University Mathematics Handbook (2015)

III. Analytic Geometry and Vectors

Chapter 8. Canonical Forms of Second-Order Surfaces in Space

8.1  Sphere

A sphere is the locus of all points in space for which the distance to point , or center is constant, and called radius.

From the formula of distance of any point on the surface of the sphere, to point , the sphere equation is obtained:

The intersection lines of a sphere with any plane are circles.
Example: The line of intersection of plane , which is parallel to plane is the following circle

when .

8.2 Ellipsoid

is an ellipsoid that is symmetric across planes , , .

Intervals on -axis, on -axis and on axis are called ellipsoid axis. The lines of intersection of ellipsoid with planes parallel to coordinate planes are ellipses.

Example: , when , is

.

Similarly, the lines of intersection with planes and are ellipses.

8.3 One-Sheet Hyperboloid

The hyperboloid is symmetric across coordinate planes , , .

The intersection lines of one-sheet hyperboloid with the plane are ellipse .

If is increasing, then the ellipses axes are increasing.

The smallest ellipse, when , is called the neck of one-sheet hyperboloid.

The intersection lines of one-sheet hyperboloid with planes and are hyperbolas.

8.4  Two-Sheet Hyperboloid

The canonical form of two-sheet hyperboloid equation indicates it is symmetric across the planes of axes .

The intersection lines of two-sheet hyperboloid with planes are ellipses.

Plane starts intersecting two-sheet hyperboloid for . In other words, between planes and , there exist no points belonging to this surface. The intersection lines with the planes at and are hyperbolas.

8.5  Cone

The intersection lines of cone with planes are ellipses.

The intersection line of the cone with plane are a pair of straight lines , and with plane , are a pair of straight lines .

A cone is formed by straight lines passing through the origin.

8.6 Elliptic Paraboloid

The intersection lines of elliptic paraboloid with planes , are ellipses that increasingly widen with increasing.

The intersection lines of it with planes , are parabolas and respectively.

8.7  Hyperbolic Paraboloid

,

Its intersection lines with planes are hyperbolas .

Its intersection lines with planes and are parabolas and respectively.

Two straight lines pass through any point in a hyperbolic paraboloid.

8.8  Cylindrical Surface

A cylindrical surface is formed by a straight line moving parallel to given straight line, along a certain curve. Any surface described by one of the equations (parallel to -axis), , (parallel to -axis), , (parallel to -axis) is cylindrical. The following are examples of cylindrical surfaces: