Finding three functions that interpolates between equitation

Question

For a past week I been trying to construct specific functions that would satisfy some the requirements, sadly it was not successful. I decided to give a shot in stack overflow.

I have three shapes in 2D world as shown in the image. Shape A with center a, Shape B with center b and shape C with center c. For this problem shape form does not matter so I will not expand on that. Other important variables are d_ab with represents minimum distance between shape A and B, d_bc - distance between B and C, d_ac - distance between A and C. Also d_ab, d_bc, d_ac ranges from 0 to 1, where 1 represents shapes are touching and 0 their too far from specific threshold.

Here is the image of it

I am trying to find three new functions:

f_a(d_ab, d_bc, d_ac) = x

f_b(d_ab, d_bc, d_ac) = y

f_c(d_ab, d_bc, d_ac) = z

Firstly, these functions with inputs d_ab, d_bc, d_ac in the left side table should result to values x, y, z right side table:

d_ab	d_bc	d_ac	x	y	z
1	0	0	(a+b)/2	(a+b)/2	c
0	1	0	a	(b+c)/2	(b+c)/2
0	0	1	(a+c)/2	b	(a+c)/2
1	1	1	(a+b+c)/3	(a+b+c)/3	(a+b+c)/3
0	0	0	a	b	c

Secondly, all other inputs d_ab, d_bc, d_ac that are not described in table (From 0 to 1) should result in interpolated values between closest ensuring continuity (Continuity I am referring there is that small change in input should not produce big difference in output, there is probably better way to describe this mathematically).

For example:

f_a(0.3, 0, 0) = x

Most likely will combine equitation's from table:

f_a(0, 0, 0) = a f_a(1, 0, 0) = (a+b)/2

Resulting into:

f_a(0.3, 0, 0) = 0.7 * a + 0.3 * (a+b)/2

I am curious is there a method to find these kind of functions, I assume there could be more than one that satisfies this requirement. But I am fine with any of it as long it has that continuity that I referred above.

I almost sure there is better way to represent this problem mathematically, but I am not that great with it.