
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a motion is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
of a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. For instance, a
plane equipped with the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
metric is a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
in which a mapping associating
congruent figures is a motion. More generally, the term ''motion'' is a synonym for
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
isometry in metric geometry, including
elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
. In the latter case,
hyperbolic motions provide an approach to the subject for beginners.
Motions can be divided into
direct
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), ...
and indirect motions.
Direct, proper or rigid motions are motions like
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s and
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s that preserve the
orientation of a
chiral shape
A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
.
Indirect, or improper motions are motions like
reflections,
glide reflections and
Improper rotations that invert the
orientation of a
chiral shape
A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
.
Some geometers define motion in such a way that only direct motions are motions.
In differential geometry
In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
is called a motion if it induces an isometry between the tangent space at a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
point and the
tangent space at the image of that point.
Group of motions
Given a geometry, the set of motions forms a
group under composition of mappings. This group of motions is noted for its properties. For example, the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformati ...
is noted for the
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s. In the plane, a direct Euclidean motion is either a translation or a
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
, while in
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
every direct Euclidean motion may be expressed as a
screw displacement according to
Chasles' theorem. When the underlying space is a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, the group of motions is a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
. 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.
The idea of a group of motions for
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
has been advanced as Lorentzian motions. For example, fundamental ideas were laid out for a plane characterized by the
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
in
American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
.
The motions of
Minkowski space were described by
Sergei Novikov in 2006:
: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 R
1,3.
History
An early appreciation of the role of motion in geometry was given by
Alhazen (965 to 1039). His work "Space and its Nature" 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.
In the 19th century
Felix Klein
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
became a proponent of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
as a means to classify geometries according to their "groups of motions". He proposed using
symmetry groups in his
Erlangen program, a suggestion that was widely adopted. He noted that every Euclidean congruence is an
affine mapping, and each of these is a
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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, the allowed movements are continuous invertible deformations that might be called elastic motions."
The science of
kinematics
In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.
Kinematics is concerned with s ...
is dedicated to rendering
physical motion into expression as mathematical transformation. Frequently the transformation can be written using vector algebra and linear mapping. A simple example is a
turn written as a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
multiplication:
where
. Rotation in
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
is achieved by
use of quaternions, and
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
by use of
biquaternions. Early in the 20th century,
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...
systems were examined. Later their
automorphism groups led to exceptional groups such as
G2.
In the 1890s logicians were reducing the
primitive notions of
synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...
to an absolute minimum.
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...
and
Mario Pieri used the expression ''motion'' for the congruence of point pairs.
Alessandro Padoa celebrated the reduction of primitive notions to merely ''point'' and ''motion'' in his report to the 1900
International Congress of Philosophy
The World Congress of Philosophy (originally known as the International Congress of Philosophy) is a global meeting of philosophers held every five years under the auspices of the International Federation of Philosophical Societies (FISP). First or ...
. It was at this congress that
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
was exposed to continental logic through Peano. In his book
Principles of Mathematics (1903), Russell considered a motion to be a Euclidean isometry that preserves
orientation.
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''. 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
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s of motion:
- 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.
- The identical mapping of space R is a motion.
- The product of two motions is a motion.
- The inverse mapping of a motion is a motion.
- 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'
- 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.
- 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
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
s on R that are
transitive so that there is a motion that maps every line to every line
Notes and references
*
Tristan Needham (1997) ''Visual Complex Analysis'', Euclidean motion p 34, direct motion p 36, opposite motion p 36, spherical motion p 279, hyperbolic motion p 306,
Clarendon Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
, .
*
Miles Reid & Balázs Szendröi (2005) ''Geometry and Topology'',
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, , {{MathSciNet, id=2194744.
External links
Motion. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.Group of motions. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.
Metric geometry
Differential geometry
Transformation (function)