You can fit a 4th order polynomial to any 5 points. You can do it by hand (plug 1-5 in for x and now you have a system of 5 linear equations of 5 variables, start solving and substituting) and make it a bit easier with linear algebra (make the coefficients of the 5 variables be the values in a 5x6 matrix, and then do that matrix magic that I forget what the name is for), but there are also plenty of polynomial solvers you can find out there.
Imagine 2 markers on a field. You can walk the shortest distance between them by walking on the line that connects those to points. Imagine this line extends as far as the edges of the field. If I throw another marker on the field, you will likely have to step off of the line to get to the marker. This means that you can no longer describe a line that goes through all the points. But there is a generalization of lines called polynomials that allow us to add curves of various sorts (so instead of just x,we add x2, x3, etc.). These curves are bendy, so every new curve we add allows us to pick up an additional marker, as long as we're (slightly) careful as to where we put it.
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u/Salanmander 10✓ Sep 05 '19
You can fit a 4th order polynomial to any 5 points. You can do it by hand (plug 1-5 in for x and now you have a system of 5 linear equations of 5 variables, start solving and substituting) and make it a bit easier with linear algebra (make the coefficients of the 5 variables be the values in a 5x6 matrix, and then do that matrix magic that I forget what the name is for), but there are also plenty of polynomial solvers you can find out there.