'Tis me, your teacher.
Here are some notes pertaining to quadric sections.
They are partially mine, partially plagiarized (not really, as I've cited them) AND I've also included links to help.
Remember that quadrics are 3-D analogs to conics.
If you're not sure what a "Conic Section" is, go dig out your precalculus text. If you do, you'll find a picture like this one (from www.dsusd.k12.ca.us)
You'll then find a bunch of equations - if 4 is a bunch - that describe each of these:
Now, I have little doubt that you can brush these right out of the old memory banks...
If you're not sure what an equation of a conic section is or how to identify it, use this simple flowchart:
·
Are both variables squared?
No:
It's a parabola.
Yes: Go to the next test....
Yes: Go to the next test....
o Do the squared
terms have opposite signs?
Yes:
It's an hyperbola.
No: Go to the next test....
No: Go to the next test....
§ Are the squared
terms multiplied by the same number?
Yes: It's a circle.
No: It's an ellipse.
No: It's an ellipse.
Now that that's out of the way... in 3-D, we find it more difficult to plot points like you can in 2-D.
Rather, we
determine what a quadric surface is by
looking at the “trace” … that is, setting one variable equal to a constant, and
determining what the surface looks like in the plane parallel to the other two axes. (Set a value for Z, look at the trace in the X-Y plane...)When you put all the traces
together, you have a picture of the surface.
There are six quadric sections you'll need to be familiar with: Ellipsoids, Hyperbolic Paraboloids, Hyperboloids of 1 sheet, Hyperboloids of 2 sheets, Elliptic Paraboloids, and Elliptic Cones
YOU SHOULD KNOW HOW TO:
Let's look at one example:
The equation
Is a quadric section. By comparing with the forms of equations in the table below, you should be able to identify it as a hyperbolic paraboloid. That means that two traces are parabolas, and one is a hyperbola.
If you set x = 0, you'll see that you're left with the equation -y^2/16 = z.
This is the trace of the surface in the YZ plane. It's a parabola which opens in the -z direction.
Likewise, if you set y = 0, you're left with x^2/9 = z. This is the trace of the surface in the XZ plane. It's a parabola which opens in the +z direction.
The final trace is found when Z = 0. Well, when Z = 0, you get a pair of lines which intersect at the origin
(y = +/- 3/4x). If is greater than or equal to 1 you start to generate hyperbolas. When z is a negative number, there is no trace!
Here's a picture of the conic section traces for the three planes where x, y and z are each set to zero (top row) and then the fourth, where z = 1.
So, now that we know what the traces look like, we can imagine what something looks like that has two parabolas and a hyperbola. It looks like this:
YOU SHOULD KNOW HOW TO:
1.Recognize
the equation of each type of quadric.
2.Describe
the trace of each quadric in relevant planes. (That means sketch - not with numbers, just shape and orientation)
Let's look at one example:
The equation
Is a quadric section. By comparing with the forms of equations in the table below, you should be able to identify it as a hyperbolic paraboloid. That means that two traces are parabolas, and one is a hyperbola.
If you set x = 0, you'll see that you're left with the equation -y^2/16 = z.
This is the trace of the surface in the YZ plane. It's a parabola which opens in the -z direction.
Likewise, if you set y = 0, you're left with x^2/9 = z. This is the trace of the surface in the XZ plane. It's a parabola which opens in the +z direction.
The final trace is found when Z = 0. Well, when Z = 0, you get a pair of lines which intersect at the origin
(y = +/- 3/4x). If is greater than or equal to 1 you start to generate hyperbolas. When z is a negative number, there is no trace!
Here's a picture of the conic section traces for the three planes where x, y and z are each set to zero (top row) and then the fourth, where z = 1.
So, now that we know what the traces look like, we can imagine what something looks like that has two parabolas and a hyperbola. It looks like this:
Ta Da!!! See - easy.
All this information, as well as several examples of how to do just that are provided in these fabulous videos listed below - which are all on You Tube. The first is a brief introduction, while the specific ones go through lots of detailed examples.
Woohoo!!!
Introduction to Quadric Surfaces
Here's a table (from your text) summarizing the parent functions as well as the major characteristics of each(pardon the watermark - I downloaded a "restricted" copy from somewhere... legal for my educational purposes, as we all have the text!
Woohoo!!!
Introduction to Quadric Surfaces
Another helpful resource - the Interactive Gallery of Quadric Sections
(You do not have to graph them in three
dimensions, but you can certainly try!)
Here's a table (from your text) summarizing the parent functions as well as the major characteristics of each(pardon the watermark - I downloaded a "restricted" copy from somewhere... legal for my educational purposes, as we all have the text!
The following is a link to an Interactive Gallery, where you can play with the parameters of each. It also gives you great pictures!
(Ain't technology grand!)
Remember - this section is nothing fancy... just keep basic traces in mind.
Have fun!
Lisa
Have fun!
Lisa
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