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Quadric Surfaces

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.

Parabolic Cylinder

Hyperbolic Cylinder

Elliptic Cylinder

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.

The Hyperbola

Ready to take a look at the third conic section? (The first two we looked at were the parabola and the ellipse). The hyperbola is created when the plane passes through both nappes of the right circular conical surface at an angle with the surface’s longitudinal axis that is less than the angle made with the axis by the generating straight line on the surface.

The hyperbola is found in many natural phenomena including an open orbit of celestial bodies or simply the curvature of light around a large gravitational body like a star. It is also the path of subatomic particles repulsed by the nucleus. In more everyday terms, the tip of a shadow cast by the sun traces out a hyperbola on the ground as the day progresses. The mathematics of hyperbolas is also used in navigation and global positioning systems.

The definition of a hyperbola that we will use to develop an algebraic expression is similar to that of the ellipse. A hyperbola is all the points in a plane the difference of whose distances from the two foci is a (positive) constant. The ellipse was simply the sum instead of the difference.

We begin with the hyperbola centered on the origin in a Cartesian-coordinate plane. As with the ellipse we use the foci, (c,0) and (-c,0). The distance from each focus to a general point (x,y) is given by the following two expressions using the two-dimensional distance formula:

Now using the above definition we will set the difference between these two to a constant number, 2a, just as we did with the ellipse. One important thing to note is since as we move about the plane the difference between these may change from negative to positive or positive to negative depending on which distance is longer. We will use the absolute value to insure we get the magnitude of the difference in distance.

As with the ellipse, using some elementary algebra we arrive at:

Now “a” represents the x-intercepts or the vertices. (Let y=0 and multiply both sides by “a” squared). With the ellipse, “a” was always greater than “c” so we factored out a negative 1 from the denominator of the second term and changed the sign between the two terms to a positive. Then we used the substitution:

In the present case, “a” will always be less than “c” so we use the substitution:

This yields the final expression:

Notice here that if we try to locate the y-intercepts by setting x=0, we get imaginary results. (Square root of negative “b” squared). Therefore this is the expression for the hyperbola with vertices on the x-axis. The expression for the hyperbola with vertices on the y-axis is:

Just as with the parabola we can use the same substitutions for translation and axes and rotation of axes to develop expressions for more general hyperbolas.

The axis that runs through the vertices is called the transverse axis. The axis that is orthogonal to the transverse axis is the conjugate axis. A hyperbola is often drawn by constructing a rectangle that is 2a by 2b in dimension with the vertices on the 2a length sides. Then diagonals are drawn through the rectangle and used as asymptotes for the hyperbolic curve. The equation for these asymptotes can be derived by solving the main equations for y.