Chapter VI. Forms of Quadric Surfaces 55. Definition of a quadric. The locus of an equation of the second degree in x,y and z. Equation of any non-composite quadric may be reduced to one of standard types, by a suitable transformation.

56. The ellipsoid. The locus of the equation It is symmetrical as to Each coordinate plane, (x,y,z)(x,y,-z); plane: z=0 Each coordinate axis, (x,y,z)(-x,-y,z); z axis The origin, (x,y,z)(-x,-y,-z)

The ellipsoid intersects with the three axes respectively, (a,0,0),(-a,0,0), X-axis; (0,b,0),(0,-b,0), Y-axis; (0,0,c),(0,0,-c), Z-axis; The above six points are called vertices. The coordinate axes included between the vertices are called axes of the ellipsoid. The point of intersection of the axes are called center. Segments from center to vertices are called semi-axes. If , the three axes joining the vertices of X-axis,Y-axis and Z-axis are called respectively, mean major axis, mean axis, and the minor axis.

If |k|=0, the ellipse are biggest. The section of the ellipsoid by the plane z=k is an ellipse whose equation are If |k|=0, the ellipse are biggest. If |k|=c, the ellipse reduces to a point If |k|>c, the ellipse is imaginary.

If a=b>c, the ellipsoid is a surface of revolution obtained by revolving the ellipse about its minor axis (Z-axis). Its called oblate spheroid (扁的回转椭圆体） If a>b=c, it is called prolate spheroid.

57. The hyperboloid of one sheet. The surface It is also symmetric as to Coordinate planes, three axes, the origin. The section by plane z=k is an ellipse of

The plane y=k’ intersects the surface in the hyperbola If |k’|<b, the transverse axis of the hyperbola is line z=0,y=k’. If |k’|=b, the intersection (is composite) reduces to two lines given by

The plane x=k” intersects the surface in the hyperbola If |k”|<a, the transverse axis of the hyperbola is line z=0,x=k”. If |k”|=a, the intersection (is composite) reduces to two lines given by

58. The hyperboloid of two sheets. The surface It is also symmetric as to Coordinate planes, three axes, the origin. The section by plane z=k is a hyperbola

The plane y=k’ intersects the surface in hyperbola The plane x=k” intersects the surface in the ellipse

59. The imaginary ellipsoid Exercises: P69. No. 1, 2 and 5.

60. The elliptic paraboloid. It is symmetric as to the planes x=0 and y=0 but not as to z=0. Notice that , so if n>0, the surface lies on the positive side,…. The section of the paraboloid by the plane z=k is an ellipse If nk<0, it is imaginary, if k=0, the origin,

The section of the paraboloid by the plane y=k’ is the parabola For all k’ these parabolas are congruent (with Z axis) As k’ increases, the vertices recede from the plane y=0 along the parabola Z X O Y

If a=b, the parabola is the surface of revolution. The section by planes x=k” are congruent parabolas (they coincide with a transition) If a=b, the parabola is the surface of revolution. By revolving the following parabola about Z-axis

61. The hyperbolic paraboloid. The surface is symmetric as to planes x=0 and y=0 but not as to z=0. The plane z=k intersects the surface in the hyperbola (assume n>0)

If k>0, the line x=0,z=k is the transverse axis and y=0, z=k is the conjugate axis. If k<0, the axes are interchanged. When k=0, the section of the paraboloid consists of two lines(see P.71, Fig.29.)

The sections by plane y=k’ are congruent parabolas The vertices of these parabolas describe the parabola The sections by planes x=k” are congruent parabolas whose vertices describe the parabola

62. The quadric cones (real) Its vertex is at the origin. It is also symmetric as to plane x=0,y=0,z=0, …. The section by the plane z=c is the ellipse

The cone is therefore the locus of a line which passes through the origin and intersects this ellipse. If a=b, the surface is the right circular cone generated by revolving the line x/a=z/c,y=0 about the Z-axis. The imaginary quadric cone is represented by

63. The quadric cylinders. The cylinders (art.43) whose equation are one of the following forms: are called elliptic, hyperbolic, imaginary, and parabolic cylinders

64. Summary. The surfaces are categorized as follows Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Imaginary ellipsoid Elliptic paraboloid

Hyperbolic paraboloid. Real quadric cone Imaginary quadric cone Quadric cylinders