Math 121 Lab 1, Part 2: 

Parametric plots of ellipses and hyperbolas

© Copyright by T.I. Lakoba,
Last modified  2022-09-01.

NOTE:  Do Part 1 before starting this Part. Answers to all Mathematica-related questions that you may have about the Exercises below can be found in Part 1.

Clear["`*"]

Exercise 1

Before starting this exercise, review the sketch that I asked you to make when reading about the canonical equation of an ellipse in Part 1. Recall that you use the canonical equation to determine the semi-axes of an ellipse. (If you have difficulty with understanding the section about ellipses in Part 1, you may also read pp. A19--A22 in Appendix C of your textbook.) However, to plot an ellipse, you need to use its parametric equations, covered in Part 1.

Below you will be asked to plot several ellipses. Before plotting each ellipse, sketch it in 3D on a piece of paper and write down its parametric equations. Add the sketch of each new ellipse to the existing sketch.

(a)   Use ParametricPlot3D to plot an ellipse, E1, whose center lies on the z-axis, whose semi-axes along the x- and y-axes equal 2 and 3 units, respectively, and which lies in the plane  z = 6. Assign a name to this plot for future use.

(b)   Repeat part (a) with one change: the new ellipse, E2, must now lie in the plane  z = -4. Assign a different name to this plot.

(c)   Plot four more ellipses, E3, E4, E5, E6, with the properties listed below. Remember to add each new ellipse to the existing combined plot of ellipses that you have constructed so far. (This is intended to give you immediate feedback of whether you are on the right track.)
 
- Ellipse E3 touches ellipses E1, E2 at points (2,0,6) and (2,0,-4) and lies in a plane parallel to the  yz-plane. 
Here are some suggestions that you may find helpful. In your sketch on paper, first mark the above two points, and then sketch E3. When writing the parametric equations of E3, you will be able to determine one of its semi-axes from your sketch so far, but not the other. Do not worry - use any number as a placeholder for that yet-undetermined semi-axis. When sketches of all the ellipses, E1 through E6, are in place, you will be able to determine that number. Follow the same suggestions when plotting E4, E5, E6.

- Ellipse E4 touches ellipses E1, E2 at points (-2,0,6) and (-2,0,-4) and lies in a plane parallel to the yz-plane. 
- Ellipse E5 touches ellipses E1, E2 at points (0,3,6) and (0,3,-4) and lies in a plane parallel to the  xz-plane. 
- Ellipse E6 touches ellipses E1, E2 at points (0,-3,6) and (0,-3,-4) and lies in a plane parallel to the  xz-plane. 
- Each of ellipses E3 and E4 also touches both of ellipses E5 and E6.

- Ellipses E3, E4 must be plotted in red; ellipses E5, E6 must be plotted in green.

When you are through sketching all the ellipses, you should see that they are arranged in a neat way on faces of a rectangular box.

 Assign different names to each of these plots.

(d)   Combine the plots of all six ellipses in the same window. 
Again: combine the existing plots, do not make a new ParametricPlot3D. A proper command for combining existing plots is found in Part 1. (You may need to set  PlotRange->All .)

Finally, examine the plot to make sure that all the conditions specified above are satisfied. If any of the conditions fails to hold, make a note discussing where you think you could have gone wrong.

Exercise 2

(a)   Before starting this exercise, review your notes on the transformation from Equation (H-2) to Equation (H-3) in Part 1 of this Lab. You will not be able to succeed in this Exercise if you have not done all of the “Practice on your own” questions about hyperbolas in Part 1. So, if that is the case, go back there and review those questions.    
Also, make sure to use the correct parametric equations of hyperbolas and not the wrong ones, as emphasized in Part 1. 

Consider an equation of a hyperbola 

     ((x^2)/(2^2)) -  ((y^2)/(3^2))  = k,
                                              
where  k  is a positive constant. Let   k =1, 2, 5,  and for each of these values make a parametric plot (in 2D) of both branches of the corresponding hyperbola.  For the parameter t use different ranges for different values of   k :
for  k=1  use the range {-3,3};
for  k=2  use the range {-2.5,2.5};
for  k=5  use the range {-2,2}.
Make all three plots to have different color (of your choice).

Assign a separate name for each plot.
I strongly recommend that you use different names for these plots than for the plots in Exercise 1. The reason is that if you decide to work intermittently between the two Exercises, the plots from different exercises would not get mixed up.

Combine the plots. (See the clarification about “combine” in the instructions for Ex. 1(d).  Again, you may need to set  PlotRange->All .)

Write a conclusion about how the value of   k  affects the plot of the hyperbola. Namely, what changes, and what seems to remain (almost) unchanged?

(b)    Repeat part (a) for the case when   k  is negative, taking    k =-1, -2, -5 . If you are lost, re-read the note in red font at the beginning of part (a).

What changes do you see in the plot (with all hyperbolas combined)  compared to such a plot in part (a)? 
Note:   Unlike in part (a), you are now asked to compare your observations with those in part (a) and not with each other for different values of  k .

Delete your output

In the main menu bar, go to “Cell” and select “Delete all output”.

The end