5. Main diameter of the quadratic curve

§ 5 main diameter of the quadratic curve

I. Definition:

Set X: Y to a non-incrementally direction of the quadratic curve F (x, y) = 0. If it is bounded by the diameter of this direction:

X (x, y) + Y (x, y) = 0 (1)

Vertical to direction X: Y, it is called the diameter of the quadratic curveMain diameterThe direction and direction of diameter (1) X: Y are called quadratic curves.Main Direction.

Note:1 ° The main diameter is actually the symmetry axis of the quadratic curve, referred to as the axis of the quadratic curve. the intersection of the axis and the curve is called the vertex of the curve.

2 ° it can be proved that X: Y in the first direction is the vertical direction of <strong> X: Y and X ': Y in the first direction.

Method 2:

1 method for determining the main direction:

X: Y in the main direction <strong> X: Y and Its Bounded Direction X ': y' =-(X + Y) :( X + Y)

Vertical <strong> xx' + YY '= 0 <strong> x': Y' =-Y: X

<Strong>-(X + Y) :( X + Y) =-Y: X

<Strong> X: Y = (X + Y) :( X + Y)

<Strong> (2)

Limit X, Y is not all 0, limit = 0

That is, λ 2-+ = 0 (3)

It can be seen that if the quadratic curve F (x, y) = 0 in the main direction X: Y, you only need to first find the root of the equation (3 ,,

After Lambda (2) is substituted and the equations are solved, the X: Y in the main direction can be obtained from its non-zero solution (X: Y.

2 main diameter method:

After finding the main direction, if the direction X: Y is not the primary direction, then X (x, y) + Y (x, y) = 0 is the main diameter. If the direction X is: if Y is no longer the direction of the curve, the curve cannot be a center quadratic curve (the direction of the center curve can only be the same as its own) and-Y: X is no longer the direction of the curve.

-Y (x, y) + X (x, y) = 0

Is the only main diameter of the curve.

Three feature equations and feature root:

1 Definition: Equation (3) called quadratic curveFeature EquationIts root is called the curveFeature root.

2 nature:

The feature root of the 1 ° quadratic curve is a real number.

In fact, △= ²-4 = (-) ² + 4² ≥ 0

The feature root of the 2 ° quadratic curve is not all 0

In fact, if not, = 0 + =-2 = 0

Required-² = 0 bytes = 0 this is impossible

3. Use feature root to study the main direction and main diameter

Theorem 1: The main direction X: Y indicates that the feature root corresponding to this direction is 0.

Certificate:If the feature root corresponding to X: Y in the main direction is λ

X + Y = λ X

X + Y = λ Y

∴ PHI (X, Y) = (X + Y) X + (X + Y) Y = λ (x² + y²)

Returns X, Y is a real number, and not all is 0, then PHI (X, Y) = 0 <then> λ = 0

Theorem 2:

(I) The center quadratic curve has at least two main diameters. Specifically, any solid diameter of the circle is the main diameter, and the non-circular center curve has only two main diameters.

(Ii) The non-center curve has only one main diameter.

Proof(I) If the quadratic curve F (x, y) = 0 is a circle, then = 0, = 0, and thus the feature root =, = 0, then (2) it is satisfied by any real direction X: Y, that is, the actual direction is the main direction, so that any real diameter of the circle is the main diameter.

For non-circular center quadratic curves, △= (-) ² + 4²> 0

Thus, the curve has different non-0 real root λ 1, λ 2, and the main direction determined by the two real roots is

: =: (λ 1-) = (λ 1 -):

: =: (λ 2-) = (λ 2 -):

It is not difficult to prove that the two main directions are vertical, and the main diameter they determine is

(X, y) + (x, y) = 0 -- edge:

And (x, y) + (x, y) = 0 -- edge:

This is only two main diameter

(Ii) For non-center curves, = 0, so the feature root λ 1 = + =0, λ = 0

The Gini curve only has an X: Y in the non-primary direction. Therefore, the primary diameter is obtained.

X (x, y) + Y (x, y) = 0

This is also the only main diameter