# Parabola from 4 approximate points

I have calculated four approximate points from a sensors to get information. I would like to deduce the closest parabola to my points. The problem is that I can't solve it to get an appropriate result.

Here are my four points: $(414, 960), (1020,2340), (1387,3300), (1848,4510)$

Also, I tried to solve it with Wolfram Alpha using this instruction: solve \begin{align*} 171396a+397440b+921600c+414d+960e+f&=0\\ 1040400a+2386800b+5475600c+1020d+2340e+f&=0\\ 1923769a+4577100b+10890000c+1387d+3300e+f&=0\\ 3415104a+8334480b+20340100c+1848d+4510e+f&=0\\ b^2=4ac \end{align*}

I wasn't able to make it works... I might not have the correct syntax. Any help will be appreciate.

• Excel will also fit a polynomial of any degree you want through a set of points. Graph them in an XY(scatter) graph and perform a regression. – Ross Millikan Jul 29 '12 at 17:42
• Plotting the data, it looks very linear. You should think about whether a parabola is better for your purposes than a straight line. The quadratic term may be generated by the measurement errors. – Ross Millikan Jul 30 '12 at 3:17

In fact there are two parabolas in the $x-y$ plane that pass through your four points (exactly).

\eqalign{&103297015086160900\,{x}^{2}\cr + &\left( -78528720845214360+8356361780\, \sqrt {31761606515} \right) xy\cr + &\left( 14930193933347471-3176347356\, \sqrt {31761606515} \right) {y}^{2}\cr +& \left( -18003514867830528700- 7272001822800\,\sqrt {31761606515} \right) x\cr +& \left( 1629063111000\, \sqrt {31761606515}+6793046116045210090 \right) y\cr +& 678223769919769543800+1052877465525600\,\sqrt {31761606515} =0\cr} and the same with $\sqrt {31761606515}$ replaced by $-\sqrt {31761606515}$

• Thank you, that is impressive. – Zonata Jul 29 '12 at 21:09

It seems you are trying to solve a problem which has no solution. Why not try a quadratic regression instead ?

EDIT: This answer is a good one, and Wolfram Alpha has a quadratic regression function you could try--just check the Examples.

The problem can be solved, as pointed by jbeuh, with a quadratic regression. I found this tool online that did the work for me (sorry, I am a lazy guy). The equation found is:

y = 2.043800872·10-4 x2 + 2.02022332 x + 84.68775484


Which gives a very good approximation of the equation.

• I got this answer, too (did not post in case OP wanted to solve themselves!) I might point out that the quadratic term does not look significant; i.e. your points seem to fall close to a line rather than a parabola. You may have a good reason to want a parabola, but a line looks like a better statistical fit. – user452 Jul 29 '12 at 18:08

Your problem is closely related to the fact that five points determine a conic section. To understand the tecnical reasons, look at this and this.

Here you can calculate your coefficients, and Geogebra has a nice surprise for you (it shows you how magic is done with the "Conic through five points tool").

Since you give only four points, it is not surprising that there are more than one conic that satisfy your condition, as Robert pointed (pun unintended).