How to find intersection point between a line and numpy meshgrid generated surface?

Question

I have a surface generated using a function, z = f(x,y), where the x,y arrays are passed to numpy meshgrid to generate a surface points. From two arbitrary points, I am using the line equation and plane equation to find the intersection point. However my script is failing to find intersection, specifically at the step when I am solving for t.

3d line equation: t = x-xo/l = y-yo/m =z-zo/n plane equation: ax+by+cz=d

My test script is below and produces NaN or Inf only. The output I am expecting is at value that I can plug it back into line equation to find my intersection point (x,y,z)

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# generate the surface
x = np.linspace(0,100,10)
y = np.linspace(0,100,10)
xv, yv = np.meshgrid(x,y)
zv = (xv**2+yv)
pos = np.vstack([xv.ravel(), yv.ravel(), zv.ravel()]).T

fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(xv,yv,zv)
plt.show()

zv_n1 = []
zv_n2 = []
p3 = []

i_range = zv.shape[0]
j_range = zv.shape[1]

# find AB, AC vectors, a surface point to later compute the normal vec and then plane equation
for i in range(i_range):
    for j in range(j_range):
        if i<i_range-1 and j<j_range-1:
            zv_a = zv[i,j+1]-zv[i,j]
            zv_b = zv[i+1,j]-zv[i,j]
            p3_arr = np.array([xv[i+1,j], yv[i+1,j], zv[i+1,j]])
            
        if i == i_range-1 and j < j_range-1:
            zv_a = zv[i,j+1]-zv[i,j]
            zv_b = zv[i-1,j]-zv[i,j]
            p3_arr = np.array([xv[i-1,j], yv[i-1,j], zv[i-1,j]])
            
        if i < i_range-1 and j == j_range-1:
            zv_a = zv[i,j-1]-zv[i,j]
            zv_b = zv[i+1,j]-zv[i,j]
            p3_arr = np.array([xv[i+1,j], yv[i+1,j], zv[i+1,j]])
            
        zv_n1.append(zv_a)
        zv_n2.append(zv_b)
        p3.append(p3_arr)
       

# sorting in pandas df
temp_df = pd.DataFrame(pd.Series(pos[:,0]), columns = ['x_val'])
temp_df['y_val'] = pd.DataFrame(pd.Series(pos[:,1]))
temp_df['z_val'] = pd.DataFrame(pd.Series(pos[:,2]))

temp_df['x1_del'] = x[1] # delta x is same for panels
temp_df['y1_del'] = y[1] # delta y is same for panels
temp_df['z1_del'] = pd.DataFrame(pd.Series(zv_n1))
temp_df['x2_del'] = x[0] # delta x is same for panels
temp_df['y2_del'] = y[0] # delta y is same for panels
temp_df['z2_del'] = pd.DataFrame(pd.Series(zv_n2))

temp_df['p3'] = pd.DataFrame(pd.Series(p3))

# compute normal vec (and thus find a,b,c coeff, and then find d value)
cross_list = []
d_val_list = []
for i in range(len(temp_df)):
    a_vec = np.array([temp_df.iloc[i]['x1_del'],temp_df.iloc[i]['y1_del'],temp_df.iloc[i]['z1_del']])
    b_vec = np.array([temp_df.iloc[i]['x2_del'],temp_df.iloc[i]['y2_del'],temp_df.iloc[i]['z2_del']])
    nor_vec = np.cross(a_vec,b_vec)
    cross_list.append(nor_vec)
    d_val = np.dot(nor_vec,temp_df.iloc[i]['p3'])
    d_val_list.append(d_val)

temp_df['normal_vec'] = pd.DataFrame(pd.Series(cross_list))
temp_df['plane_d_val'] = pd.DataFrame(pd.Series(d_val_list))

# create two points and find the line vector
p1 = np.array([0,60,0])
p2 = np.array([0,60,5000])
p_vec = p2-p1

# solve for t - rearrange line equations (t = X-x/l = Y-y/m = Z-z/n) to (X =tl+x) and so on..
# and plug in plane equation ax+by+cz=d
for i in range(len(temp_df)):
    t = (temp_df.iloc[i]['plane_d_val'] - \
        temp_df.iloc[i]['normal_vec'][0] * temp_df.iloc[i]['p3'][0] - \
        temp_df.iloc[i]['normal_vec'][1] * temp_df.iloc[i]['p3'][1] - \
        temp_df.iloc[i]['normal_vec'][2] * temp_df.iloc[i]['p3'][2])/(\
            temp_df.iloc[i]['normal_vec'][0] * p_vec[0] +\
            temp_df.iloc[i]['normal_vec'][1] * p_vec[1] +\
            temp_df.iloc[i]['normal_vec'][2] * p_vec[2])
    print(t)

# after finding t, plug in line equation to solve for X

I would appreciate some guidance with this problem. Any recommendation for alternate methods/libraries are also welcome, as the sole purpose is to identify the intersection point.

Thank you,

Answer 1

so I figured out my issue with the code. Was using the point on the plane instead of point from the line.

Also this test script is not useful for curved surface functions. And it would seem prudent just to solve the equations mathematically..

Anyways updated code is below:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

x = np.linspace(0,100,10)
y = np.linspace(0,100,10)
xv, yv = np.meshgrid(x,y)
zv = (xv**2+yv)

fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(xv,yv,zv)
plt.show()


xv_n1 = []
xv_n2 = []

yv_n1 = []
yv_n2 = []

zv_n1 = []
zv_n2 = []

p3 = []

i_range = zv.shape[0]
j_range = zv.shape[1]

count = -1
for i in range(i_range):
    for j in range(j_range):
        count += 1
        if i<i_range-1 and j<j_range-1:
            # now x
            xv_a = xv[i,j+1]-xv[i,j]
            xv_b = xv[i+1,j]-xv[i,j]
            # now y
            yv_a = yv[i,j+1]-yv[i,j]
            yv_b = yv[i+1,j]-yv[i,j]
            #now z
            zv_a = zv[i,j+1]-zv[i,j]
            zv_b = zv[i+1,j]-zv[i,j]
            p3_arr = np.array([xv[i+1,j], yv[i+1,j], zv[i+1,j]])
            
        if i == i_range-1 and j < j_range-1:
            # now x
            xv_a = xv[i,j+1]-xv[i,j]
            xv_b = xv[i-1,j]-xv[i,j]
            # now y
            yv_a = yv[i,j+1]-yv[i,j]
            yv_b = yv[i-1,j]-yv[i,j]
            #now z
            zv_a = zv[i,j+1]-zv[i,j]
            zv_b = zv[i-1,j]-zv[i,j]
            p3_arr = np.array([xv[i-1,j], yv[i-1,j], zv[i-1,j]])
            
        if i < i_range-1 and j == j_range-1:
            # now x
            xv_a = xv[i,j-1]-xv[i,j]
            xv_b = xv[i+1,j]-xv[i,j]
            # now y
            yv_a = yv[i,j-1]-yv[i,j]
            yv_b = yv[i+1,j]-yv[i,j]
            #now z
            zv_a = zv[i,j-1]-zv[i,j]
            zv_b = zv[i+1,j]-zv[i,j]
            p3_arr = np.array([xv[i+1,j], yv[i+1,j], zv[i+1,j]])
            
        xv_n1.append(xv_a)
        xv_n2.append(xv_b)
        
        yv_n1.append(yv_a)
        yv_n2.append(yv_b)
        
        zv_n1.append(zv_a)
        zv_n2.append(zv_b)
        p3.append(p3_arr)
       
pos = np.vstack([xv.ravel(), yv.ravel(), zv.ravel()]).T

temp_df = pd.DataFrame(pd.Series(pos[:,0]), columns = ['x_val'])
temp_df['y_val'] = pd.DataFrame(pd.Series(pos[:,1]))
temp_df['z_val'] = pd.DataFrame(pd.Series(pos[:,2]))

temp_df['x1_del'] = x[1]
temp_df['y1_del'] = y[1]
temp_df['z1_del'] = pd.DataFrame(pd.Series(zv_n1))
temp_df['x2_del'] = x[0]
temp_df['y2_del'] = y[0]
temp_df['z2_del'] = pd.DataFrame(pd.Series(zv_n2))

temp_df['p3'] = pd.DataFrame(pd.Series(p3))


cross_list = []
d_val_list = []
for i in range(len(temp_df)):
    a_vec = np.array([temp_df.iloc[i]['x1_del'],temp_df.iloc[i]['y1_del'],temp_df.iloc[i]['z1_del']])
    b_vec = np.array([temp_df.iloc[i]['x2_del'],temp_df.iloc[i]['y2_del'],temp_df.iloc[i]['z2_del']])
    nor_vec = np.cross(a_vec,b_vec)
    cross_list.append(nor_vec)
    d_val = np.dot(nor_vec,temp_df.iloc[i]['p3'])
    d_val_list.append(d_val)

temp_df['normal_vec'] = pd.DataFrame(pd.Series(cross_list))
temp_df['plane_d_val'] = pd.DataFrame(pd.Series(d_val_list))


p1 = np.array([0,60,6000])
p2 = np.array([100,60,6000])
p_vec = p2-p1

for i in range(len(temp_df)):
    t = (temp_df.iloc[i]['plane_d_val'] - \
        temp_df.iloc[i]['normal_vec'][0] * p1[0] - \
        temp_df.iloc[i]['normal_vec'][1] * p1[1] - \
        temp_df.iloc[i]['normal_vec'][2] * p1[2])/(\
            temp_df.iloc[i]['normal_vec'][0] * p_vec[0] +\
            temp_df.iloc[i]['normal_vec'][1] * p_vec[1] +\
            temp_df.iloc[i]['normal_vec'][2] * p_vec[2])

p1 = np.array([0,60,6000])
p2 = np.array([100,60,6000])
p_vec = p2-p1

for i in range(len(temp_df)):
    t = (temp_df.iloc[i]['plane_d_val'] - \
        temp_df.iloc[i]['normal_vec'][0] * p2[0] - \
        temp_df.iloc[i]['normal_vec'][1] * p2[1] - \
        temp_df.iloc[i]['normal_vec'][2] * p2[2])/(\
            temp_df.iloc[i]['normal_vec'][0] * p_vec[0] +\
            temp_df.iloc[i]['normal_vec'][1] * p_vec[1] +\
            temp_df.iloc[i]['normal_vec'][2] * p_vec[2])

if np.isnan(t) == True:
    q = np.nan
elif t == 0:
    q = np.nan
elif np.isinf(t) == True:
    q = np.nan
else:
    x = t*p_vec[0]+p2[0]
    y = t*p_vec[1]+p2[1]
    z = t*p_vec[2]+p2[2]
    q = np.array([x,y,z])
    
print(q)