Conic Sections – Computer Lab 1

(This is not a race. Take your time and learn from this lab.)

Go to the following website:

On the 3D Interactive Conics Graph:

1. Set y = 0, x = 0, and rotation angle = 0 ( the angle of the plane relative to the x-y axies.)

Look at the intersection from the side (so the plane looks like a line), and from the top (like you’re looking in the top of the cone), and the bottom, tilting the cone a little and practice rotating the plane. Try changing x to x=100 and x=-100.

2. Move y to y = -50 and then to y = + 50. (If it appears to be the wrong direction, flip the cone over.) Now look at the figure from all directions (sideways, top, bottom, a bit tilted). Notice from the top the intersection looks like a ______, but from the side like a line segment, and from halfway tilted it looks like a tilted ______(which seems to look like a/an ______. But we know it’s really a ______and that it only looks different ways because of the angle from which we’re looking at it. Video games, etc, need to concern themselves with the angles things are viewed from (projections) to give the appearance of 3 dimensions.

3. Move y to y = 30 and the angle to 30 degrees. Look from the side, the top, rotating the plane around. What figure do you think this is? ______. Is there any way you can move the plane/cone, so it looks like a circle? ______. (i.e., is there a projection where it looks like a circle?). Change the rotation angle of the plane to a lot of different settings between –33 degrees and 33 degrees rotating the figure and looking at it from all angles and pulling the plane from side to side as well.

4. Try y = 40 and x = 0 and try various rotations of the angle of the plane. What angle seems to be the same angle as the side of the cone? ______. It’s only that way in this demonstration, but it’s a common choice. Look at the figure from the side, top, bottom, and various tilts of the cone and plane. What is the name of the figure this intersection makes? ______. Try y at various values. Does the figure seem to reach a limit at some value? What is the figure? ______, and what is that value of y at the limit? ______

5. Now try y = 40 and making the angle of rotation of the plane even larger. What figure does the intersection make? ______. Look at it from all directions and twists right and left. Try making the angle even larger and smaller (but not below the angle of the side of the plane). See how it changes the shape of the figure. Now try various values of y. Does the figure seem to reach a limit at some value? What is the figure? ______, and what is that value of y at the limit? ______

Now let’s learn A LOT about analytic geometry. Go to

1. Above you used #6.c. on this page to visualize all of the conic sections.

Explore the slope of the sine curve:

Read this entire article:

At the end of the article is an interactive tool to see the slope of the (tangent to) sine curve at any point, x. The slope on the sine curve at point A is plotted on the same graph as point B. As you drag point A along the sine curve (changing x), a plot is formed of the slopes to the sine curve (the points B) for each value of x. What graph does the plot of the slopes to the sine curve seem to make? ______. Later, in calculus, you will prove analytically the same thing that you just saw in this demonstration.