Orientation-accurate illustration

In mathematics , images are referred to as orientational images if they do not change the direction of rotation.

Orientation-accurate (or orientation-preserving) maps can be viewed in different general contexts, for example in the elementary geometry of the plane, in linear algebra for maps between vector spaces and in differential geometry for maps between manifolds.

Orientation-maintaining images of the plane

A self-mapping of the plane is called true to orientation if it contains the sense of rotation of polygons.

Among the congruence maps , the parallel displacements and rotations are orientationally true, while reflections and sliding reflections are not orientationally true.

Among the similarity pictures are turning dilations orientation-preserving while folding stretching the orientation not be obtained.

Linear mappings of vector spaces

A linear mapping of a vector space to itself is called true to orientation if it maps positively oriented bases onto positively oriented bases.

This is exactly the case when the determinant of the mapping is positive, and in particular does not depend on which orientation was chosen on the vector space.

Example: The linear mapping is orientationally accurate if and only if the dimension of the vector space is even. ${\displaystyle f(x)=-x}$

Representations of manifolds

A diffeomorphism of an orientable , differentiable manifold is called orientation-true if its differential maps positively oriented bases of the tangent space into positively oriented bases of at every point . ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle x}$ ${\displaystyle T_{x}M}$ ${\displaystyle T_{f(x)}M}$

Example: The self-mapping of the unit sphere, which is defined by , is orientationally true if and only if is odd. ${\displaystyle f(x)=-x}$ ${\displaystyle S^{n}}$ ${\displaystyle n}$

Web links

• Orientation Preserving (MathWorld)