Wilson Williams - Rhea

                                              Quadric Surfaces 

1. Here is the general equation of an ellipsoid

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

An ellipse is a generalization of a circle, and ellipsoid is a generalization of a sphere. Planet earth is not a true sphere. It is an ellipsoid. You will know if it is an ellipsoid if it is wider than it is taller.

2. Here is the general equation of a cone

x^2/a^2 +y^2/b^2 = z^2/c^2

A cone is a shape that tapers smoothly from a base to a point called the apex or vertex.

3. Here is the general equation of a cylinder

x^/a^2+y^b^2= 1

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment.

4. Here is the equation of a hyperboloid of one sheet

x^2/a^2+y^2/b^2-z^2/c^2=1

The variable with the negative in front of it will give the axis along which the graph is centered.

5. Here is the equation of a hyperboloid on two sheets

-x^2/a^2-y^2/b^2 + z^2/c^2 = 1

The only difference between the hyperboloid of one sheet and the hyperboloid of two sheets is the signs in front of the variables. They are exactly the opposite.

6. Here is the equation of a elliptic paraboloid

x^2/a^2 +y^2/b^2 = z/c

As with cylinders this has a cross section of an ellipse and if it will have a cross section of a circle. When we deal with these we’ll generally be dealing with the kind that have a circle for a cross section.

7. Here is the equation of a Hyperboloic Paraboloid

x^2/a^2 -y^2/b^2 = z/c

These graphs are vaguely saddle shaped and as with the elliptic paraoloid the sign of c will determine the direction in which the surface “opens up”. The Quadric Surfaces are sometimes not l

Quadric surfaces also can be interesting and are helpful sometimes when solving problems. Perspectivly speaking lets not look at quadric surfaces for solving problems, lets look at quadric surfaces in the light of exploring the concept and the euclidean space. A euclidean space is a 3 dimensional plane that has (x,y,z) coordinates. While on the other hand there is the Cartesian coordinate system where there is only a (x,y) axis.

We use quadric surfaces to represent real objects. Intuitively speaking one way to think of the euclidean plane is as a set of points satisfying certain relationships. For example, there are two operations. One is translation, shifting of the plane so that every point is shifted in the same direction and same distance. The other is rotation about a fixed point in the plane, in which every point in th plane turns about that fixed point through the same angle. This example is representable int autodesk. I created the shapes in autodesk. It all depends on which way is best for you to learn the concept. In reality, One way to think of it is like an open space with 3 cordinates. But if we really think of it in reality it would be an infinite amount of coordinates. So we will just create a coordinate system with 3 coordinates that go 90 degrees around a point. In essence the point must be 3 dimensional as well, because we cannot create a space that forms around the sphere. So the space is already there, we are just using coordinates for our recording. So technically a 2 dimensional space cannot be real in reality, while a 3 dimensional space cannot be real on paper. So it is very hard to relate the euclidean coordinates to reality of the infinite amount of coordinates that there are really there. Nontheless it would be a general discription of the points in a 3 dimensional space. Broadly speaking the world is infinite and our feeble minds cannot comprehend so many coordinates so we use what we can comprehend which is a 3 coordinate system.

Wilson Williams