Kvadraticissa

Kvadraticissa is a term used in some mathematical texts to denote a class of two-variable quadratic forms and the corresponding plane curves known as conic sections. In its common usage, a kvadraticissa is defined by a general quadratic equation in two variables: ax^2 + bxy + cy^2 + dx + ey + f = 0, where the real coefficients satisfy at least one a, b, c nonzero.

The quadratic part can be represented by the symmetric matrix Q = [[a, b/2], [b/2, c]]. The discriminant Δ = b^2 - 4ac determines the basic type of conic: Δ < 0 yields an ellipse (including circles); Δ > 0 yields a hyperbola; Δ = 0 yields a parabola or a degenerate case. Nondegenerate kvadraticissa have a nonzero determinant of the 3x3 augmented matrix; degeneracy yields intersecting lines, a single point, or no real points.

Classification is invariant under invertible linear changes of coordinates; diagonalizing the quadratic part reduces the equation to a simpler canonical form, revealing axes and orientation.

Examples include: ellipse x^2/4 + y^2/9 = 1; hyperbola x^2 - y^2 = 1; parabola y = x^2.

Kvadraticissa objects appear in analytic geometry, computer graphics, and CAD, and they are central to solving intersections of conics and to modeling orbits and optical paths in some theoretical frameworks.

Origin and usage notes: the term is not universally standardized; in many texts these objects are simply called conic sections or quadratic forms. Kvadraticissa is used mainly as an illustrative label in certain curricula.