Quadric Surfaces

Quadric surfaces are defined as functions of the form: \[ Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 \] There are many surfaces that can be created from this equation. There are many factors that can affect the shape of the created surface. The signs of the coefficients can result in completely different shapes and the magnitude of these coefficients stretch and shrink the surface. One example of these quadric surfaces is a hyperboloid of one sheet. This shape visually appears as a hyperbola that has been rotated around an axis. This equation can be represented in two main ways: implicitly as a function of x, y, and z, and as a parameterization. For this case, the parameterization will be used.

Paratmerizations of the Surface

The parameterization of this surface is a fun one: \[ f(s,t) = ( (e^{.2s} + e^{-.2s})\cos{30t}, .5(e^{.2s} + e^{-.2s})\sin{30t}, .5(e^{.2s} + e^{-.2s}) \] Euler's constant is a result of the use of the hyperbolic trig functions used to parameterize the original function. In terms of hyperbolic trig functions the parameterization can be represented as: \[ f(s,t) = (2\cosh(.2s)\cos(30t), \cosh(.2s)\sin(30t), \sinh(.2s)) \] At this point, we have a parameterization that gives the quadric surface:

This hyperboloid of one sheet is a single piece that opens up on the z-axis. The surface is twice as long along the x-axis as compared to the y-axis.

Taking a Slice

For any quadric surface, a plane, or some other line or curve, can be uesd to cut through the surface. The result is a slice of the 3d shape. This slice is interesting as it the shape of this slice can vary greatly throughout the same quadric surface. Let's cut this hyperboloid into two different pieces using a plane. The plane that will be used to slice is: \[ x = z \] This 45 degree plane will take out a nice hyperbola that contains componenets of the hyperboloid on both the negative and positive sides of every axis. This hyperbola will also have symmetry about the x-axis. Here is a picture of the hyperboloid with the slice shown:

The Resulting Slice

The slice mentioned above was created by setting x equal to z. This will most easily be done by formulating the implicit equation of the surface. From the parameterization, it can be seen that the radius in the x direction is 2, and the radius in the y direction is 1. It can also be determined that the coefficient for the z value is also 1, since there are no constants in the parameterization. This means that the formula in terms of x, y, and z is: \[ \frac{x^2}{4} + y^2 - z^2 =1 \] Setting z = x gives: \[ \frac{x^2}{4} + y^2 - x^2 = -3\frac{x^2}{4} + y^2 = 1 \] Which is the equation of a hyperbola. Hyperboloids are interesting as they can generate many different shapes depending on the hyperboloid itself and the position of the slice from the hyperboloid. Under the right conditions, the possible slices of an ellipse include circles, ellipses, parabolas, and hyperbolas. This box of chocolates makes this specific surfaces quite appealing for taking a slice. I wanted to take out a hyperbola from this surfaces as it is the most interesting. This slice takes the a piece from either side of the x-axis to generate a symmetrical section. This piece also has the property that when rotated around the z-axis in an elliptical path, it recreates the original surface. This is true for any hyperobla with symmetry about the x-axis.