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I have seven linear regression lines, each with a different value for $z$, i.e. $$\text{eq1}: y = 3x + 2,\, z = 0,\qquad \text{eq2}: y = 2.5x + 3,\, z = 2$$

All of the lines have the same $x$ range (lets say from $x=1$ to $x=10$). With the start and end points of each of the $7$ lines known, I would like to first make a $3$rd degree interpolation over the start points and again for the end points, this will result in two curves, parallel in the $z$ direction (extending back).

Then with these two parallel curves, I would like to create a plane that consists of a straight line from each point on the two curves. In other words, with the two curves extending into the $z$ plane, taking the two values for when $z=0$: find the straight line that connects these two points, then again for when $z=0.1$ etc.

This would make up a plane, and I would like to know the equation of that plane by just plugging in the $7$ initial straight line equations. Or if easier, plugging in the two $3$rd degree polynomial equations for the start/end points.

Thanks in advance!

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