interpolation
I have 2 points X,Y (for example [5,10] and [20,30]) and I need to interpolate points between these 2 points in order that all this points are spaced by 1 measurement unit.
Let's pretend I am using cm (as my measurement unit) and I have a point at [5,10] and another at [20,30]. How can I know the first point in this interpolation so it's spaced only 1cm from [5,10]? Walking 1cm each step, I would like to know every coordinate of points till I reach the last point [20,30].
I have come up with a simpler equation while formulating my question.
Idea is to interpolate between X and Y axis proportionally to distance to them.
Let me guide you through with an example:
// Input
X = 0.5; Y = 0.1
// This is base interpolation that prefers closer edge
x = A * (1-Y) + B * X + C * Y + D * (1-X); // Sum = 2
// Now calculate proportion between distances to edges
distX = 0.5 - Abs(X-0.5); // Distance to edge 0..0.5 range
distY = 0.5 - Abs(Y-0.5); // Distance to edge 0..0.5 range
if distX + distY > 0.001 then //0.001 is to deal with FP precision loss
begin
distSum = distX + distY;
coefX := distX / distSum; // 0..1 range
coefY := distY / distSum; // 0..1 range
end
else
begin
coefX := 0.5;
coefY := 0.5;
end;
// Our example values
coefX = 0.5 / 0.6 = 0.83
coefY = 0.1 / 0.6 = 0.17
// Add X/Y weights into equation, to get Sum = 1
x = A * (1-Y) * coefX + B * X * coefY + C * Y * coefX + D * (1-X) * coefY;
// Our example values
x = A * 0.747 + B * 0.085 + C * 0.083 + D * 0.085; // Sum = 1
Sorry I'm not familiar with Excel formulas close enough to build a diagram, but it looks much alike this:
Benefits are: no need for iterations, knowing ABCD values and X Y position of x is enough.
Your data seems to be pretty smooth. A simple solution will be the so-called bilinear interpolation. For a given (PSI, GPM) pair you can find the four neighboring known data points: in the case of (2300, 4.5), the neighbors are 2030, 2400, and 4, 5.5.
You perform two independent linear interpolations on PSI, and then a final linear interpolation on GPM using these two interpolated values. (Note that reversing the order of interpolation, GPM then PSI, you get the same results.)
Best Answer
Given two points $A$ and $B$, this answer will put points on the line segment between $A$ and $B$ so that the first point is $1$ unit from $A$, the second point is $2$ units from $A$, and so forth until the last point, which is a whole number of units from $A$ and one unit or less from $B$.
Suppose the Cartesian $(x,y)$ coordinates of the points are $A = (x_A, y_A)$ and $B = (x_B, y_B)$. Let $d$ be the distance between these two points; by the Pythagorean Theorem, $$d = \sqrt{(x_A - x_B)^2 + (y_A - y_B)^2}.$$
Since $B$ is at a distance $d$ from $A$, to move $d$ units from $A$ toward $B$ we add $x_B - x_A$ to $x_A$ and $y_B - y_A$ to $y_A$ to get the new $(x,y)$ coordinates. To move just $1$ unit we want to move $\frac1d$ times as far, that is, the point $1$ unit from $A$ is $(x_1,y_1)$ where \begin{align} x_1 = x_A + \frac1d(x_B - x_A),\\ y_1 = y_A + \frac1d(y_B - y_A). \end{align} The next point is at $\frac2d$ of the distance from $A$ to $B$, the next at $\frac3d$ the distance, and so forth. In general the $n$th point that we place along the segment from $A$ to $B$ should be at coordinates $(x_n,y_n)$ where \begin{align} x_n = x_A + \frac nd(x_B - x_A),\\ y_n = y_A + \frac nd(y_B - y_A). \end{align} We do this for each integer $n$ such that $1 \leq n < d$.