NYU MATH-UA 123.005 Calculus III Spring 2012

This Demonstration shows the intersections of planes perpendicular to the axes with plots of quadratic forms, which are paraboloids, hyperbolic paraboloids, ellipsoids, cones, or hyperboloids of one or two sheets.

It's centered on Somerville, MA, but you can enter latitude and longitude or a ZIP code to focus on any point in the US. The curved lines represent points with the same elevation. Lines close together mean a steep slope (either upward or downward depending on the actual elevations). Concentric curves mean a peak or valley. What indicates a saddle point?

The graph of the equation ax^2 + by^2 + cz^2 + dxy + exz + fyz +gx + hj + i z + j=0 is shown for your settings of the parameters.

Set d=e=f=0 to keep things centered at the origin. To get the paraboloids you want one of (g,h,i) to be nonzero. To get the hyperboloids set g=h=i=0 and let a, b, c be mixed signs. To get the ellipsoids make sure a, b, and c are all negative. See if you can find a setting where a single parameter changes the surface from a hyperboloid of two sheets, to a cone, to a hyperboloid of one sheet.

In honor of the helix

Curvature and Torsion Choose among several curves and see the rotation of the Frenet-Serret frame as you move the slider. From this you can perceive the curvature and torsion of the curve. Associated objects (such as the circle of curvature, evolute, and osculating sphere, as well as two views of the Frenet frames) may be displayed.

I was wrong in class when I said the sexy d was used by Leibniz. It goes back only to the late 18th century. Also, Wikipedia does not list "sexy d" as a known pronunciation of the symbol.