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A normal plane is any plane containing the normal vector of a
A normal plane is any plane containing the normal vector of a surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
at a particular point.
The normal plane also refers to the plane that is perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
to the tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
of a space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
; (this plane also contains the normal vector) see Frenet–Serret formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
.
Normal section
The normal section of asurface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
at a particular point is the curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
produced by the intersection of that surface with a normal plane.
The curvature of the normal section is called the '' normal curvature''.
If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the '' principal curvatures''.
If the surface is saddle shaped the maxima of both sides are the principal curvatures.
The product of the principal curvatures is the Gaussian curvature of the surface (negative for saddle shaped surfaces).
The mean of the principal curvatures is the ''mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
'' of the surface; if (and only if) the mean curvature is zero, the surface is called a ''minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
''.
See also
* Earth normal section *Normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
* Normal curvature
* Osculating plane
* Principal curvature
* Tangent plane (geometry)
References
Geometrical optics Surfaces Differential geometry {{differential-geometry-stub