How can a parametric parabola connecting points from different spaces be determined, given the possibility of different coefficients for each line?

A parametric parabola is usually defined by equations of the form:

x(t) = a_x t^2 + b_x t + c_x
y(t) = a_y t^2 + b_y t + c_y
z(t) = a_z t^2 + b_z t + c_z

where the coefficients can differ for each coordinate (space). To determine such a parabola connecting given points, you need to:

1. Set a parameter t, for example, t=0 for the first point, t=1 for the second.
2. For each coordinate, form a system of equations based on the known points.
3. Solve the system for the coefficients a, b, c.

If there are more points than coefficients, the least squares method can be used for approximation.

Example for 2D (x, y) with two points (x0, y0) and (x1, y1):

import numpy as np

# points
points = np.array([[x0, y0], [x1, y1]])
# parameters t
t = np.array([0, 1])

# Matrix for quadratic function: [t^2, t, 1]
T = np.vstack([t**2, t, np.ones_like(t)]).T

# Solve for x and y separately
coeff_x = np.linalg.lstsq(T, points[:,0], rcond=None)[0]
coeff_y = np.linalg.lstsq(T, points[:,1], rcond=None)[0]

# coeff_x and coeff_y contain [a, b, c] for x(t) and y(t)

Thus, you can define a parametric parabola with different coefficients for each coordinate, connecting points from different spaces.