How to find a point value on a 2D surface?

Question

I was struggling for a while... Can anyone help me find a formula to solve this problem for any point? Thanks!

UPDATE: I was trying to simplify, but in our case, each number in the corner represents driving time minutes. And each side of this square is 5 mile straight line on a map. What we are trying to achieve, is to approximate driving time to any point on this square (part of the map)

For example, if chosen point is close to 40, then 40 will have most influence on the point's value, but still all 4 numbers should be taken into account.

I'm thinking that this can be represented as a 3D contour graph, where those values are Z dimension.

Answer 1

If I have interpreted your problem correctly, then I believe it will be

in general

$$ Z=(1-p_2)[20(1-p_1)+40p_1] +p_2[45(1-p_1)+90p_1]$$

but I must caution that your problem is vaguely stated so I cannot be sure if this is what you are looking for.

If $C_1=20,C_2=40,C_3=45,C_3=90 $ this formula can be re-written

$$ (1-p_1)(1-p_2)C_1+p_1(1-p_2)C_2+p_2(1-p_1)C_3+p_1p_2C_4 $$

and represents a double direct proportion on a flat surface.

Answer 2

Moving horizontally across a full edge seems to multiply by $2.25$ while moving vertically across a full edge seems to multiply by $2$, so perhaps the answer is $$20 \times \left(\dfrac{45}{20}\right)^{0.23}\times \left(\dfrac{40}{20}\right)^{0.56}$$