§ 6 Quadric surface _
1. Spherical
[ Equation, center and radius of a sphere ]
|
Equations and Graphics |
Center and Radius |
||
|
or
( Spherical coordinate equation . In the formula, j is the longitude , and q is the co-latitude )
|
|
Center G (0,0,0 ) Radius R |
|
|
or ( j and q in the spherical coordinate equation are the same as above )
|
|
Center G ( a,b,c ) Radius R
|
|
|
Equations and Graphics |
Center and Radius |
||
|
|
ball center radius |
||
[ The tangent and normal of the sphere ] Suppose a plane P passes through a point M on the sphere and is perpendicular to the radius GM , then P is called the tangent of the sphere at M. The straight line MG is called the normal of the sphere at the point M.
Let the spherical equation be
Then the tangent equation of the sphere at the point M ( 
) is
The normal equation of the sphere at the point M ( ) is
[ Intersection angle of two spheres ] Set two spheres
=0
=0
The intersection angle of two spheres refers to the angle between the two tangent planes at the intersection point , denoted by q , then
Since the coordinates of the intersection point are not included in the formula, the intersection angles of the points on the intersection of the two spheres must be equal .
The orthogonal condition of the two spheres is
[ Spherical bundle · Root surface of two spheres ] set
In the formula and as defined in formula (1) , and are parameters , there are
A certain value of the pair represents a sphere . When all values are taken , the whole of the spheres represented is called a spherical bundle . When it is a plane , it is called the root surface of the two spheres , and its equation is
The root plane is perpendicular to the connecting center line of the sum , the center of any sphere in the bundle is on the connecting center line , and the ratio of the sub-connecting center lines is .
[ Spherical sink·root axes of three spheres ] Let the sum be as defined by formula (1) , and let
Assume
where are two independent parameters , then we have
A pair of definite values of the pair represents a spherical surface , and when all values are taken , the whole of the represented spherical surface is called a spherical sink .
The root surfaces of each pair of spheres in the three spheres are
and
These three planes intersect in a straight line called the root axis .
Second, the ellipsoid
|
Equations and Graphics |
basic elements |
Features |
|
[ ellipsoid ]
When a=b , it is an ellipsoid of revolution ( curve on the Ozx plane It is spherical when a=b=c |
vertex Spindle Principal planes and their equations : Oxy plane z=0 Oyz plane x=0 Ozx plane y=0 The equation of the main axis : AA’ y=z=0 BB’ z=x=0 CC’ x=y=0 Center O (0,0,0) Diameter plane through the center of the plane |
The intersection of any plane and the ellipsoid is an ellipse ( a circle in special cases ). The midpoints of a set of chords parallel to a given direction d lie in a plane , which is a diametrical plane , which is conjugate to the direction d . The three principal planes are diametrical planes that are conjugated to the principal axes respectively . The volume of the ellipsoid : |
Three, hyperboloid
|
Equations and Graphics |
basic elements |
Features |
|
[ Single leaf hyperboloid ]
[ Futaba Hyperboloid ] When a = b , it is [ Rotating Hyperboloid ] ( curve on the Oxz plane obtained by rotating around the z -axis ) |
Spindle Center O (0,0,0) Principal planes and their equations : Oxy plane z = 0 Oyz plane x = 0 Ozx plane y = 0 |
The intersection of a plane parallel to the z -axis with a hyperboloid is a hyperbola ( for a single-lobe hyperboloid , it may be a pair of intersecting lines ). The intersection of a plane parallel to the Oxy plane and a hyperboloid is an ellipse . There are two families of straight generatrixes on a single-leaf hyperboloid , and their equations are ( l is a parameter )
and ( m is a parameter ) |
4. Paraboloid
|
Equations and Graphics |
basic elements |
Features |
|
[ ellipse paraboloid ] When a = b , it is a paraboloid of revolution ( The curve on the Ozx plane is obtained by rotating around the z - axis ) [ Hyperbolic Paraboloid ]
|
Vertex O (0,0,0) main axis z axis Principal planes and their equations : Oyz plane x = 0 Ozx plane y = 0 |
The intersection of an elliptic paraboloid with a plane parallel to the z -axis is a parabola ; the intersection with a plane parallel to Oxy is an ellipse . volume
volume
The intersection of a hyperbolic paraboloid with a plane parallel to Oyz ( or a plane parallel to Ozx ) is a parabola ; the intersection with a plane parallel to Oxy is a hyperbola . The shape of the hyperbolic paraboloid is saddle-shaped , so it is also called the saddle surface . There are two families of straight generatrix on the hyperbolic paraboloid , and their equations are and |
( 
(
5. Cone and Cylinder
|
Equations and Graphics |
basic elements |
Features |
|
[ Elliptical Cone ] When a = b , it is a conical surface ( The line on the Oxz plane is obtained by rotating around the z axis ) |
main axis z axis vertex origin O a, b are the semi-axes of the intersection ( ellipse ) between the plane of z=c and the cone (ellipse)
|
The intersection of the elliptical cone with the plane z = h parallel to Oxy is an ellipse It intersects the Oxy plane at the origin O.
|
|
[ elliptical cylinder ] When a = b , it is a cylindrical surface |
The equation of the directrix is The number of directions of the busbar is (0,0,1) |
The intersection of an elliptical cylinder with any plane parallel to Oxy is the same ellipse |
|
[ Hyperbolic Cylinder ]
|
The equation of the directrix is The number of directions of the busbar is (0,0,1)
|
|
|
Equations and Graphics |
basic elements |
Features |
|
[ Parabolic Cylinder ] |
The equation of the directrix is The number of directions of the busbar is (0,0,1) |
|
|
[ Asymptotic Cone ] Secondary cone is a hyperboloid asymptotic cone of |
|
Similar to the asymptote of a hyperbola, the intersection of each plane passing through the z -axis and the hyperboloid is a pair of conjugate hyperbolas , and the intersection with the conical surface is two straight lines , that is, the asymptote of the pair of hyperbolas. . |
6. General quadratic surface
1. General properties of quadric surfaces
For the ellipsoids, hyperboloids, paraboloids, etc. listed above , their equations are quadratic about x , y, and z . The general quadratic equation for x, y, and z is in the form of
The surface it represents is called a general quadratic surface . Here are some common properties of these surfaces .
[ Intersection point of a straight line and a quadric surface ] A straight line intersects a quadratic surface at two points ( real , imaginary , coincident ). Or the line is all on the surface , at this time it is called the straight generatrix of the quadric surface or busbar .
[ Intersection of plane and quadratic surface ] The intersection of any plane and a quadratic surface is a quadratic curve .
[ Diameter plane and center of quadric surface ] The midpoint of the chord parallel to the known direction of a quadric surface is on a plane , called the diameter plane , which bisects a set of parallel chords . Set the number of directions in the known direction For l , m , n , the equation of the diameter plane is
or rewritten as
When l , m , n vary , this equation represents a plane , and the diameter planes of the quadric form a plane . Any plane in the plane passes through the intersection of the following three planes :
If the intersection point is not on the surface , it is called the center of the quadric surface , and if the intersection point is on the surface , it is called the vertex of the quadric surface . All quadric surfaces with a center are called centered quadrics , and the rest are called quadrics is a centerless quadratic surface .
[ Principal plane and principal axis of quadric surface ] If the diameter plane is perpendicular to the chord bisected by it , it is called the principal plane ( symmetry plane ) . Any two principal planes are perpendicular to each other , and their intersection is the principal axis .
[ Tangent plane and normal of quadric surface ] The equation of the tangent plane of the quadric surface at a point M ( ) is
The straight line perpendicular to the tangent plane of the quadratic surface at point M is called the normal of the surface at point M , and its equation can be written as
[ Circular section of quadric surface ] If the intersection of a plane and a quadric surface is a circle , then the plane is called the circular section of the surface .
If the quadric surface is not a spherical surface , then through a point in space , the quadric surface has six circular sections ; there are generally two real circular sections and four imaginary circular sections ; and several of the six circular sections are coincident .
2. Invariants of quadric surfaces
By the general equation of quadratic surface
(1)
The coefficients of are composed of the following four functions :
It is called the invariant of quadratic surface , that is, after coordinate transformation , these quantities are invariant . The determinant is called the discriminant of quadratic equation (1) .
3. Standard equations and shapes of quadric surfaces
|
invariant _ |
The equation after coordinate transformation |
Curve shape |
||
|
Concentric Quadratic Surface |
D>0 |
where A , B , C , are characteristic equations
The three characteristic roots of |
A , B , C , when the sign is different, it is a single-leaf hyperboloid A , B , C , no track when the same sign |
|
|
D<0 |
A , B , C , the same sign is an ellipsoid A , B , C , if the sign is different, it is a double-leaf hyperboloid |
|||
|
D=0 |
A , B , C , no track when the same sign A , B , C , when the sign is different, it is a quadratic cone |
|||
|
D = 0 Centerless Quadric |
D<0 |
|
Ellipse Paraboloid ( When both A and B are positive , the negative sign is taken before the square root ; when both A and B are negative , the positive sign is taken before the square ) |
|
|
D>0 |
hyperbolic paraboloid |
|||
|
D=0 |
|
|
When A , B have the same sign, there is no track |
|
|
J = 0 |
|
parabolic cylinder a pair of parallel planes no track A pair of coincident planes |
||
: A , B , C , 
: A , B , C , 
