Quadratic curvature for algebraic curves (cont) | Differential Geometry 15 | NJ Wildberger

We continue in developing fundamental formulas that deal with curvature for surfaces in three dimensional space, given by algebraic equations. Our approach continues to be that the normal paraboloid to such a surface at a point is the key object that encodes the quadratic metrical information, including curvatures. However we want formulas that deal with the general situation, not only the simpler case when the tangent plane is horizontal.

[It appears this video has been unlisted lately, even though it is some years old! Better late than never to get it back out there ....]

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[bernardoxbm]

Answer for the problem 23:00

The centers of the circle in different cases are

For Blue geometry
C_b = -(l^2 + m^2)/(2 (n m^2 - 2 p l m +q l^2)) [l, m]

For Red geometry
C_r = (l^2 - m^2)/(2 (n m^2 - 2 p l m +q l^2)) [m, l]

For Green geometry
C_g = (2 l m)/(2 (n m^2 - 2 p l m +q l^2)) [l, -m]

[ericbischoff]

small detail: at 16:10 it's W', not W

[lixinfan]

As fro the approximation (Y) of the curve P,  how does Y differ from the parabola Z since they both approximate the curve P?  

[lixinfan]

Algorithm starting from 14:00 is nice. 

[brendawilliams]

I enjoy math and don’t use a computer. Professor wildberger. Thankyou.

[postbodzapism]

So have you tried out yourself at that research problem yourself, Professor? Or that we really can publish a research paper based on that topic?