy_graph_num_neighbors
    # Nodes = number of points, max # of edges = number of points * num_neighbors
    proxy_graph = build_proxy_graph(proxy_n_dist, proxy_n_indices)
    # Now, using the geodesic_num_neighbors, get geodesic neighborshood for tangent space construction
    tangent_spaces = generate_tangent_spaces(proxy_graph, flattened_points)
    node_to_use = 2968
    # 3rd vector of tangent space is normal to plane
    # plot_tangent_plane_at_point(pointset, flattened_points, node_to_use, tangent_spaces[node_to_use][:, 2])
    path = geodesic_single_path_dijkstra(flattened_points, proxy_graph, tangent_spaces, 250, 249750)
    plot_path_on_surface(pointset, flattened_points, path)
</code></pre>

<p>Note that I installed and set up mayavi to get a decent output image (matplotlib doesn't have real 3d rendering and consequently, its plots suck).  I did however leave the matplotlib code in if you want to use it.  If you do, just remove the scaling by .1 in the path plotter and uncomment the plotting code. Anyways, here's an example image for z=x^2+y^2.  The white line is the geodesic path:</p>

<p><a href="https://i.stack.imgur.com/FaL9p.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FaL9p.jpg" alt="Example output"></a></p>

<p>You could also fairly easily adjust this to return all the pairwise geodesic distances between nodes from dijkstra's algorithm (look in the appendix of the paper to see the minor modifications you'll need to do this).  Then you could draw whatever lines you want on your surface.</p>
How to plot geodesic curves on a surface embedded in 3D?
<p>I have in mind <a href="https://www.youtube.com/watch?v=NfqrCdAjiks" rel="nofollow noreferrer">this video</a>, or this <a href="http://www.physikdidaktik.uni-karlsruhe.de/software/geodesiclab/a3.html" rel="nofollow noreferrer">simulation</a>, and I would like to reproduce the geodesic lines on some sort of surface in 3D, given by a function f(x,y), from some starting point.</p>

<p>The <a href="https://cs.stanford.edu/people/jbaek/18.821.paper2.pdf" rel="nofollow noreferrer">midpoint method</a> seems computationally and code intense, and so I'd like to ask if there is a way to generate an approximate geodesic curve based on the normal vector to the surface at different points. Each point has a tangent vector space associated with it, and therefore, it seems like knowing the normal vector does not determine a specific direction to move forward the curve.</p>

<p>I have tried working with Geogebra, but I realize that it may be necessary to shift to other software platforms, such as Python (or Poser?), Matlab, or others.</p>

<p>Is this idea possible, and can I get some ideas as to how to implement it?</p>

<hr>

<p>In case it provides some ideas as to how to answer the question, there previously was an answer (now unfortunatley erased) suggesting the midpoint method for a terrain with the functional form z = F(x,y), starting with the straight line between the endpoints, splitting in short segments [I presume the straight line on the XY plane (?)], and lifting [I presume the nodes between segments on the XY plane (?)] on the surface. Next it suggested finding "a midpoint" [I guess a midpoint of the segments joining each consecutive pairs of projected points on the surface (?)], and projecting "it" [I guess each one of these midpoints close, but not quite on the surface(?)] orthogonally on the surface (in the direction of the normal), using the equation Z + t = F(X + t Fx, Y + t Fy) [I guess this is a dot product meant to be zero...</p>

<p><img src="https://i.stack.imgur.com/rkO9C.png" alt="enter image description here"></p>

<p>(?)], where (X,Y,Z) are the coordinates of the midpoint, Fx, Fy the partial derivatives of F, and t the unknown [that is my main issue understanding this... What am I supposed to do with this t once I find it? Add it to each coordinate of (X,Y,Z) as in (X+t, Y+t, Z+t)? And then?]. This is a non-linear equation in t, solved via <a href="https://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow noreferrer">Newton's iterations</a>.</p>

<hr>

<p>As an update / bookmark, Alvise Vianello has kindly posted a Python computer simulation of geodesic lines inspired on <a href="http://www.physikdidaktik.uni-karlsruhe.de/software/geodesiclab/a3.html" rel="nofollow noreferrer">this</a> page <a href="https://github.com/amv213/Geodesic-Lines-3D" rel="nofollow noreferrer">on GitHub</a>. Thank you very much!</p>