§1. Spherical surface

1. Spherical equation, sphere and radius graph equation

Sphere and radius

1° Standard equation:

x²+y²+z²=r²

2° parameter equation

(Φ is longitude, theta is latitude)

3° spherical coordinate equation R=r sphere: G (0,0,0)

Radius: R

1° (x-a) ²+ (y-b) ²+ z-c (²=r²)

2° parameter equation:

(φ is longitude, θ is latitude) center of the Center: G (A,B,C)

Radius: R

X²+y²+z²+2px+2qy+2rz+d=0

P²+q²+r²>d Sphere: G (-p,-q,-r)

Radius:

2. Tangent plane and normal of spherical surface

If the plane p passes through a point m on the spherical surface and is perpendicular to the radius of GM, the plane p is called the tangent planes of the spherical surface in point m. The linear mg is called the normal of the spherical surface in point m.

Set spherical equation to X²+y²+z²+2px+2qy+2rz+d=0

The tangent plane equation of the spherical surface at the point is x0x+y0y+z0z+p (x+x0) +q (y+y0) +r (z+z0) +d=0

The normal equation of the spherical surface at the point is

3. The angle of two spherical surfaces, the orthogonal conditions of two spheres

(1) The intersection of two spherical surfaces is the angle between the two tangent planes at the intersection.

Set two spherical s1:x²+y²+z²+2p1x+2q1y+2r1z+d1=0

S2:x²+y²+z²+2p2x+2q2y+2r2z+d2=0

If the angle of their intersection is Theta, there is

The upper type does not contain the coordinates of the intersection point, so the angles of the two spheres are equal at the intersection points of the intersecting lines.

(2) by the cosine expression of the intersection, two spherical orthogonal conditions are obtained

2p1p2+2q1q2+2r1r2-d1-d2=0

§2 Ellipsoidal surface

1. ellipsoidal surface equation

(1) Standard equation:

(a,b,c>0)

Parametric equation:

(Angular φ,θ as shown in the picture)

(2) Special case when the a=b is a rotating ellipsoidal surface

It is the OXZ plane curve (ellipse): The rotation around the z-axis

Note: The case of b=c or a=c is similar when a=b=c is spherical: x²+y²+z²=a²

2. Basic element vertex: spindle:

According to the size of a,b,c, it is called the long axis, the middle axis, the short axis respectively. The radius of the spindle is called the half axis, which is similar to the part of the long half axis, the middle half axis and the short half axis. Main plane oxy plane: z=0; Oyz plane: x=0; OZX plane: y=0 center O (0,0,0) diameter: Through the center of the chord diameter plane: Through the center of the plane

§3 Double Curved surface

1. Single leaf double curved surface

(1) Standard equation

(a,b,c>0)

(2) Basic elements

• Vertex

• Spindle (in accordance with the size of the a,b is called Real long axis and real short axis)

• Center O (0,0,0)

• Main plane oxy plane: z=0; Oyz plane: x=0; OZX plane: y=0

(3) The equation of the ruled system bus

The single leaf hyperboloid is a ruled surface and has two straight generatrix on each point on the curve plane. The two-family straight bus equations on a single leaf surface are:

And

(4) intersection of planar and single-leaf hyperboloid

• The intersection of a plane parallel to a z-axis and a single hyperboloid is a hyperbolic curve, and a pair of intersecting lines is a special case.

• The intersection of a plane perpendicular to the z axis and a single hyperboloid is an ellipse, in particular, the intersection of a oxy plane and a surface:

A waist circle called a hyperboloid of two surfaces. (as shown on the right)

2. Double-leaf double curved surface

(1) Standard equation

(a,b,c>0)

(2) Basic elements

• Vertex C,c´ (0,0,±c)

• Spindle (according to the a,b size is called real long axis and real short axis)

• Center O (0,0,0)

• Main plane oxy plane: z=0; Oyz plane: x=0; OZX plane: y=0

(3) When the a=b is rotated double surface

When A=b, the two-leaf hyperboloid are rotated around the z-axis by the hyperbola on the OXZ plane.

(4) intersection of planar and two-leaf hyperboloid

• The intersection lines of planar and double curved surfaces parallel to the z axis are hyperbolic

• The intersection of planar z=k (|k|≥c) perpendicular to the z axis and two-leaf hyperboloid is elliptical. Special case (|K|=C) for one point.

from:http://202.113.29.3/nankaisource/mathhands/