SUNY Geneseo Department of Mathematics
Wednesday, February 1
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Applications for Geneseo scholarships are open from now to March 1 at https://geneseo.academicworks.com
It’s one (supposedly) easy general application, based on which the scholarship folks will try to match as many students to as much money as they can.
One of the other things I do besides math is involvement in rolling out Geneseo’s new general education program.
As part of that, I’m interested in how students view general education, so that we can hopefully find ways to explain and advise for the new program that make sense to students.
You can help, by filling out the survey online.
It should be quick (under 10 minutes), and anonymous, yet very helpful.
Based on section 1.6 of the textbook.
We have a new notion of “cylinder”: a shape produced by drawing straight lines through any curve, not just through a circle.
We also have a general equation for the whole family of “quadric surfaces”: Ax2 + By2 + Cz2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
And we’ve seen lots of extensions to what you know about plotting with Mathematica…
ContourPlot3D for plotting implicit surfaces and Plot3D for plotting explicit ones.
ContourPlot3D has 4 arguments, namely the equation for the surface and the ranges of x, y, and z values to plot overPlot3D has 3 arguments, the formula for calculating z from x and y, and the ranges of x and y values to plot overAxesLabel option, which lets you label the axes with short bits of text.What are rulings? Thinking about a cylinder as any surface you get by drawing a series of parallel lines through some curve, the rulings are the parallel lines. For example, if you draw a curve on a whiteboard and visualize a cylinder by curling a piece of notebook paper along that curve, the lines on the paper are rulings (as long as you hold the paper so the lines run towards the board, not parallel to it).
The focal point example? The example is motivated by the fact that if you think of a parabola as reflective, rays of light parallel to the axis but arriving anywhere on the parabola will all reflect towards its focus, which is a point on the axis. Furthermore, equations for parabolas can be put into a form x2 = 4py, in which p is a number that is the distance from the vertex (i.e., tip) of the parabola to the focus. The answer to the question starts by turning the problem into a 2-dimensional one, arguing that since the parabola has a circular cross-section (i.e., circular traces perpendicular to the z axis), you can find the focus by just finding the focus of, say, the trace in the xz plane. Then they use the x2 = 4py (or 4pz, since that’s the actual variable in the question) to solve for p.
An even better way of seeing traces in Mathematica occurred to me after class yesterday, using Mathematica’s ability to plot several surfaces at once.
Namely, plot the surface you want a trace of, cutting off one variable’s range at the plane of the trace, like we did yesterday, but also plot the plane.
To plot multiple surfaces, list all the equations you want plotted inside curly braces in a ContourPlot3D command.
You can download a notebook from Canvas to see this idea in action.
For the 2 paraboloids, z is an “explicit” function of x and y, i.e., the surface is defined by an equation that you can plug x and y values into and get a z value out. You can plot such surfaces in Mathematica using the Plot3D function, which is a bit simpler to use than ContourPlot3D:
Plot3D[ function, {x, lowX, highX}, {y, lowY, highY} ]
For an example, see the notebook with today’s other Mathematica examples
Is it possible to have more than 1 cylinder passing through the same curve?
Yes, because there’s no requirement for the rulings to have any particular relationship to the curve, they just have to be parallel to each other. Rulings that hit the curve at different angles will give rise to different cylinders:
Our book gives the general form for an ellipsoid as x2/a2 + y2/b2 + z2/c2 = 1. Is 7x2 + 28y2 + 14z2 = 7 a valid equation for an ellipsoid? Plot it with Mathematica to check your thinking.
Even though the coefficients don’t seem to be in the right places, the equation is an ellipsoid, as you can see by rewriting to have coefficients in the expected form, and by seeing that it does indeed look ellipsoidal when plotted.
The general equation for a hyperboloid of 1 sheet is x2/a2 + y2/b2 - z2/c2 = 1. What governs the width of narrowest part of such a hyperboloid? Test your ideas by plotting hyperboloids with Mathematica.
The a and b coefficients determine the width, in particular larger values lead to thicker hyperboloids. Intuitively, larger a and b mean that x and y have to be bigger in order to still produce 1 when divided by a and b and having z2/c2 subtracted. Slightly more formally, you can also realize that the narrowest section is where z = 0, and recognize the trace of that narrow section is a circle, if a and b are equal:
In the equation for a circle, the divisor of x2 and y2 is the square of the radius, so larger values of a and b correspond to larger circles.
The notebook of Mathematica examples for today ends with some examples of this.
So-called “rectangular” xyz coordinates aren’t the only ways to describe and plot things in 3 dimensions.
Friday I want to look at 2 often-helpful alternatives: cylindrical and spherical coordinates.
Please read section 1.7 in the textbook to get a start on them.