A Brief Catalogue of the Quadric Surfaces The curves in the xy-plane defined by equations in x and y of the second degree are the conic sections:circle,ellipse,parabola,hyperbola. The surfaces in three-dimensional space defined by equations in x,y,z of the second degree, (*) Ax2+By2+C22+Dxy+Exz+Fyz++y++K=0 are called the quadric surfaces. Equation (*contains terms in xy,xz,yz.These terms can be eliminated by a suitable change of coordinates.Thus,for our purposes,the quadric surfaces are given by equations of the form Ax2+By2+C2+Dx+Ey+F+H=0 with 4,B,C not all zero.(If A,B,C are all zero,we don't have an equation of the second degree.) Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The curves in the xy-plane defined by equations in x and y of the second degree are the conic sections: circle, ellipse, parabola, hyperbola. Equation (∗) contains terms in xy, xz, yz. These terms can be eliminated by a suitable change of coordinates. Thus, for our purposes, the quadric surfaces are given by equations of the form Ax2 + By2 + Cz2 + Dx + Ey + Fz + H = 0 with A, B,C not all zero. (If A, B,C are all zero, we don’t have an equation of the second degree.) The surfaces in three-dimensional space defined by equations in x, y, z of the second degree, (∗) Ax2 + By2 + Cz2 + Dxy + Exz + Fyz + Hx + I y + Jz + K = 0, are called the quadric surfaces
A Brief Catalogue of the Quadric Surfaces The quadric surfaces can be viewed as the three-space analogs of the conic sections.They fall into nine distinct types. 1.The ellipsoid. 2.The hyperboloid of one sheet. 3.The hyperboloid of two sheets. 4.The elliptic cone 5.The elliptic paraboloid. 6.The hyperbolic paraboloid 7.The parabolic cylinder 8.The elliptic cylinder. 9.The hyperbolic cylinder. Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The quadric surfaces can be viewed as the three-space analogs of the conic sections. They fall into nine distinct types. 1. The ellipsoid. 2. The hyperboloid of one sheet. 3. The hyperboloid of two sheets. 4. The elliptic cone. 5. The elliptic paraboloid. 6. The hyperbolic paraboloid. 7. The parabolic cylinder. 8. The elliptic cylinder. 9. The hyperbolic cylinder
A Brief Catalogue of the Quadric Surfaces The Ellipsoid x2 v2 c=1 a+ ellipsoid Figure 15.2.1 Main Meny C0
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Ellipsoid 2 2 2 2 2 2 1 x y z a b c + + =
A Brief Catalogue of the Quadric Surfaces The Hyperboloid of One Sheet x2 y222 hyperboloid of one sheet Figure 15.2.2 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Hyperboloid of One Sheet 2 2 2 2 2 2 1 x y z a b c + − =
A Brief Catalogue of the Quadric Surfaces The Hyperboloid of Two Sheets x2 y222 a+b-=-1 (0,0,c (0,0,-c) hyperboloid of two sheets Figure 15.2.3 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Hyperboloid of Two Sheets 2 2 2 2 2 2 1 x y z a b c + − = −
A Brief Catalogue of the Quadric Surfaces The Elliptic Cone x2.y2 3 b223 elliptic cone Figure 15.2.4 Main Meny☐ C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Elliptic Cone 2 2 2 2 2 x y z a b + =
A Brief Catalogue of the Quadric Surfaces The Elliptic Paraboloid x2.y2 a2+6= elliptic paraboloid Figure 15.2.5 Main Menu cmew59时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Elliptic Paraboloid 2 2 2 2 x y z a b + =
A Brief Catalogue of the Quadric Surfaces The Hyperbolic Paraboloid x2 y2 a262= hyperbolic paraboloid Figure 15.2.6 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Hyperbolic Paraboloid 2 2 2 2 x y z a b − =
A Brief Catalogue of the Quadric Surfaces The Parabolic Cylinder x2 =4cy parabolic cylinder Figure 15.2.7 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Parabolic Cylinder x 2 = 4cy
A Brief Catalogue of the Quadric Surfaces The Elliptic Cylinder x a+护1 elliptic cylinder Figure 15.2.8 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. A Brief Catalogue of the Quadric Surfaces The Elliptic Cylinder 2 2 2 2 1 x y a b + =