Quadric Surfaces in 3 dimensional Space
Student project for MA265
by Wilson Williams
12/16/2011
To start simply, let’s define what a quadric surface is. First we can define quadric. Quad means 4. And we obviously know what a surface is. It can be considered a 2 dimensional space that is not necessarily flat. So by using the words we can define what a quadric surface is. A quadric surface a 4 dimensional surface. Now technically speaking, this is incorrect by philosophers, and mathematicians of past. But what way can we best define a word that will help us best understand the meaning of it. What also relates to the word Quadric Surface is the word dimension. What is a dimension? Broadly speaking, a dimension is an open space. Now how can you describe something as an open space? A dimension can be used to describe the infinite space. For example, in a 2 dimensional space you are only describing the 2 dimensions. Not to include that there are not the rest of the infinite dimensions, but that is as far as you are going to describe a particular space. Ok, it may be easier to describe what a dimension is not. A dimension is not, not a space at all. Meaning it has to be a space. Now speaking on reality with a 2 dimensional space, you cannot see 2 dimensional spaces in reality, because it is flat. On the other hand, in a 3 dimensional space you cannot see an object in 2 dimensions. Or else, you will not see the whole object. Prospectively speaking a 2 dimensional space is not real. And a 3 dimensional space is not fake. You cannot view a 2 dimensional space in reality. When I say reality I mean in the physical. You cannot touch a 2 dimensional object. A surface is considered something flat. Unless you put a number in front of it. In that sense, meaning an object with 4 surfaces. But when we use a quadric surface we only are describing it in on a 2 dimensional space (paper, computer). Technically speaking there is an infinite amount of surfaces that are in a quadric surface but just so we can define and visually look at the object we only use quadric surface. Quadric surfaces can be closely related to physics, which is usually seeking the undefined. Mathematically speaking quadric surfaces are used to define usually a problem of some sort. So in essence it is like using physics concepts to apply mathematics as well as mathematics to apply physics concepts? Graphs are used to describe mathematical equations. In this case we are describing a mathematical equation in quadric surfaces. Here is the general equation for quadric surfaces.
Ax^2+By^2+CZ^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0
Ellipsoid – x^2/a^2+y^2/b^2+z^2/c^2=1
Hyperboloid of one sheet – x^2/a^2+y^2/b^2-z^2/c^2=1
Hyperboloid of 2 sheets – x^2/a^2+y^2/b^2-z^2/c^2=-1
Elliptic parabolic – x^2/a^2+y^2/b^2=z/c
Hyperbolic parabolic x^2/a^2-y^2/b^2=z/c
There is also a website that you can go to get a view of what quadric spaces look like. http://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx
Here is table showing the relationship between physics and mathematics. Also how the graphs relate to one another conceptually speaking about the 2 topics.
Another Great tool to learn about Quadrics is to look up on the search engine Rhea page and type in the topic on Assimi search engine.
It is also helpful thttp://www.youtube.com/watch?v=uTUB_a-tyWI&feature=results_main&playnext=1&list=PLA1398719815F6BB0o look at these YouTube videos.
A fun activity and learning exercise is to go onto to Autodesk and create the quadric shapes. This is a great exercise for visual learners. Tip- If you would like to get a better understanding on the concept as a whole you can look up several things. One would be the idea of relativity and space in general. This topic can go on and on but relativity is one of the topics that can fit into this discussion
There are hundreds of ways to learn about Quadrics in 3D space. The most important thing is you should find which way is best for you to learn. If you are a more visual person, I would recommend taking a look at the graphs. If you are a more organized and structure learner, then I would recommend looking at tables and charts. If you are a conceptual learner, I would recommend reading the Essay and learning the philosophical aspect of it. Overall looking at all of the different ways to learn about quadrics is important, because it will make you a better-rounded learner.
