In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve. If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula
Attributes | Values |
---|
rdfs:label
| - Total absolute curvature (en)
- Вариация поворота кривой (ru)
|
rdfs:comment
| - Вариация поворота кривой — интеграл кривизны кривой по её длине. (ru)
- In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve. If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula (en)
|
dcterms:subject
| |
Wikipage page ID
| |
Wikipage revision ID
| |
Link from a Wikipage to another Wikipage
| |
sameAs
| |
dbp:wikiPageUsesTemplate
| |
has abstract
| - In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve. If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula where s is the arc length parameter and κ is the curvature.This is almost the same as the formula for the total curvature, but differs in using the absolute value instead of the signed curvature. Because the total curvature of a simple closed curve in the Euclidean plane is always exactly 2π, the total absolute curvature of a simple closed curve is also always at least 2π. It is exactly 2π for a convex curve, and greater than 2π whenever the curve has any non-convexities. When a smooth simple closed curve undergoes the curve-shortening flow, its total absolute curvature decreases monotonically until the curve becomes convex, after which its total absolute curvature remains fixed at 2π until the curve collapses to a point. The total absolute curvature may also be defined for curves in three-dimensional Euclidean space. Again, it is at least 2π (this is Fenchel's theorem), but may be larger. If a space curve is surrounded by a sphere, the total absolute curvature of the sphere equals the expected value of the central projection of the curve onto a plane tangent to a random point of the sphere. According to the Fáry–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4π. (en)
- Вариация поворота кривой — интеграл кривизны кривой по её длине. (ru)
|
prov:wasDerivedFrom
| |
page length (characters) of wiki page
| |
foaf:isPrimaryTopicOf
| |
is Link from a Wikipage to another Wikipage
of | |
is foaf:primaryTopic
of | |