Guide line

Geometric Academic Language

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stay Conic curve In the unified definition of constant The locus of e (e>0) points is called conic curve. This fixed line is called the directrix. When 0<e<1, the track is ellipse ； When e=1, the track is parabola ； e> At 1, the track is hyperbola 。 Parabolic directrix is related to p value.

In the general theory of space surface, a surface can be seen as a trajectory formed by a family of curves moving along its directrix. For a curve family to generate a surface, the directrix is a space curve that intersects every curve in the curve family.

Chinese name: Guide line
Foreign name: Directrix
Applicable fields: Conic curve

Applied discipline: geometry
Research discipline: analytic geometry
Definition: Ratio of distance from curve point to fixed point and fixed line

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The intersection of the directrix of a conic curve and the axis of symmetry is called the directrix of a conic curve. The straight line passing through the conic curve whose focus is perpendicular to the axis of symmetry (the major axis and the real axis in the ellipse and hyperbola respectively) intersects the conic curve at two points A and B. Line AB is called the path of the conic curve.^[3]

When cutting a cone with a plane, the plane is perpendicular to the axis of symmetry, and the circle is obtained; Gradually incline the plane to get an ellipse; When the plane is inclined to be parallel to only one generatrix of the cone, the parabola is obtained; The hyperbola can be obtained by intercepting the plane parallel to the symmetric axis of the cone. Because of this, the ancient Greek mathematician Apolloni once called the ellipse "deficient curve", the parabola "homogeneous curve", and the hyperbola "hypercurve".^[4]

Directrix equation

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ellipse

Alignment: vertical The line on the line with the major axis

ellipse ： (x^2/a^2)+(y^2/b^2)=1（a>b>0）

Directrix equation Is: x=± a ^ 2/c^[1]

hyperbola

hyperbola ：(x^2/a^2)-(y^2/b^2)=1

The directrix equation is: x=± a ^ 2/c^[1]

parabola

1、 parabola ：y^2=2px

Directrix equation Is: x=- p/2

2. Parabola: y ^ 2=- 2px

The directrix equation is: x=p/2^[1]

Geometric properties

Alignment to vertex The distance of is Rn/e, from the alignment to focus The distance of is P=Rn (1+e)/e=L0/e.

When the eccentricity e is greater than zero, P is a limited quantity, and the distance from the guide line to the focus is P=Rn (1+e)/e=L0/e.

When the eccentricity e is equal to zero, P is infinite, and P is a non universal amount. It is unreasonable to define conic curve with infinity.

The reason for the limitation of the definition in the textbook is that we do not understand the geometric properties of the guide line. When e is equal to zero, the guide line is infinite. The guide line is not general and appropriate limitations Amount of. The textbook uses the guide line to define the reason why the conic curve does not contain a circle.^[1]

Guide line on cone

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definition

In the general theory of space surfaces, a surface can be seen as a trajectory formed by the movement of a family of curves along their directrix. For a curve family to generate a surface, the directrix is a space curve that intersects every curve in the curve family. The determination of the directrix equation is of great value to the study of the geometric characteristics and shape of surfaces. On the one hand, determining the equation of a directrix is the premise of establishing the surface equation. On the other hand, the geometric characteristics of the surface with a given equation can also be studied through a directrix equation on it.

Definition 1, for a curve Γ in space and a point A not on the curve Γ, the surface formed by a family of lines passing through point A and intersecting with the curve Γ is called a cone, and these lines are called cone a bus or bus bar The curve Γ is called the directrix of the cone.

Lemma 1, an n (n>0) degree homogeneous equation about x-a, y-b, z-c represents a cone with A (a, b, c) as its vertex.

Lemma 2, with

Is the guide line, and the cone equation of the vertex at the origin is

。

Lemma 3, the intersection line between the cone and the plane passing through the apex of the cone is either a vertex or a straight line.^[2]

Characteristics of the General Cone Collinear Equation

From definition 1, we can see that any curve in space that is not through a vertex and intersects with each straight generatrix of the cone can be used as the guide line of the cone. Therefore, in particular, take a plane that is not through a vertex and intersects with each straight generatrix, and its intersection line with the cone can be used as the guide line of the cone. The following theorem combines the geometric characteristics of the guide line to give an analytical formula of the guide line.

Theorem 2, let the cone