دانشنامه ریاضی و کامپیوتر

Quadratic Curve

The general bivariate quadratic curve can be written

Define the following quantities:

			(2)
			(3)
			(4)
			(5)

Then the quadratics are classified into the types summarized in the following table (Beyer 1987). The real (nondegenerate) quadratics (the ellipse and its special case the circle, hyperbola, and parabola) correspond to the curves which can be created by the intersection of a plane with a (two-nappes) cone, and are therefore known as conic sections.

curve
coincident lines	0	0		0
ellipse (imaginary)
ellipse (real)
hyperbola
intersecting lines (imaginary)	0
intersecting lines (real)	0
parabola		0
parallel lines (imaginary)	0	0
parallel lines (real)	0	0

It is always possible to eliminate the cross term by a suitable rotation of the axes. To see this, consider rotation by an arbitrary angle . The rotation matrix is

---

 

			(6)
			(7)

 

so

 

			(8)
			(9)
			(10)
			(11)
			(12)

 

Plugging these into (◇) and grouping terms gives

 

 

Comparing the coefficients with (◇) gives an equation of the form

 

 

where the new coefficients are

 

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)

 

The cross term can therefore be made to vanish by setting

 

			(21)
			(22)

 

For to be zero, it must be true that

 

 

The other components are then given with the aid of the identity

 

 

by defining

 

 

so

 

			(26)
			(27)

 

Rotating by an angle

 

 

therefore transforms (◇) into

 

 

Completing the square,

 

 

 

Defining , , and gives

 

 

If , then divide both sides by . Defining and then gives

 

 

Therefore, in an appropriate coordinate system, the general conic section can be written (dropping the primes) as

 

 

Consider an equation of the form where . Re-express this using and in the form

 

 

Therefore, rotate the coordinate system

 

 

so

 

 

and

 

			(38)
			(39)
			(40)

 

Therefore,

 

			(41)
			(42)
			(43)
			(44)
			(45)

 

From (39) and (45),

 

 

the same angle as before. But

 

			(47)
			(48)
			(49)

 

so

 

 

Rewriting and copying (◇),

 

			(51)
			(52)
			(53)

 

Adding (52) and (53) gives

 

			(54)
			(55)

 

Note that these roots can also be found from

 

 

	(57)
	(58)
	(59)
	(60)

 

The original problem is therefore equivalent to looking for a solution to

 

 

 

which gives the simultaneous equation s

 

 

Let be any point with old coordinates and be its new coordinates. Then

 

 

and

 

			(65)
			(66)

 

If and are both , the curve is an ellipse. If and are both , the curve is empty. If and have opposite signs, the curve is a hyperbola. If either is 0, the curve is a parabola.

 

To find the general form of a quadratic curve in polar coordinates (as given, for example, in Moulton 1970), plug and into (◇) to obtain

 

 

 

Define . For ,we can divide through by ,

 

 

Applying the quadratic formula gives

 

 

where

 

			(71)
			(72)

 

Using the trigonometric identities

 

			(73)
			(74)

 

it follows that

 

			(75)
			(76)
			(77)

 

Defining

 

			(78)
			(79)
			(80)
			(81)
			(82)

 

then gives the equation

 

 

(Moulton 1970). If , then (◇) becomes instead

 

 

Therefore, the general form of a quadratic curve in polar coordinates is given by