Analytic geometry: Sixth chapter two quadric surface (2) parabolic two conic cylindrical surface general two times surface _ analytic geometry

§4 Parabolic

1. Elliptical Paraboloid

(1) Standard equation

(2) Basic elements

* Vertex O (0,0,0)

* Spindle Z-axis

* Main plane Oyz plane: X=0;ozx plane: y=0

(3) When a=b, the elliptical parabolic surface is rotated by the parabola of the OZX plane around the z axis.

(4) intersection of planar and elliptic parabolic surfaces

* The intersection of planar and elliptic parabolic surfaces parallel to the z-axis is a parabola

* The planar z=k () perpendicular to the axis () and the intersection line of the elliptic parabola are ellipses, and the special case (k=0) is a little

2. Hyperbolic paraboloid

(1) Standard equation

(2) Basic elements

* Vertex O (0,0,0)

* Spindle Z-axis

* Main plane Oyz plane: X=0;ozx plane: y=0

(3) The equation of the ruled system bus

The hyperbolic paraboloid is a ruled surface and has two straight bars on each point of the curved surfaces.

The two-family straight bus equations for hyperbolic parabolic surfaces are:

And

§ 52 conic and cylindrical surface

1. Definition of Cone

Set γ to a curve on the plane Pi, and V for a little outside of pi. When the moving point P moves along the gamma motion, the trajectory of the linear VP (line L) is called "V" as the vertex, gamma is the bottom line, and L is the cone of the bus (as shown on the right).

The cone is a ruled surface.

2. Oval Cone

(1) Standard equation

(2) Basic elements (e.g. right) vertex O (0,0,0) Spindle Z-alignment γ

(3) When the a=b is a rotational surface

When A=b, the ellipse cone is a conical plane, which is ozx the straight line around the z axis.

(4) The intersection of plane and ellipse cone is parallel to the plane of Oxy: the intersection of z=k and Ellipse cone is elliptical:

In particular, the intersection of planar z=0 and curves with the original point O parallel to the Oyz or OZX plane with the ellipse cone is a hyperbola (k≠0) or a pair of straight lines (k=0)

(5) The elliptical cone is the asymptotic cone of two-leaf hyperboloid and single hyperboloid

Oval Cone:

Double-leaf double curved surface:

Single-leaf hyperboloid: (a,b,c>0) The intersection of three surfaces and planar z=k is elliptical, and when k→∞, three ellipses are infinitely close, i.e. three surfaces are infinitely close. The intersection of each plane of the z-axis and the hyperboloid is a pair of conjugate hyperbola, and the intersection line of the cone is two straight lines, that is, the asymptotic line of the hyperbolic curve.

3. Definition of cylindrical surface

Set γ to be a spatial curve, L is a fixed line. The surface of a set of straight lines that intersects with gamma and parallel to L is called a cylindrical surface. The line called Γ as the cylindrical surface, and the straight lines parallel to l on the cylinder are referred to as Busbar.

Obviously, the cylindrical s is the surface that can be seen as the L-parallel motion along the gamma.

The cylinder is a ruled surface.

4. Elliptic cylinder, hyperbolic cylindrical surface, parabolic cylindrical name graph equation alignment and bus and planar intersection ellipse

Cylindrical surface

When a=b, a cylindrical surface:

X2+Y2=A2 guideline:

Bus direction number: (0,0,1) and parallel to the Oxy plane z=k intersection is an ellipse:

Double Song

Cylindrical surface

Alignment:

Bus direction number: (0,0,1) and parallel to the Oxy plane z=k intersection lines are hyperbola:

Throwing objects

Cylindrical surface

Alignment:

Bus direction number: (0,0,1) and parallel to the Oxy plane Z=k intersection line are parabolic lines:

　

§6 General two-time surface

1. General equation of two times surface

The two-time equation of x,y,z

The two-time surface that is represented is called a general two-time surface.

2. General properties of two-time surfaces

(1) The intersection point of a line and a two-time surface

A straight line with a two-time curve is personally at two o'clock (real, imaginary, coincident); or the line is all on the surface, at this point it is called the straight bus or bus bar of the two-time surface.

(2) intersection of planar and two-time surfaces

The intersection of any plane with a two-time surface is a two-time curve.

(3) the diameter plane and center of two times surface

The midpoint of a two-time surface parallel to a known direction is on a plane, called a diameter plane, which splits a set of parallel strings and sets the direction number of the known direction to L,m,n, then the equation of the diameter plane is

(AL+HM+GN) x+ (HL+BM+FN) y+ (GL+FM+CN) z+ (PL+QM+RN) =0

or rewritten as

(ax+hy+gz+p) l+ (hx+by+fz+q) m+ (gx+fy+cz+r) n=0

When l,m,n changes, this equation represents a planar one, by which the diameter plane of the two-time surface is formed into a plane, and the plane puts any plane through the intersection of the following planes:

Ax+hy+gz+p=0

Hx+by+fz+q=0

Gz+fy+cz+r=0

If the intersection is not on the surface, it is called the center of the two-time surface, or the vertex of the two-time surface if the intersection is on the surface. Where the center of the two-time surface is called a two-time surface, the rest is called the unintentional two-time surface.

(4) The main plane and the spindle of two times surface

If the diameter plane is perpendicular to the string to which it is divided, it is called the main plane (the symmetric plane), each two-time surface has at least one real master plane, and the two main planes of the non-rotating two-time surfaces are perpendicular to each other, and their intersection lines are the main axes.

(5) Circular cross-section of two times surface

If the intersection of a plane with a two-time surface is a circle, the plane is called a circular cross-section of the surface.

If the two-time surface is not spherical, then a point in space, two times surface has six circular cross-section; there are generally two real circles.

section, four imaginary circular sections, and several of the six circular sections are coincident.

3. Plane and normal of two times surface

The section equation of the two quadric surface at a point m (x0,y0,z0) is

Ax0x+by0y+cz0z+f (y0z+z0y) +g (z0x+x0z) +h (x0y+y0x) +p (x+x0) +q (y+y0) +r (z+z0) +d=0

The straight line between the point M and the section of the two surface is called the normal of the surface at point M, and its equation can be written as

4. Invariant of two times surface

The general equation of the two-time surface

(1)

The coefficients consist of the following four functions:

The invariant of a two-time surface, that is, after a coordinate transformation, these quantities are invariant, and the determinant δ is called the two-time equation (1).

discriminant type.

5. Two times surface standard equation and shape invariant coordinate transformation equation curve shape

D≠0

Two times curved surface δ>0

A,b,c in the formula, as the characteristic equation

The three characteristic root a,b,c is a single leaf hyperboloid.

A,b,c when the same number is no trajectory δ<0 a,b,c the same number as the ellipsoidal surface

A,b,c with two-leaf hyperbolic δ=0 a,b,c with no trace at the same number

A,b,c, two conic

D=0

Centerless two times surface

Δ<0

　

Elliptical paraboloid

(When the a,b are positive, the square root takes the minus sign, the a,b is negative, and the square root takes a plus) δ>0 the hyperbolic paraboloid.

Δ=0

J≠0

A,b the same number as elliptic cylinder or no trajectory, a,b is a hyperbolic cylindrical surface

A,b is a pair of intersection plane, A,b no trace when the same number

J=0

Parabolic cylindrical surface

A pair of parallel planes

No Trace

A pair of coincident plane

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