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normal section
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(Definition)
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Normal sections
Let $P$ be a point of a surface
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(1) |
where $F$ has the continuous first and second order partial derivatives in a neighbourhood of $P$ . If one intersects the surface with a plane containing the surface normal at $P$ , the intersection curve is called a normal
section.
Normal curvatures
When the direction of the intersecting plane is varied, one gets different normal sections, and their curvatures at $P$ , the so-called normal curvatures, vary having a minimum value $\varkappa_1$ and a maximum value $\varkappa_2$ . The arithmetic mean of $\varkappa_1$ and $\varkappa_2$ is called the mean curvature of the surface at $P$ .
By the suppositions on the function $F$ , examining the normal curvatures can without loss of generality be reduced to the following: Examine the curvature of the normal sections through the origin, the surface given in the form
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(2) |
where $z(x,\,y)$ has the continuous first and second order partial derivatives in a neighbourhood of the origin and $$z(0,\,0) = z'_x(0,\,0) = z'_y(0,\,0) = 0.$$ Indeed, one can take a new rectangular coordinate system with $P$ the new origin and the normal at $P$ the new $z$ -axis; then the new $xy$ -plane coincides with the tangent plane of the surface (1) at $P$ . The equation (1) defines the function of (2).
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"normal section" is owned by pahio.
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Cross-references: equation, tangent plane, normal, rectangular coordinate, origin, without loss of generality, function, mean curvature, arithmetic mean, normal curvatures, curve, surface normal, plane, intersects, neighbourhood, partial derivatives, continuous, surface, point
There are 4 references to this entry.
This is version 7 of normal section, born on 2007-07-30, modified 2007-07-30.
Object id is 9820, canonical name is NormalSection.
Accessed 1619 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) | | | 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions) | | | 53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space) |
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Pending Errata and Addenda
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