Monthly Archives: July 2011
Previously we discussed various 2-dimensional geometric shapes or conic sections like the parabola, hyperbola, and ellipse. Now let us consider the 3-dimensional cylinder or cylindrical surface. In mathematics, a general cylindrical surface is defined as follows:
Let C be a curve in a plane, and let L be a line that is not in a parallel plane. The set of points on all lines that are parallel to L and intersect C is a cylinder.
Notice that the above cylindrical surfaces can be irregular and open or closed. When we extend the parabola, hyperbola, and ellipse directly into three dimensions, we simply get parabolic, hyperbolic, and elliptic cylinders. These cylindrical surfaces have the same algebraic expression as the corresponding 2-dimensional conic sections.
Now if we extend the equations into 3-dimensional space by algebraically adding 3rd dimension terms, we get the analogous surfaces of the paraboloid, hyperboloid, and the ellipsoid. The elliptic paraboloid below is given by the equation:
If we simply change the sign of one of the terms above we get the hyperbolic paraboloid below given by:
The hyperboloid has two general forms and one special degenerate form. The first form seen below is called the hyperboloid of one sheet. It is given by:
The special degenerate form of the hyperboloid of one sheet given by:
Is the double-napped cone.
The second form of the hyperboloid is called the hyperboloid of two sheets and is given by:
Finally we have the ellipsoid which is given by:
A special case of the ellipsoid where a=b=c is the sphere.
Click here to see a plethora of different and interesting algebraic surfaces.