SUNY Geneseo Department of Mathematics
Tuesday, January 31
Math 223
Spring 2023
Prof. Doug Baldwin
Problem 3 on problem set 1 involves generalizing the distance formula to 4 dimensions.
Some of you are starting to make appointments with me (thank you!), so I demonstrated how to do it through Google Calendar.
If you missed or want to review the demonstration, this video from CIT is old, but still helpful for the main points about using Google Calendar to schedule meetings:
The first math colloquium of the semester is this Thursday (February 2)!
Prof. Gary Towsley (SUNY Geneseo department of mathematics, emeritus).
“The Langlands Program, Then and Now”
3:00 PM, Newton 201.
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
Traces are cross-sections through a surface parallel to a coordinate plane. They can give you a way to see 2-dimensional structure in a 3-dimensional surface.
The relations between various quadric formulas and the graphs they produce are interesting, but not always intuitive.
What are the connections between quadric surfaces and conic sections, and how do traces fit into that?
The “conic sections” are curves that you can produce by slicing a cone (or a pair of cones, but mathematicians actually think of the pair as “a cone”) with a plane. There are 3 conic sections you can produce this way, namely
Traces of quadric surfaces are usually (but not absolutely always) conic sections.
You can see traces of some quadrics by plotting them in Mathematica, but with one dimension cut off where you want the trace. You can download this notebook to see and experiment with some examples.
One way to understand where some of these shapes come from is to think about the equations you get if you move one of the variables to the other side of the standard equation. For example, for a hyperboloid of 1 sheet…
After moving the originally negative z term, you have an equation for a circle, whose radius is always positive, but increases as z gets further from 0. Thus a shape that has circular cross-sections, and that flares outward as it moves away from the origin in the z direction.
Continue the discussion of quadrics, cylinders, and things you can do with them through Mathematica.
No new reading, but review section 1.6 if you want.