PaulS:

You can't have a surface in 2D space.

Sure you can... (you can’t have a volume)

For example this describes the unit square: `x >= 0 and x <= 1 And y >= 0 and y <= 1`

or the inequation `x[sup]2[/sup]+y[sup]2[/sup] <= 1`

is the unit disc

(if XY axes are linear and Cartesian axes of a direct orthonormed system)

Here OP is in 3D space

There are an infinite number of surfaces in 3D space joining OP’s curves.

One way to describe the curves is to have a param t varying between 0 and 1. Say curve1 is defined by the F_{1} function, with an interval of definition [x1_{s}, x1_{e}], and say curve2 is defined by F_{2} in interval of definition [x2_{s}, x2_{e}] then you can “walk” along the curves by doing

f_{1}(t) = F_{1}(x1_{s} + t.(x1_{e} - x1_{s}))

f_{2}(t) = F_{2}(x2_{s} + t.(x2_{e} - x2_{s}))

Now you have one unique t param in a simple [0,1] interval defining Z1 through f_{1}. (same for Z2)

As you know points for a given curve are all in a plane with Y being constant, points for the curve1 in 3D space are defined with t in [0,1]

X1 = x1_{s} + t.(x1_{e} - x1_{s})

Y1 = a constant (0 for example)

Z1 = f_{1}(t)

Same for X2,Y2,Z2 where Y2 would be d+Y1 constant (so just d if you picked 0), d being the distance between your two curves’ planes

Now for a given t in [0,1] you have two points in 3D space: (X1,Y1,Z1) on Curve1 and (X2,Y2,Z2) on Curve2

You can link them with whatever curve function you want, easiest one being a straight line as you drew.

For that it’s easy you just need another param k varying between [0,1] and the line between (X1,Y1,Z1) and (X2,Y2,Z2) is the set of points (X3,Y3,Z3) (depending on k)

X3 = X1 + k(X2-X1)

Y3 = Y1 + k(Y2-Y1)

Z3 = Z1 + k(Z2-Z1)

So now you have Your surface as a function S(t,k) defined for each of the three axis, with t and k varying between 0 and 1

Note you could imagine other constructs like a Bézier curve between the two extremes points with control points being also calculated smartly (possibly based on the first or second derivative of your functions to take into account speed or acceleration along the curve for example).