How to calculate the vertex of a parabola given three points

Asked 2009-04-04T20:30:45.553

I have three X/Y points that form a parabola. I simply need to calculate what the vertex of the parabola is that goes through these three points. Preferably a quick way as I have to do a LOT of these calculations!

The "Ask A Scientist" website provides this answer:

The general form of a parabola is given by the equation: A * x^2 + B * x + C = y where A, B, and C are arbitrary Real constants. You have three pairs of points that are (x,y) ordered pairs. Substitute the x and y values of each point into the equation for a parabola. You will get three LINEAR equations in three unknowns, the three constants. You can then easily solve this system of three equations for the values of A, B, and C, and you'll have the equation of the parabola that intersects your 3 points. The vertex is where the first derivative is 0, a little algebra gives: ( -B/2A , C - B^2/4A ) for the vertex.

It would be nice to see actual code that does this calculation in C# or C++. Anybody?

curve-fitting

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2 Answers

Thanks David, I converted your pseudocode to the following C# code:

public static void CalcParabolaVertex(int x1, int y1, int x2, int y2, int x3, int y3, out double xv, out double yv)
{
    double denom = (x1 - x2) * (x1 - x3) * (x2 - x3);
    double A     = (x3 * (y2 - y1) + x2 * (y1 - y3) + x1 * (y3 - y2)) / denom;
    double B     = (x3*x3 * (y1 - y2) + x2*x2 * (y3 - y1) + x1*x1 * (y2 - y3)) / denom;
    double C     = (x2 * x3 * (x2 - x3) * y1 + x3 * x1 * (x3 - x1) * y2 + x1 * x2 * (x1 - x2) * y3) / denom;

    xv = -B / (2*A);
    yv = C - B*B / (4*A);
}

This is what I wanted. A simple calculation of the parabola's vertex. I'll handle integer overflow later.

answered 2009-04-04T21:16:09.110

This is really just a simple linear algebra problem, so you can do the calculation symbolically. When you substitute in the x and y values of your three points, you'll get three linear equations in three unknowns.

A x1^2 + B x1 + C = y1
A x2^2 + B x2 + C = y2
A x3^2 + B x3 + C = y3

The straightforward way to solve this is to invert the matrix

x1^2  x1  1
x2^2  x2  1
x3^2  x3  1

and multiply it by the vector

y1
y2
y3

The result of this is... okay, not exactly all that simple ;-) I did it in Mathematica, and here are the formulas in pseudocode:

denom = (x1 - x2)(x1 - x3)(x2 - x3)
A = (x3 * (y2 - y1) + x2 * (y1 - y3) + x1 * (y3 - y2)) / denom
B = (x3^2 * (y1 - y2) + x2^2 * (y3 - y1) + x1^2 * (y2 - y3)) / denom
C = (x2 * x3 * (x2 - x3) * y1 + x3 * x1 * (x3 - x1) * y2 + x1 * x2 * (x1 - x2) * y3) / denom

Alternatively, if you wanted to do the matrix math numerically, you'd typically turn to a linear algebra system (like ATLAS, though I'm not sure if it has C#/C++ bindings).

In any case, once you have the values of A, B, and C as calculated by these formulas, you just have to plug them into the expressions given in the question, -B / 2A and C - B^2/4A, to calculate the coordinates of the vertex.¹

Note that if the original three points have coordinates that make denom a very large or very small number, doing the calculation directly might be susceptible to significant numerical error. In that case it might be better to modify it a bit, to avoid dividing by the denominators where they would cancel out anyway:

denom = (x1 - x2)(x1 - x3)(x2 - x3)
a = (x3 * (y2 - y1) + x2 * (y1 - y3) + x1 * (y3 - y2))
b = (x3^2 * (y1 - y2) + x2^2 * (y3 - y1) + x1^2 * (y2 - y


                        
                        
                            answered 2009-04-04T20:45:54.427



        
        
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