4 § 6 parabolic

§ 6 parabolic

 

Example: The surface is rotated by a parabolic line around the axis.

This surface is calledRotating parabolicTo promote the surface:

IElliptical parabolic:

1,Definition: In a Cartesian system, the image represented by the equation (a, B> 0) (1) is calledElliptical parabolic(1) A standard equation called an elliptical parabolic dish.

Note: In the right angle system, the equation or the image is also an elliptical parabolic.

2,Properties and shapes:

(I) symmetry: elliptical parabolic table (1) about the Z axis, plane, and plane symmetry. In ch6, we will know that the elliptical parabolic table has no symmetric center.

(Ii) Boundary: from (1) ZZ = 0, the upper of the ∴ oval parabolic table (1) is located, and it is unbounded.

(Iii) intersection with the coordinate axis and intersection with the Coordinate Plane

(1) The three coordinate axes are handed over to the origin -- vertex; (1) the three coordinate axes are handed over

, That is

(2), (3), (4)

(2), (3) are parabolic, And the vertices are origin points. The open direction is the forward direction of the Z axis. The axes of symmetry are Z axes, and (4) are the origin.

(Iv) intersection with a plane parallel to the coordinate plane:

First, (1) it is handed over to the plane parallel to the plane, that is

() (5)

At that time, (5) is the origin;

At that time, (5) was an elliptic, and its vertex was (0, ± B, k) ε (2), (± a, 0, k) ε (3 ).

It can be seen that an elliptical parabolic table (1) is composed of a series of "Parallel" edges on the top of the plane. The vertices of these rectangles change on parabolic (2) and (3.

 

 

(Fig. 4.6)

In addition, an elliptical parabolic table (1) is handed over to a plane parallel to a plane, that is

(6)

Right, (6) are all completely equal parabolic. Its vertex (, 0,) ε (3) symmetric axis returns the Z axis, and the opening direction toward the Z axis forward (consistent with the opening direction of (3)

Finally, if we use the plane parallel to the plane to truncate (1), the cut line is similar. From this, we can obtain the geometric characteristics of the elliptical parabolic table as follows:

An Elliptical parabolic dish is a trajectory formed by moving one parabolic line along another certain parabolic line. During the moving process, the vertex of the dynamic parabolic line is always on the fixed parabolic line, and the opening direction is the same as that of the fixed parabolic open direction, and their planes are always vertical (4.6 ).

 

IIHyperbolic Parabolic:

1,Definition: In a Cartesian system, the image represented by the equation (a, B> 0) (1) is calledHyperbolic Parabolic(1) A standard equation called a hyperbolic parabolic table.

Note: In the right angle system, the equation or the image is also a hyperbolic parabolic dish.

2,Properties and shapes:

(I) symmetry: hyperbolic parabolic table (1) about the Z axis, plane, and plane symmetry. In ch6, we will know that the hyperbolic parabolic table has no symmetric center.

(Ii) Boundary: from (1) known hyperbolic parabolic table (1) is unbounded.

(Iii) intersection with the coordinate axis and intersection with the Coordinate Plane

(1) The three coordinate axes are handed over to the origin -- vertex; (1) the three coordinate axes are handed over

, That is

(2), (3), (4)

(2), (3) are parabolic, and its vertices are origin points. The opening direction is forward to the Z axis and negative to the Z axis. The axes of symmetry are all Z axes, and (4) are two intersecting lines.

(Iv) intersection with a plane parallel to the coordinate plane:

First, (1) it is handed over to the plane parallel to the plane, that is

(5)

At that time, (5) was (4 );

At that time, (5) was a hyperbolic curve, and its vertex was (± a, 0,K) ε (3 ).

When (5) is still a hyperbolic curve, its vertex is (0, ±,K) ε (2)

It can be seen that the hyperbolic parabolic table (1) is composed of a series of "Parallel" hyperbolic curves parallel to the surface. the vertices of these hyperbolic tables change on the parabolic table (2) and (3.

In addition, the hyperbolic parabolic table (1) is handed over to the plane parallel to the plane, that is

(6)

Right, (6) are all completely equal parabolic, its vertex (k, 0,) ε (3) symmetric axis ∥ Z axis, open direction toward Z axis negative (and (3) in the opposite direction)

Finally, if we use a plane parallel to the plane to truncate (1), the cut line is similar. Therefore, the geometric characteristics of the hyperbolic parabolic table are as follows:

A hyperbolic parabolic dish is a trajectory formed by moving one parabolic line along another certain parabolic line. During the moving process, the vertex of the dynamic parabolic line is always on the fixed parabolic line, and the opening direction is opposite to the opening direction of the fixed parabolic line, and their planes are always vertical (4.7 ).

 

 

 

 

(Fig. 4.7)