Quadric surfaces.

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

Example 3.2       

Take a circle $\mathcal{C}$in the xy-plane. Draw lines parallel to the z-axis and intersecting $\mathcal{C}$. The surface we get is a cylinder whose basis is the circle $\mathcal{C}$. In Fig 1(a), the equation of the circle is x²+y²=1; as a point of the space belongs to the cylinder 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 $\mathcal{P}$in the xy-plane. Draw lines parallel to the z-axis and intersecting $\mathcal{P}$. The surface we get is a cylinder whose basis is the parabola $\mathcal{P}$(Fig 1(b)). The equation of the parabolic cylinder is y=x² too.

 

   

Figure 1: Cylinders.

 

Example 3.3   The surface whose equation is

$$\frac {x^2}{a^2}+\frac {y^2}{b^2} + \frac {z^2}{c^2} = 1$$

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).

   

Figure 2: Ellipsoids.

 

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.

 

  

Figure 3: An elliptic paraboloid ( z=x²+4y²).

 

Example 3.5   The surface whose equation is

$$\frac {x^2}{a^2}+\frac {y^2}{b^2} - \frac {z^2}{c^2} = 1$$

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 a 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).

   

Figure 4: Hyperboloids.

 

The surface whose equation is

$$\frac {x^2}{a^2}- \frac {y^2}{b^2} - \frac {z^2}{c^2} = 1$$

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 a 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

$$\frac {x^2}{a^2}- \frac {y^2}{b^2} = \frac {z}{c}$$

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 a hyperbola.

 

  

Figure 5: An hyperbolic paraboloid (z=x²-y²).

 

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

$$\frac {x^2}{a^2}+ \frac {y^2}{b^2} = \frac {z^2}{c^2}$$

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.

 

  

Figure 6: The cone whose equation is x²+y²=z².