4 § 7 straight lines of a Quadratic Surface

§ 7 straight lines of a Quadratic Surface

 

IDefinition: A surface formed by a set of continuously changing straight lines is calledStraight SurfaceEach line is called its bus.

Note: The cylindrical surface and cone surface are both straight surface, but the elliptical surface, dual-blade dual-surface and elliptical parabolic surface are not straight surface.

Is the single-leaf dual-surface and hyperbolic parabolic surface a straight line surface? The answer is yes.

IIStraight Lines of Single-blade dual-surfaces:

Single-leaf dual-surface (1)

(1) equivalent to () = (2)

Namely: =: (3)

For λ = 0, Equations

(4)

Indicates that the line is always running. In addition

(5) and

(6)

It also indicates a straight line. Obviously, every persistent line in a straight-line family composed of (4)-(6) is on a single-leaf dual-surface (1. In addition, there are

Note that the values of 1 + and 1-are not all 0.

1 ° if 1 + = 0

When Lambda =

Zyl (4)

At that time, then 1-= 0, then ε (5)

2 ° if 1 + = 0, 1-= 0

When λ = 0, then ε (4)

At that time, there were records (6)

Surface: the single-leaf dual-surface is composed of the straight-line family (4)-(6). The single-leaf dual-surface is a straight surface. Similarly

μ =0 (4 ′)

(5 ′)

And (6 ′)

A straight-line family can also form a single-leaf dual-surface (1). For convenience of memory, (4)-(6) and (4')-(6') are written in the following unified form:

U, U' is not all 0 (7)

V and V' are not all 0 (7 ′)

(7) (7') is the u family of Single-leaf dual-surface (1) and the v Family direct bus.

3.Straight Lines of the hyperbolic parabolic table:

For a hyperbolic parabolic table (1)

Yes (2)

It is similar to the case of a single-leaf dual-surface. It can be proved that the linear family:

U is any real number (3)

Can constitute a hyperbolic parabolic table (1) and

V is an arbitrary real number (3 ′)

It can also be formed that the hyperbolic parabolic Region (1) is a straight surface, which is called (3), and (3') is a hyperbolic parabolic Region (1 ).

U and v Direct busbars.

 

ThuStraight busbars of Single-blade dual-curved surface and hyperbolic parabolic shape:

Theorem 1: Straight busbars with single-blade dual-surfaces have the following properties:

(I) any two straight busbars in the same family have different faces, and any two straight busbars in different families have common faces;

(Ii) over a single-leaf dual-surface (1) a single point has only one straight bus in a family;

(Iii) a plane passing through a straight bus must also pass through a straight bus belonging to another generation;

(Vi) any three straight busbars in the same family cannot be parallel to the same plane.

Certificate:(I) any dual-Curved Single-blade (1) Dual-u direct bus

Li I = 1, 2

Then

=-('-') = 0

Parallel, the same surface can prove any two straight busbars in different families.

(Ii) omitted.

(Iii) any u-family direct bus

:

After plane π:

V' [u (

That is

∴ π over a v Family direct bus

:

(Vi) any three u direct busbars: I = 1, 2, 3

Then the direction vector is {(,,}

Determined

• =

  =

  =-= 0
  

The three straight lines in the same plane travel unevenly.

Theorem 2: Straight busbars of the hyperbolic parabolic shape have the following properties:

(I) any two straight busbars in the same family have different planes and any two straight busbars in different families have intersection;

(Ii) any point above the hyperbolic parabolic table, which has only a persistent busbar in the family;

(Iii) All straight busbars in the same family are parallel to a certain plane.

Certificate:Verify only (iii) direct bus to u family

Its Direction vector is {} bytes. The direct bus is parallel to the plane bx + ay = 0.