Welcome to an introduction to the six basic quadric surfaces.
The goals of this video are to define a quadric surface
and also to give an example of the six basic quadric surfaces.
After this introductory video, we'll take a look at each of the six basic quadric surfaces in more detail.
A quadric surface is given by a degree-two equation in the following form.
However, in my set of videos, we won't be dealing with the rotation of these quadric surfaces
so we won't see equations with an XY term, XZ term, or a YZ term.
The six basic types of quadric surfaces are: the ellipsoid, the hyperboloid in one sheet
the hyperboloid of two sheets, elliptical cone
elliptical paraboloid, and the hyperbolic paraboloid.
The degree and sign of the terms as well as which terms are present
determine which of the six quadric surfaces is given.
When it comes to graphing a quadric surface, it's often helpful to graph the XY, XZ, and YZ traces.
These are the intersections of the surface with the three planes in the XYZ coordinate system.
To determine the XY trace, we'll set Z equal to zero.
To determine the XZ trace, we'll set Y equal to zero.
And to determine the YZ trace, we'll set X equal to zero.
We'll take a look at this in much more detail in the next several videos.
Here is the form of the quadric surface that creates what is called an ellipsoid.
And here are the characteristics:
All three degree-two terms are present.
All three degree-two terms are positive when the equation is equal to positive one.
And all three traces are an ellipse.
Let's take a look at another graph of this ellipsoid.
Here it is again. If we take a look at it looking down on the XYplane
as we see here, we see the shape is an ellipse.
If we take a look at it from the
XZ direction
it's also an ellipse
If we take a look at it from the YZ direction, again it's also an ellipse.
So this is what we're talking about when we refer to the different traces of the quadric surface.
This is the ellipsoid.
Now let's take a look at the hyperboloid of one sheet.
Here's the equation in the most general sense.
Notice all three degree-two terms are present
However, two of the degree-two terms are positive
as we see here for the X squared and Y squared term
but one of the squared terms is negative as we see here
for the Z squared term.
Now, it doesn't always have to be the Z squared term that's negative but two are positive and one is negative.
For a hyperboloid of one sheet
one of the traces will be an ellipse
two of the traces will be hyperbolas
and the axis is parallel to the negative variable.
So here we see a negative Z squared term
and the axis for this is parallel to the Z-axis
Let's take a closer look at this graph as well.
So again one trace is an ellipse as we see in this case looking at the XY plane.
If we take a look at the XZ trace, it would look something like this.
We can see the hyperbola here and here.
If we take a look at the YZ trace, in this case, it's also a hyperbola.
This is a hyperboloid of one sheet.
Let's take a look at the hyperboloid of two sheets.
Notice all three degree-two terms are present
But now, two of the degree-two terms are negative
and only one is positive
when the equation is equal to positive one.
So If we have two negative terms, it's going to be a hyperboloid of two sheets.
And again, we have two traces that are hyperbolas, and one trace that is an ellipse.
Now the elliptical trace may not be in the exact plane, but it will be parallel to the third plane.
And the axis will be parallel to the positive term
meaning, in this case, the axis is parallel to the Z-axis because the Z term is positive.
Let's take a closer look.
Here it is. Notice the XY trace.
We're looking down on the XY plain, we see an ellipse.
Looking at the XZ trace in this direction here
We see the hyperbola, as well as in the YZ trace.
So we see here.
This is the hyperboloid of two sheets.
Next, we have the elliptical cone.
Again, notice all degree-two terms are present
where two of them are positive and one of them is negative, but now
the equation is equal to 0.
We have two traces that are hyperbolas
and one trace that is an ellipse.
And the axis is parallel to the negative variable.
Here's the graph of the elliptical cone.
Again in this case the XY trace is elliptical.
The XZ trace is a hyperbola.
And so is the YZ trace.
This is the elliptical cone.
Next, we have the elliptical paraboloid.
Notice now there are only two degree-two terms and one degree-one term.
Here the Z term is degree-one, but again , it doesn't have to be the Z term
it could be the X or the Y term that's degree-one.
Two traces will be parabolas and one trace will be an ellipse.
And the axis will be parallel to the degree-one variable.
Here's the elliptical paraboloid.
Notice in this case, the XZ trace is a parabola
as well as the YZ trace.
And the XY trace is elliptical
This is the elliptical paraboloid.
And the last one we'll talk about is the hyperbolic paraboloid.
Notice again there are two degree-two terms where now one is positive and one is negative.
And there's one degree-one term.
One trace is a hyperbola
and two traces are parabolas
And the axis is parallel to the degree-one variable.
In this case the XZ trace is the
parabolic trace as well as the
YZ trace. Notice one parabola opened up and one opens down
And the XY trace, as we see here, is hyperbolic.
This is the graph of a hyperbolic paraboloid.
As I mentioned at the beginning of the video, we'll take a look at each of these individualy
in the next six videos. Thank you for watching.