General Equations for Conic Sections

The general form of the implicit equation of the second degree

(3.12)

generates various two-dimensional curves called conic sections. Fig.3.2 shows three types of conic sections – parabola, hyperbola and ellipse. Circle is a special case of the ellipse.

In determining the coefficients a, b, c, d, e and f we can produce different conic sections. If a section is given with respect to the local coordinate system and passes through its origin, then f = 0. To draw a curve through the data points we use the boundary conditions.

If c = 1, then the curve segment between two points is determined by five independent conditions, which are calculated from the remaining coefficients a, b, d, e and f. For example, you can specify the location of extreme points, the slope of the curve in them, and an intermediate point on the curve.

If b = 0, c = 1, then the analytic representation of the curve obtained by using only four additional conditions, such as those end-points and the slope of the curve in them. Curve for a = 1, b = 0 and c = 1 is even easier

(3.13)

Fig.3.2. Conic Sections

Three conditions for the calculation of d, e and f are two end points and the slope of the curve in one of them, or the two end points and the third point on the curve.

If a = b = c = 0 is obtained by a straight line its equation is

(3.14)

Conic sections are at the center (an ellipse and a hyperbola) or off-center (a parabola). In addition, there are a number of degenerate forms, which are central.

Thus, the equation is a parabola with b² - 4ac = 0, and the central section with b² - 4ac <> 0. If a section is central and b² - 4ac <0, the equation represents an ellipse, and if b² - 4ac> 0 the equation represents a hyperbole.

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