Welcome to a video on graphing
the hyperboloid of one sheet.
To graph a quadric surface,
it is often helpful to look at the xy, xz and yz traces.
This is where the surface intersects these three planes.
To determine the xy-trace, we'll set z to zero.
To determine the xz-trace, we'll set y equal to zero.
And then last, to determine the yz-trace,
we'll set x equal to zero.
Once we have these traces,
we should be able to make a nice graph of the surface
but in this video we'll take a look
at the surface using Maple.
Remember the terms present in the equation
of a quadric surface tells us
what type of quadric surface we have.
For a hyperboloid, notice that
all three degree two terms are present
but when the equation is equal to one,
two of the terms are positive
and one of the terms is negative.
When we take a look at the traces of a hyperboloid,
one trace will be an ellipse
and then lastly, the axis of the hyperboloid
will be parallel to the axis of the negative variable.
Notice the z term is negative
and the axis is a vertical line.
Let's take a look at an example.
Here we should notice right away
that this does fit the form for a hyperboloid.
Now let's go and take a look at the traces.
To determine the xy-trace, we'll set z equal to zero
so we'll have x squared divided by 16.
Plus y squared divided by 25
equals one.
And this should remind us of an ellipse
and notice that the larger denominator is under the y term,
so the major axis will be vertical.
And this ellipse is also centered at the origin
so a squared is equal to 16,
b squared is equal to 25.
So since b is five, we'll go up five units on the y-axis
and down five units on the y-axis.
Notice this is scaled by twos.
This will be the endpoints of the major axis.
A squared is equal to 16,
so a is equal to four,
so I'll go four units to the right
and four units to the left.
And the ellipse passes through these four points
so will look something like this.
This would be the xy-trace.
Now let's take a look at the xz-trace.
We'll set y equal to zero,
that will give us x squared over 16,
minus z squared over nine, equals one.
Well, we know from our study of conic sections
this is a hyperbola.
What we're going to do is construct the rectangle,
sketch the asymptotes and then sketch the hyperbola.
Since the positive part is the x part,
this hyperbola will open left and right along the x-axis.
It's centered at the origin
and since a squared is equal to 16,
we'll move four units to the right
and four units to the left.
And since c squared is equal to nine,
c would be three, so I'll plot a point
up three units and down three units.
We'll use these four points to sketch a rectangle.
And then we'll sketch the diagonals of the rectangle.
This will be the asymptotes.
And again, we know because the positive part is the x part,
this hyperbola will open left and right
so here's one vertex opening to the right.
Here's the other vertex opening to the left.
This would be the xz-trace.
And now for the yz-trace,
we're going to set x equal to zero
so we'll have y squared over 25
minus z squared over nine,
equals one.
Again, we have another hyperbola
since now the y-axis is horizontal.
This hyperbola will open left and right
and since b squared is equal to 25, b is five,
so we'll plot a point five units to the right
and five units to the left.
Again this is scaled by twos
and c squared is nine so c is three,
so three units up and three units down.
Here's our rectangle.
Sketch the diagonals.
And now we can sketch the hyperbola,
here and here.
So if we slice this surface along the xy, xz and yz planes,
the intersection of the plane and the surface
would look like this.
We call these the traces.
And if we put these traces together to form a surface,
it would be a hyperboloid of one sheet as we see here.
Let's go and take a look at a dynamic graph
so that we can verify our traces.
Again, if we take a look at just the xy-plane,
we can see the elliptical trace here.
If we take a look at just the xz-plane,
it would look like this.
Again, you can see the hyperbola opening left and right.
And then if we take a look at the yz-plane,
here, again you can see the hyperbola
opening left and right along the y-axis this time.
This is the hyperboloid of one sheet.
I hope you found this example helpful.
Thank you for watching.