# Find equations of specific lines lies on a saddle shaped 3d surface

I have equations for saddle-shaped surface (likely hyperbolic paraboloid) in $$3D$$ space. Example image

In such cases, I want to know the equations of two lines which are,

1. Lies on the surface of the given equation
2. has constant $$z$$

For example, for the following values,

$$z = a + bx + cy + dxy+ex^2+fy^2$$

$$a = 1.3907,$$

$$b = -0.087591,$$

$$c = -0.25811,$$

$$d = 0.033397,$$

$$e = 0.0027985,$$

$$f = 0.00089385$$

The shape of the surface and such line would be like this image ( wolfram alpha link for the surface )

How can I get an equation of such a line?

• A line may not have constant $z$. In this case, it could be any conic. – Anubhab Ghosal Dec 31 '18 at 8:01
• @AnubhabGhosal Yes, there could be many lines over the given surface with non-constant z. However, my question is, I want to know only the equation of lines having constant z on the surface of the given equation. And according to my instinct, (of course, it could be wrong) There would be only two lines if given surface is kind of hyperbolic paraboloid. – Sunkyue Kim Dec 31 '18 at 8:27
• All contours(curves with constant z) are conics in this case. The condition that the contour is a pair of straight lines can be found by equating the discriminant to 0. – Anubhab Ghosal Dec 31 '18 at 8:41
• @AnubhabGhosal Thank you for your answer. However, unfortunately, I don't know how to calculate discriminant on a multivariate case. Where can I find relevant information? In addition, If it is okay for you, may I ask you how can I find the saddle point of the given hyperbolic surface? Sorry for bothering you since I'm not good at math. – Sunkyue Kim Dec 31 '18 at 8:58
• – Anubhab Ghosal Dec 31 '18 at 9:20