Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
Essentials
Table 3.3.1 lists the quadric surfaces, surfaces described by equations quadratic in the three variables x,y,z, that is, equations of the form
α1 x2+α2 y2+α3 z2+β1 x+β2 y+β3 z+γ=0
Quadric
Standard Form
Ellipsoid
x−x02a2+y−y02b2+z−z02c2=1
Hyperboloid of 1 sheet
x−x02a2+y−y02b2−z−z02c2=1
Hyperboloid of 2 sheets
x−x02a2−y−y02b2−z−z02c2=1
Cone
z−z02c2=x−x02a2+y−y02b2
Elliptic Paraboloid
z−z0c=x−x02a2+y−y02b2
Circular Paraboloid
z−z0c=x−x02a2+y−y02a2
Hyperbolic Paraboloid
z−z0c=x−x02a2−y−y02a2
Circular Cylinder
x−x02+y−y02=r2
Elliptic Cylinder
x−x02a2+y−y02b2=1
Parabolic Cylinder
y−y0=a x−x02
Table 3.3.1 Quadric surfaces
Standard form is typically obtained by completing the square in each of the three variables, as applicable. The point x0,y0,z0 is generally a central point for the quadric surface.
Table 3.3.2 provides additional details for some of the quadric surfaces listed in Table 3.3.1.
The hyperboloid of one sheet is characterized by a single minus sign in front of one of the squared terms. The orientation of the central axis for this surface is determined by the variable before which there is the minus sign.
The hyperboloid of two sheets is characterized by two minus signs in front of two of the squared terms. The orientation of the central axis for this surface is determined by the variable before which there is the single plus sign.
The orientation of the central axis for the cone is determined by the variable on the left: this variable can be any one of the three, with the other two on the right having plus signs before them.
As for the cone, the orientation of the central axis of a paraboloid is determined by the variable on the left: this variable can be any one of the three, with the other two on the right having plus signs before them. The difference between the cone and the paraboloid is that the variable isolated on the left is not squared for the paraboloid.
The central axis for a cylinder is determined by the variable that is "missing" from the equation. Consequently, any contour in the plane can be "extruded" along an axis perpendicular to the plane of the curve, thereby forming a cylinder whose cross section has the shape of the planar curve.
Table 3.3.2 Notes pertinent to the quadrics listed in Table 3.3.1
Examples
In Examples 3.3.(1-12), put the given equation into standard form for a quadric surface, identify the surface, draw its graph, and discuss the nature of the level curves and plane sections.
Example 3.3.1
4x2+4y2−z2−8 x+8 y+4 z+4=0
Example 3.3.2
2 x2−y2−4 x−2 y−z+2=0
Example 3.3.3
z2−4 x2−4y2−4 z+8 x−8 y−8=0
Example 3.3.4
3 x2− 6 x− y+ 1=0
Example 3.3.5
16x2+9y2+36z2−32 x+18 y−144 z+25=0
Example 3.3.6
x2+y2−2 x+2 y−z+3=0
Example 3.3.7
9x2+4y2−18 x+16 y−11=0
Example 3.3.8
z2−4 x2−2y2−4 z+8 x−4 y−6=0
Example 3.3.9
2 x2+y2−4 x+2 y−z+4=0
Example 3.3.10
4x2+4y2−z2−8 x+8 y+4 z=0
Example 3.3.11
4x2+2y2−z2−8 x+4 y+4 z+2=0
Example 3.3.12
4x2+2y2−z2−8 x+4 y+4 z−2=0
Example 3.3.13
Give an equation for the elliptic cylinder whose central axis is the line through 1,2,3, parallel to the y-axis. The cross sections perpendicular to the central axis should be ellipses with a vertical semi-major axis of length 3 and a horizontal semi-minor axis of length 1. Graph this cylinder for y∈1,5.
Example 3.3.14
For z∈0,3, draw a cylinder whose axis is parallel to the z-axis, and whose cross sections are the shape of the curve defined by the equation x2+y2+x=x2+y2.
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