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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 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 $\mathcal{P}$ 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 $\mathcal{P}$ (cd Fig  1(b)). The equation of the parabolic cylinder is y=x2 too.


   
Figure 1: Cylinders.
\begin{figure}
\mbox{\subfigure[$x^2+2y^2=1$ ]{\epsfig{file=Cylinder1.eps,height...
... {x^2}{2}=0$ ]{\epsfig{file=ParabolicCylinder.eps,height=6cm} }
}\end{figure}

Example 3.3   The surface whose equation is

\begin{displaymath}\frac {x^2}{a^2}+\frac {y^2}{b^2} + \frac {z^2}{c^2} = 1
\end{displaymath}

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.
\begin{figure}
\mbox{
\subfigure[]{\epsfig{file=Ellipsoid.eps,height=4cm} }
\q...
... \qquad
\subfigure[]{\epsfig{file=Ellipsoid2.eps,height=4cm} }
}\end{figure}

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).
\begin{figure}
\mbox{\epsfig{file=EllipticParaboloid.eps,height=8cm} }
\end{figure}

Example 3.5   The surface whose equation is

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

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).

   
Figure 4: Hyperboloids.
\begin{figure}
\mbox{\subfigure[One sheet]{\epsfig{file=hyperboloid1.eps,height=...
...bfigure[Two sheets]{\epsfig{file=hyperboloid2.eps,height=7cm} }
}\end{figure}

The surface whose equation is

\begin{displaymath}\frac {x^2}{a^2}- \frac {y^2}{b^2} - \frac {z^2}{c^2} = 1
\end{displaymath}

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

\begin{displaymath}\frac {x^2}{a^2}- \frac {y^2}{b^2} = \frac {z}{c}
\end{displaymath}

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.


  
Figure 5: An hyperbolic paraboloid (z=x2-y2).
\begin{figure}
\mbox{\epsfig{file=HyperbolicParaboloid.eps,height=6cm} }
\end{figure}

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

\begin{displaymath}\frac {x^2}{a^2}+ \frac {y^2}{b^2} = \frac {z^2}{c^2}
\end{displaymath}

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.
\begin{figure}
\mbox{\epsfig{file=cone2.eps,height=6cm} }
\end{figure}


next up previous contents
Next: Parametrized surfaces. Up: Surfaces. Previous: Surfaces.
Noah Dana-Picard
2001-05-30