Sub-sampling the minimum number of points to accurately reproduce a curve

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Michael Granatosky on 8 Jan 2020
Edited: David Goodmanson on 8 Jan 2020
Hello,
Currently some students of mine are working on a bioinspired robot. We are using poistion data collected directly from a moving animal and putting those values onto servo motors. Recently, we have come across a bit of an over-sampling issue that is casuing some computational issues.
Anyway, when I collect data from an animal I can get kinematic movements of position versus time. For this question, let's assume 100 data points. (Sorry about the sideways orientation of the photos).
Together these 100 data points can be graphed as a continuous curve
Now my problem is that if I make our robot hit all 100 data points the movement is highly accurate, but causes the servo motors to be very jerky and a lot of computational power. So my hope is to downsample/ subsample the number of points I have, but still retain the shape of that curve. For example:
If I reduce the data down to two points I get an abysmal representation of the original curve. What about 3 points?
Still not great...
However, by the time I hit 9 points, we're doing pretty good.
Essentially, I'd like some advice on 1) what this form of downsampling/ subsampling is even called; and 2) any ideas on some MATLAB code that can help me maximize accuracy while limiting data.

David Goodmanson on 8 Jan 2020
Edited: David Goodmanson on 8 Jan 2020
Hi Michael, take a look at the interp1 function. There are several choices for the type of interpolation, and assuming a curve that looks like your example, not too wild, the spline option (cubic spline) is one of the most accurate.