Invariant - Algebraic Invariance

Algebraic invariance refers to combinations of coefficients from certain functions that remain constant when the coordinate system in which they are expressed is translated, or rotated. An example of this kind of invariance is seen in the behavior of the conic sections (cross sections of a right circular cone resulting from its intersection with a plane). The general equation of a conic section is ax² + bxy + cy² + dx + ey + f = 0. Each of the equations of a circle, or an ellipse, a parabola, or hyperbola represents a special case of this equation. One combination of coefficients, (b²-4ac), from this equation is called the discriminant. For a parabola, the value of the discriminant is zero, for an ellipse it is less than zero, and for a hyperbola is greater than zero. However, regardless of its value, when the axes of the coordinate system in which the figure is being graphed are rotated through an arbitrary angle, the value of the discriminant (b²-4ac) is unchanged. Thus, the discriminant is said to be invariant under a rotation of axes. In other words, knowing the value of the discriminant reveals the identity of a particular conic section regardless of its orientation in the coordinate system. Still another invariant of the general equation of the conic sections, under a rotation of axes, is the sum of the coefficients of the squared terms (a+c).