Motion (geometry)

Source: Wikipedia, the free encyclopedia.
A glide reflection is a type of Euclidean motion.

In

surjective isometry in metric geometry,[2] including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions
provide an approach to the subject for beginners.

Motions can be divided into

direct
and indirect motions. Direct, proper or rigid motions are motions like
orientation of a chiral shape
. Indirect, or improper motions are motions like
orientation of a chiral shape
. Some geometers define motion in such a way that only direct motions are motions[citation needed].

In differential geometry

In differential geometry, a diffeomorphism is called a motion if it induces an isometry between the tangent space at a manifold point and the tangent space at the image of that point.[3][4]

Group of motions

Main article: Isometry group

Given a geometry, the set of motions forms a

screw displacement according to Chasles' theorem. When the underlying space is a Riemannian manifold, the group of motions is a Lie group. Furthermore, the manifold has constant curvature if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry.[5]

The idea of a group of motions for special relativity has been advanced as Lorentzian motions. For example, fundamental ideas were laid out for a plane characterized by the quadratic form in

American Mathematical Monthly.[6]
The motions of Minkowski space were described by Sergei Novikov in 2006:[7]

The physical principle of constant velocity of light is expressed by the requirement that the change from one
inertial frame
to another is determined by a motion of Minkowski space, i.e. by a transformation
preserving space-time intervals. This means that
for each pair of points x and y in R1,3.

History

An early appreciation of the role of motion in geometry was given by

Alhazen (965 to 1039). His work "Space and its Nature"[8] uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space. He was criticised by Omar Khayyam who pointed that Aristotle had condemned the use of motion in geometry.[9]

In the 19th century

projective transformation; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruences. The term motion, shorter than transformation, puts more emphasis on the adjectives: projective, affine, Euclidean. The context was thus expanded, so much that "In topology, the allowed movements are continuous invertible deformations that might be called elastic motions."[10]

The science of

turn written as a complex number
multiplication: where . Rotation in space is achieved by use of quaternions, and Lorentz transformations of spacetime by use of biquaternions. Early in the 20th century, hypercomplex number systems were examined. Later their automorphism groups led to exceptional groups such as G2.

In the 1890s logicians were reducing the

Principles of Mathematics (1903), Russell considered a motion to be a Euclidean isometry that preserves orientation.[11]

In 1914

D. M. Y. Sommerville used the idea of a geometric motion to establish the idea of distance in hyperbolic geometry when he wrote Elements of Non-Euclidean Geometry.[12]
He explains:

By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a displacement of the whole space, or, if we are dealing with only two dimensions, of the whole plane. A motion is a transformation which changes each point P into another point P ′ in such a way that distances and angles are unchanged.

Axioms of motion

László Rédei gives as axioms of motion:[13]

  1. Any motion is a one-to-one mapping of space R onto itself such that every three points on a line will be transformed into (three) points on a line.
  2. The identical mapping of space R is a motion.
  3. The product of two motions is a motion.
  4. The inverse mapping of a motion is a motion.
  5. If we have two planes A, A' two lines g, g' and two points P, P' such that P is on g, g is on A, P' is on g' and g' is on A' then there exist a motion mapping A to A', g to g' and P to P'
  6. There is a plane A, a line g, and a point P such that P is on g and g is on A then there exist four motions mapping A, g and P onto themselves, respectively, and not more than two of these motions may have every point of g as a fixed point, while there is one of them (i.e. the identity) for which every point of A is fixed.
  7. There exists three points A, B, P on line g such that P is between A and B and for every point C (unequal P) between A and B there is a point D between C and P for which no motion with P as fixed point can be found that will map C onto a point lying between D and P.

Axioms 2 to 4 imply that motions form a group.

Axiom 5 means that the group of motions provides

transitive
so that there is a motion that maps every line to every line

Notes and references

  1. ISBN 0-534-00034-7
  2. ISBN 0-471-41825-0
  3. ^ A.Z. Petrov (1969) Einstein Spaces, p. 60, Pergamon Press
  4. ISBN 978-0-387-97663-1
  5. ISBN 0-387-52000-7
  6. American Mathematical Monthly
    91(9):543–9, group of motions: p 545
  7. ISBN 0-8218-3929-2
  8. ^ Ibn Al_Haitham: Proceedings of the Celebrations of the 1000th Anniversary, Hakim Mohammed Said editor, pages 224-7, Hamdard National Foundation, Karachi: The Times Press
  9. ISBN 978-0-470-63056-3
    .
  10. ^ Ari Ben-Menahem (2009) Historical Encyclopedia of the Natural and Mathematical Sciences, v. I, p. 1789
  11. Principles of Mathematics
    p 418. See also pp 406, 436
  12. ^ D. M. T. Sommerville (1914) Elements of Non-Euclidean Geometry, page 179, link from University of Michigan Historical Math Collection
  13. ^ Redei, L (1968). Foundation of Euclidean and non-Euclidean geometries according to F. Klein. New York: Pergamon. pp. 3–4.

External links
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