Paraboloids are three-dimensional objects that are used in many science, engineering and architectural applications. In this lesson, we explore the elliptic paraboloid and the hyperbolic paraboloid.
The ‘Oid’ in a Paraboloid
In the word asteroid ‘oid’ means like , put together with ‘aster’, or star, and you get star-like. Similarly a paraboloid is an object resembling a parabola, which will be explained in the next section. In this lesson we explore the two types of paraboloids: the elliptic paraboloid and the hyperbolic paraboloid.
Looking at Elliptic Paraboloids
We have two words here: ellipse and parabola. The ellipse is a circle that has been stretched in one direction. The parabola is the curve that looks like the particular orientation of the capital letter U.
Writing the equation for an ellipse, you see
Unlike a circle which has one radius, an ellipse has two: the ‘a’ and the ‘b’ in the equation. First, we will examine how to go from the equation of the ellipse, to the equation of an elliptic paraboloid. Check the following:
It’s the same equation as the ellipse except the 1 on the right-hand side is now a z. We are in three dimensions. In addition to x and y we have a z. For a moment, look at the left-hand side of the equation. Do you see only a positive result even for negative x and/or y? An option is to allow for the curve to open in the negative z direction. We do this by writing –z on the right-hand side. Here’s the plot of the elliptic paraboloid with the +z. Please note the figure has been cut off at some positive value of z. In reality, the elliptic paraboloid continues to infinity in the z direction.
‘Paraboloid’ means like a parabola. Let’s look at the elliptic paraboloid very carefully. Imagine you are at the arrow point on the x-axis and you are looking towards the origin along the x-axis. The origin is where the three axes cross. Do you see a parabola? Another strategy is to visualize a ‘slice’ of the elliptic paraboloid. The slice is parallel to the y–z plane. The slice cuts the x-axis at some point. Is the following parabola what you see?
Sitting at the arrowhead of the y-axis and looking towards the origin gives us another view. The slice parallel to the x–z plane will pass through the y-axis at some point. From this viewpoint, the x-axis is increasing from right to left, but there’s still another parabola from this view:
Did you notice this second parabola is wider? Does this make sense from the 3D plot of the elliptic paraboloid?
Imagine being in deep space and looking down at the orbit of an asteroid. We would see an ellipse. The same thing happens as we move to the top of the z-axis and look down from overhead. We will see an ellipse in the x–y plane. Positive y is to the right, and positive x is down.
Just like the asteroid which burns as bright as a star when it enters the atmosphere, you are becoming the ‘rock star’ of paraboloids! There is one more paraboloid type to look at, the hyperbolic paraboloid.
Looking at Hyperbolic Paraboloids
The equation for the hyperbola is a lot like that of an ellipse, except there’s a minus sign instead of the plus.
This type of hyperbola will have a right and left opening. To get the hyperbola to open up and down, we reverse the order of x and y. Just as before, to get the equation for the three-dimensional paraboloid, we replace the 1 on the right-hand side with a z. Replacing z with –z is an option although it has the same effect as reversing the order of the x and y terms.
Here’s a plot for the hyperbolic paraboloid, keeping in mind that the plot actually goes to infinity.
Once again we search for parabolas. First, we sight down the x-axis into the y–z plane. A parabola appears:
The other parabola is in the x–z plane when we take a ‘slice’ parallel to this plane and cut the y-axis:
Did you notice how this parabola is in the opposite direction? Does this make sense when you look at the 3D plot of the hyperbolic paraboloid?
The last view is from above. In a plane parallel to the x–y plane we see a hyperbola:
And just like an asteroid, the view of these paraboloids can be pretty spectacular!
Paraboloids are three-dimensional figures resembling parabolas. In this lesson we have explored the two types of paraboloids: the elliptic paraboloid and the hyperbolic paraboloid.