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Definition 3.1   Surfaces defined by polynomials of degree 2 in three variables are called quadrics.

Example 3.2

Take a circle in the xy-plane. Draw lines parallel to the z-axis and intersecting . The surface we get is a cylinder whose basis is the circle . In Fig  1(a), the equation of the circle is x2+y2=1; as a point of the space belongs to the cyclinder if, and only if, its projection onto the xy-plane is a point of the circle, this equation defines the cylinder.

Let y=x2 be the equation of a parabola in the xy-plane. Draw lines parallel to the z-axis and intersecting . The surface we get is a cylinder whose basis is the parabola (cd Fig  1(b)). The equation of the parabolic cylinder is y=x2 too.

Example 3.3   The surface whose equation is

where a,b,c are given positive numbers, is called an ellipsoid. Its intersection with any plane, parallel to a coordinate plane, is an ellipse.

Substitute -x instead of x (resp. -y instead of y, resp. -z instead of z); the equation is not modified, thus the ellipsoid is symmetric about the yz-plane (resp. the xz-plane, resp. the xy-plane).

If a=b=c, the surface is a sphere.

Example 3.4   The surface whose equation is

z=ax2+by2

where a,b are given positive numbers, is called a paraboloid. Its intersection with a plane parallel to the xy-plane is a circle; its intersection with a plane parallel to another coordinate plane is a parabola.

Example 3.5   The surface whose equation is

where a,b,c are given positive numbers, is called a one sheet hyperboloid. It has one component, and its intersection with a plane, parallel to the xy-plane, is either empty or an ellipse. Its intersection with a plane, parallel to another coordinate plane, is an hyperbola.

Substitute -x instead of x (resp. -y instead of y, resp. -z instead of z); the equation is not modified, thus the ellipsoid is symmetric about the yz-plane (resp. the xz-plane, resp. the xy-plane).

The surface whose equation is

where a,b,c are given positive numbers, is called a two sheet hyperboloid. It has two components; its intersection with a plane parallel to a coordinate plane is either empty or an ellipse or an hyperbola. Substitute -x instead of x (resp. -y instead of y, resp. -z instead of z); the equation is not modified, thus the ellipsoid is symmetric about the yz-plane (resp. the xz-plane, the xy-plane).

Example 3.6   The surface whose equation is

where a,b,c are given positive numbers, is called an hyperbolic paraboloid. It has one component; its intersection with a plane parallel to a coordinate plane is either a parabola or an hyperbola.

Substitute -x instead of x (resp. -y instead of y); the equation is not modified, thus the ellipsoid is symmetric about the yz-plane (resp. the xz-plane).

The surface whose equation is

where a,b,c are given positive numbers, is called cone. It has one component, and it is made of straight lines through its vertex. Its intersection with a plane parallel to a coordinate plane is either an ellipse or the union of two lines.

Next: Parametrized surfaces. Up: Surfaces. Previous: Surfaces.
Noah Dana-Picard
2001-05-30