CODEWITHSUNDEEP
excelinterpolationcurve-fitting
david_10001
david_10001

Reputation: 492

Interpolated curves between existing curves do not look correct

I have a chart that has several existing curves on it that I have tried to interpolate new curves in between. I have used linear interpolation in the form of y = ((x - x1)(y2 - y1) / (x2 - x1)) + y1, however the new curves look out of place.

You can see in the picture that every second line (from the bottom) is the interpolated line. While the very second line data points are exactly centered between the first and third data points in the y axis, the third line data points are not centered between the second and fourth y data points, making the graph look skew.

So I am thinking linear interpolation may not be what I am after here. Can someone recommend another method that would create curves between the existing ones, but replicates the same form?

enter image description here

Upvotes: 1

Views: 764

Answers (2)

Michael
Michael

Reputation: 4848

An unconventional approach may be a piece-meal style of interpolation. It may be possible to model the 3 regions of different gradients separately.

Start by identifying the 2 lines that would be drawn through the 2 sets of kinks, creating 3 regions of space. The vertical line would stop at the horizontal line near the bottom right corner of the graph.

For each region (and potentially for each value of x in each region) determine the gradient of the lines. When you're doing your interpolation of a new line, for each starting point (x1, y1), look at which region it falls in. Use the gradient of that region as a significant factor when determining the next point. Keep doing this until you reach a region boundary. When the interpolated point crosses into a different region, then use the gradient of that region as a significant factor to interpolate the next point.

It will be quite pointy if you did this strictly, so graph with some smoothing (or incorporate a smoothing factor using weighted averages of the gradients as you transition between regions, but this could be a whole lot of effort without necessarily closer results!)

Upvotes: 0

Michael
Michael

Reputation: 4848

Sudden changes in gradient are hard to interpolate. When you're at the point where you want an interpolated line to suddenly change gradient, there is no information from the points in close proximity that give information as to where the sudden change in gradient should occur.

To replicate the pattern, you actually need to copy the gradient of the line below then smoothly transition to the gradient of the line above. Visually we can see that it should occur half way between the change in gradients for the lines above and below on either side, but detecting the locations of those changes is not trivial.

The points where the sudden change in gradient are occurring are separated by a large change in the x-axis by only a small change in the y-axis. When calculating y-values for x-values in between the the changes in gradient you get the aberrations. I suggest trying to interpolate x-values based on y-values instead. For each curve, for each small arbitrary step in the y-axis, find/calculate the closest x-values from the curve on either side and take the average to plot your interpolation.

Upvotes: 1

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