Lesson plan
Focal surface
(white) z=x^2-y^2,\; 0\le x,y\le 0.5 (blue). At the center point of the monkey saddle the Gauss curvature is 0, otherwise negative. For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines. As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each normal direction to the surface. At points where the Gaussian curvature is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature. If \vec p is a point of the given surface, \vec n the unit normal and k_1,k_2 the principal curvatures at \vec p , then : \vec b_1(\vec p)=\vec p+\frac{\vec n}{k_1} \quad and \quad \vec b_2(\vec p)=\vec p+\frac{\vec n}{k_2}\ are the corresponding two points of the focal surface.
Article summary
(white) z=x^2-y^2,\; 0\le x,y\le 0.5 (blue). At the center point of the monkey saddle the Gauss curvature is 0, otherwise negative. For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines. As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each normal direction to the surface. At points where the Gaussian curvature is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature. If \vec p is a point of the given surface, \vec n the unit normal and k_1,k_2 the principal curvatures at \vec p , then : \vec b_1(\vec p)=\vec p+\frac{\vec n}{k_1} \quad and \quad \vec b_2(\vec p)=\vec p+\frac{\vec n}{k_2}\ are the corresponding two points of the focal surface.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.