Assertion: curvature's curvature | /r/HasProperty | extrinsic
Facets:
value: in mathematics
type: OTHER
value: purely
type: DEGREE
value: due to its embedding in three dimensions
type: CAUSE
value: in the sense
type: OTHER
value: unlike gauss curvature
type: OTHER
Source sentences:
text: Unlike Gauss curvature, the mean curvature is extrinsic and depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
text: The curvature is intrinsic in the sense that it is a property defined at every point in the space, rather than a property defined with respect to a larger space that contains it.
text: The mean curvature is an extrinsic measure of curvature equal to half the sum of the principal curvatures, k 1 + k 2
text: (Athough defined extrinsically, Gaussian curvature is an intrinsic notion.)
text: On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.
text: The curvature of a curve is an extrinsic geometric property, telling how it is bent in the plane, or bent in space.
text: Gaussian curvature is an intrinsic geometric property: it stays the same no matter how a surface is bent, as long as it is not distorted, neither stretched or compressed.
text: Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in Euclidean space.
text: The curvature we see in a 2-torus as an inner tube is extrinsic due to its embedding in three dimensions.
text: Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the shape operator are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides.
text: In mathematics, the mean curvature of a surface 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.