如何快速计算插值二维数组上的一维积分？

Question

Given is a geometrical object, for simplification a semisphere with a certain radius. This is displayed as a 2D matrix with the Z data being the height. Assuming that I cut the object along any line, I want to calculate the area of the cut. My solution is to interpolate the semisphere using scipys RectBivariateSpline to accurately display it.

import numpy as np
import scipy.interpolate as intp

radius = 15.
gridsize = 0.5
spectrum = np.arange(-radius,radius+gridsize,gridsize)
X,Y = np.meshgrid(spectrum,spectrum)
Z = np.where(np.sqrt(X**2+Y**2)<=radius, np.sqrt(radius**2-np.sqrt(X**2+Y**2)**2), 0)

spline = intp.RectBivariateSpline(x = X[0,:], y = Y[:,0], z = Z)

#Example coordinates of the cut
x0 = -4.78
x = -6.73
y0 = -15.
y = 15.

However, the RectBivariateSpline only offers an area integral (which can be quickly checked by setting x0 = x or y0 = y). On the other hand the UnivariateSpline only takes in 1D array, which would only work if my cut happened to be along one specific vector of the matrix Z.

Since I want to perform this operation a few thousand times, I would need a comparably quick way to solve the integral (numerically or analytically doesn't matter as long as the error is somewhat negligible). Does anyone have an idea on how to do this?

Answer 1

It turned out, that, for my case, it was sufficient to sample the spline along my cut (using numpy's arange to gather equally spaced points) and then by integrating via the Simpson rule, which only requires a number of points with a sufficiently low distance (which can be controlled via arange's step parameter).