Get the domain of a second degree bivariate curve

Question

I am writing a program which needs to sample one point from a second degree bivariate curve of the form:

$$ Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0 $$

To get this sample point I only need to feed a value of $x$ to the formula, but I have to make sure it's in the domain of course. The problem is the coefficients are unknown and have to be evaluated at run time.

So, is there a formula for calculating the domain (or range)?

hmakholm left over Monica · Accepted Answer · 2013-09-01T14:55:52.840

1

Once you choose an $x$, what's left is a quadratic equation in $y$. The discriminant of that equation is $$ \tag{1} \Delta_x =(D+Ex)^2-4C(Ax^2+Bx+F)$$ which rearranges to a quadratic polynomial in $x$: $$ \tag{2} \Delta_x = (E^2-4AC)x^2 + (2DE-4BC)x + (D^2-4CF) $$ You can use the quadratic formula on $(2)$ to find the out which ranges of $x$ make $\Delta_x$ nonnegative.

edited Sep 01 '13 at 14:55

answered Sep 01 '13 at 14:50

hmakholm left over Monica

286,031

Is it possible for such a curve not to existe in $R^2$? Or will it always existe for any combination of $A,B,C,D,E,F$? – qed Sep 01 '13 at 15:53
@qed: The solution set can be empty -- say, for $A=F=1$ and $B=C=D=E=0$ the equation reduces to $x^2+1=0$. – hmakholm left over Monica Sep 01 '13 at 16:10

Get the domain of a second degree bivariate curve

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