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I have 5 points which I would like to fit onto a curve. Mostly this is just for fun, but trying to do it is hopefully teaching me and my daughter some mathematics. We're using shape-building to learn about the properties of equations.

We would like the curve to be concave to the left like a parabola and we would like it to hit each point.

We tried a linear regression of a quadratic function and the software did not hit all points. Is that a limitation of the software or are we simply trying to do something that a parabola cannot do?

The points are:

1, 2
1.5, 1.9
2, 1.5
1.5, 1.1
1, 1

Essentially a sideways, symmetrical parabola-like shape.

I tried higher order functions but as you might expect, the shape that the software produced was not concave to the left.

I'd like to know both whether there exists a perfect-fit equation which is concave forever, and also whether there exists a perfect-fit equation which is simply concave in the region 1,1

In general I'd like to understand whether every concave curve can be described as a single equation and if so, how.

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  • $\begingroup$ A parabola is determined by three parameters, so it can fit exactly at most three points. To find a curve passing through five points you need a polynomial of fourth degree. $\endgroup$ – Intelligenti pauca Apr 24 '17 at 17:14
  • $\begingroup$ A parabola whose axes are not parallel to coordinate axes is fixed by 4 points. $\endgroup$ – Narasimham Apr 24 '17 at 17:35
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The points given are symmetrical about $y=1.5$ appear to form a curve with a stationary point at $(2,1.5)$.

You would get a pretty good fit with a $2^{\text{nd}}$ order curve $(R^2=0.96)$ using $$2-x=3.7457(y-1.5)^2$$

and an even better one with a $4^{\text{th}}$ order curve $(R^2\approx 1)$ using
$$2-x=9.7222(y-1.5)^4+1.5694(y-1.5)^2$$

as shown in the diagram below.

enter image description here

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  • $\begingroup$ That's just what I was looking for. What technique should I have used to find it? Perhaps a linear regression of x = a*(y-b)^4 + c*(y-d)^3+e*(y-f)^2+g*(y-h)+i ? $\endgroup$ – Paul Prescod Apr 24 '17 at 20:04
  • $\begingroup$ I use desmos and it seems like you do too. I typed in the following to try and get desmos to find the constants but something weird happened. I transformed your equation to: x_3~a(y_3-b)^4+c(y_3-d)^2+f Interestingly, it came up with the right numbers but drew the parabola opening downwards. I wonder if that's just a desmos bug? $\endgroup$ – Paul Prescod Apr 24 '17 at 20:49
  • $\begingroup$ Try fitting a 4th order polynomial with $x$ as the dependent variable and $y$ the independent. $\endgroup$ – Hypergeometricx Apr 26 '17 at 16:24
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A parabola opening to the left has the equation $$x = Ay^2 + By + C, \quad A < 0.$$

This gives you 3 parameters from which to try to vary the curve, but you would like to pass through 5 points. You can either

  1. reduce the number of points, e.g. leaving points 1,3,5 or 2,3,4 to get the shape you want exactly (but it won't pass through the other 2 points you left out); or
  2. find values of $A,B,C$ to make the curve as close as possible to all 5, but then quite likely, it won't pass through any of them altogether...
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Five points determine a conic,its equation has 5 constants.

$$ ax^2+ 2 h xy + by^2 +dx+ ey =1$$

Four points determine a parabola, its equation has 4 constants:

$$ ax^2+\sqrt{ab } xy + by^2 +dx+ ey =1$$

By Cramer's rule (arranged in a matrix) we can find the parabola passing through four given points ( as also the conics through 5 points).

In the figure below(drawn using Geogebra) 4 fixed points$(A,B,C,D)$ are placed on a known convenient parabola $ y = x^2/4+1 $ shown.

Fun with parabolas

A fifth variable point placed on a slider (p1,p2,P,p3,p4) forms ellipses when placed outside parabola and when placed inside it, form a hyperbolas.

The locus of a fifth point to remain on a parabola set up through 4 prescribed points is the same parabola.

So from the data you gave there are 5 such combinations/cases. In each case place the fifth point on a slider (with a reasonable slider domain) and see all possibilities as above.

I believe all possible convex/concave shapes can be obtained this way.

Next, I took the 5 points you gave and it is seen to makes a narrow ellipse, but not narrow enough to make a parabola with arms going to $ \infty $ to the left. So I perturbed the second point on a circle with first point as center. When the second point is placed a bit below the location you have given, (all the other four points being fixed ) a parabola can be captured. Exact value for symmetric case is given below.

EDIT1:

After what is mentioned above if you wish to have a parabola opening to left side with its axis displaced parallel to x-axis $k=1.5, $ passing through $( 1,1), (1,2), (2,1.5) $ the one and only possibilitycorresponding to $x=1.5$ y-coordinates should be

$$ (y_1= 1.85355, y_2= 1.14645) $$

symmetric to $y= 1.5$

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