Neliökäyriä

Neliökäyriä, or quadratic curves, are geometric shapes defined by a second-degree polynomial equation in two variables. The most common and simplest examples are the conic sections: the circle, ellipse, parabola, and hyperbola. These shapes arise from the intersection of a plane and a double cone. A general second-degree equation in two variables can be written as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The coefficients A, B, and C determine the type of curve. For instance, if B^2 - 4AC < 0, the curve is an ellipse or a circle (a special case of an ellipse). If B^2 - 4AC = 0, it is a parabola. If B^2 - 4AC > 0, it is a hyperbola. Degenerate cases can result in lines or points. Neliökäyriä have numerous applications in various fields, including physics (describing projectile motion or orbits), engineering (designing bridges or lenses), and computer graphics. Understanding their properties is fundamental in analytic geometry.