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A linear transformation is hyperbolic if none of its eigenvalues has modulus 1. This means that can be written as a direct sum of two -invariant subspaces and (where stands for stable and for unstable) such that there exist constants , , and with(1)(2)for , 1, ....

A transformation from one reference frame to another moving with a constant velocity with respect to the first for classical motion. However, special relativity shows that the transformation must be modified to the Lorentz transformation for relativistic motion. The forward Galilean transformation isand the inverse transformation is

The rotation operator can be derived from examining an infinitesimalrotationwhere is the time derivative, is the angular velocity, and is the cross product operator.

The turning of an object or coordinate system by an angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's rotation theorem states that an arbitrary rotation can be parameterized using three parameters. These parameters are commonly taken as the Euler angles. Rotations can be implemented using rotation matrices.Rotation in the plane can be concisely described in the complex plane using multiplication of complex numbers with unit modulus such that the resulting angle is given by . For example, multiplication by represents a rotation to the right by and by represents rotation to the left by . So starting with and rotating left twice gives , which is the same as rotating right twice, , and . For multiplication by multiples of , the possible positions are then concisely represented by , , , and .The rotation symmetry operation for rotation by is denoted "." For periodic arrangements of points..

Rodrigues' rotation formula gives an efficient method for computing the rotation matrix corresponding to a rotation by an angle about a fixed axis specified by the unit vector . Then is given by(1)(2)(3)where is the identity matrixand denotes the antisymmetric matrix with entries(4)Note that the entries in this matrix are defined analogously to the differentialmatrix representation of the curl operator.Note that(5)so applying the rotation matrix given by Rodrigues' formula to any point on the rotation axis returns the same point.

An infinitesimal transformation of a vector is given by(1)where the matrix is infinitesimal and is the identity matrix. (Note that the infinitesimal transformation may not correspond to an inversion, since inversion is a discontinuous process.) The commutativity of infinitesimal transformations and is established by the equivalence of(2)(3)(4)(5)Now let(6)The inverse is then , since(7)(8)(9)Since we are defining our infinitesimal transformation to be a rotation, orthogonalityof rotation matrices requires that(10)but(11)(12)(13)so and the infinitesimal rotation is antisymmetric. It must therefore have a matrix of the form(14)The differential change in a vector upon application of the rotation matrix is then(15)Writing in matrix form,(16)(17)(18)(19)Therefore,(20)where(21)The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the rotating frame. However,..

The symmetry operation corresponding to a rotation followed by an inversion operation, also called a rotoinversion. This operation is denoted for an improper rotation by , so the crystallography restriction gives only , , , , for crystals. The mirror plane symmetry operation is , etc., which is equivalent to .

The four parameters , , , and describing a finite rotation about an arbitrary axis. The Euler parameters are defined by(1)(2)(3)where is the unit normal vector, and are a quaternion in scalar-vector representation(4)Because Euler's rotation theorem states that an arbitrary rotation may be described by only three parameters, a relationship must exist between these four quantities(5)(6)(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by(7)(8)(9)and(10)The Euler parameters may be given in terms of the Eulerangles by(11)(12)(13)(14)(Goldstein 1980, p. 155).Using the Euler parameters, the rotation formulabecomes(15)and the rotation matrix becomes(16)where the elements of the matrix are(17)Here, Einstein summation has been used, is the Kronecker delta, and is the permutation symbol. Written out explicitly, the matrix elements are(18)(19)(20)(21)(22)(23)(24)(25)(26)..

According to Euler's rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices , , and , then a general rotation can be written as(1)The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix as(2)The so-called "-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles , where 1. the first rotation is by an angle about the z-axis using , 2. the second rotation is by an angle about the former x-axis (now ) using , and 3. the third rotation is by an angle about the former z-axis (now ) using . Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use , Tuma (1974) says is..

In geometry, the term "enlargement" is a synonym for expansion.In nonstandard analysis, let be a set of urelements, and let be the superstructure with individuals in : 1. , 2. , 3. . Let be a superstructure monomorphism, with and for . Then is an enlargement of provided that for each set in , there is a hyperfinite set that contains all the standard entities of .It is the case that is an enlargement of if and only if every concurrent binary relation satisfies the following: There is an element of the range of such that for every in the domain of , the pair is in the relation .

The game of billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight line, but the mathematical study of billiards generally consists of reflections in which the reflection and incidence angles are the same. However, strange table shapes such as circles and ellipses are often considered. The popular 1959 animated short film Donald in Mathmagic Land features a tutorial by Donald Duck on how to win at billiards using the diamonds normally inscribed around the edge of a real billiard table.Many interesting problems can arise in the detailed study of billiards trajectories. For example, any smooth plane convex set has at least two double normals, so there are..

In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77): 1. The locus of the center of a variable circle, tangent to a fixed circle and passing through a fixed point inside that circle, is an ellipse. 2. If a variable circle is tangent to a fixed circle and also passes through a fixed point outside the circle, then the locus of its moving center is a hyperbola. 3. If a variable circle is tangent to a fixed straight line and also passes through a fixed point not on the line, then the locus of its moving center is a parabola. Let be a smooth regular parameterized curve in defined on an open interval , and let and be points in , where is an -dimensional projective space. Then has a reflection property with foci and if, for each point , 1. Any vector normal to the curve at lies in the vector space span of the vectors and . 2. The line normal to at bisects one of the pairs of opposite angles formed by the intersection of the lines joining..

The operation of exchanging all points of a mathematical object with their mirror images (i.e., reflections in a mirror). Objects that do not change handedness under reflection are said to be amphichiral; those that do are said to be chiral.Consider the geometry of the left figure in which a point is reflected in a mirror (blue line). Then(1)so the reflection of is given by(2)The term reflection can also refer to the reflection of a ball, ray of light, etc. off a flat surface. As shown in the right diagram above, the reflection of a points off a wall with normal vector satisfies(3)If the plane of reflection is taken as the -plane, the reflection in two- or three-dimensional space consists of making the transformation for each point. Consider an arbitrary point and a plane specified by the equation(4)This plane has normal vector(5)and the signed point-plane distance is(6)The position of the point reflected in the given plane is therefore given by(7)(8)The..

The map(1)(2)which leaves the parabola(3)invariant.

for . Fixed points occur at , 1/2, and order 2 fixed points at . The natural invariant of the map is

A class of area-preserving maps ofthe form(1)(2)which maps circles into circles but with a twist resulting from the term.

Exchanges branches of the hyperbola .(1)(2)

A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined asNote that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).Examples of transformations are summarized in the following table.TransformationCharacterizationdilationcenter of dilation, scale decrease factorexpansioncenter of expansion, scale increase factorreflectionmirror line or planerotationcenter of rotation, rotation angleshearinvariant line and shear factorstretch (1-way)invariant line and scale factorstretch (2-way)invariant lines and scale factorstranslationdisplacement..

Let and , thenis a Möbius transformation, where is the complex conjugate of . is a conformal mapping self-map of the unit disk for each , and specifically of the boundary of the unit disk to itself. The same holds for .Any conformal self-map of the unit disk to itself is a composition of a Möbius transformation with a rotation, and any conformal self-map of the unit disk can be written in the formfor some Möbius transformation and some complex number with (Krantz 1999, p. 81).

An entire Cremona transformation is a birational transformation of the plane. Cremona transformations are maps of the form(1)(2)in which and are polynomials. A quadratic Cremona transformation is always factorable.

A map from a domain is called a map of class if each component ofis of class ( or ) in , where denotes a continuous function which is differentiable times.

Min Max Re Im for , where is the floor function. The natural invariant of the map is

A bijective map between twometric spaces that preserves distances, i.e.,where is the map and is the distance function. Isometries are sometimes also called congruence transformations. Two figures that can be transformed into each other by an isometry are said to be congruent (Coxeter and Greitzer 1967, p. 80).An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map. Two geometric figures related by an isometry are said to be geometrically congruent (Coxeter and Greitzer 1967, p. 80).If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections..

A linear transformation of period two. Sincea linear transformation has the form,(1)applying the transformation a second time gives(2)For an involutory, , so(3)Since each coefficient must vanishseparately,(4)(5)(6)Equation (5) requires . Taking in turn requires that , giving , i.e., the identity map, while taking gives , so(7)which is the general form of a line involution.

The shear matrix is obtained from the identity matrix by inserting at , e.g.,(1)Bolt and Hobbs (1998) define a shear matrix as a matrix(2)such that(3)(4)

Also known as the a Lorentz transformation or Procrustian stretch, a hyperbolic transformation leaves each branch of the hyperbola invariant and transforms circles into ellipses with the same area.(1)(2)

An automorphism is an isomorphism of a system of objects onto itself. The term derives from the Greek prefix (auto) "self" and (morphosis) "to form" or "to shape."The automorphisms of a graph always describea group (Skiena 1990, p. 19).An automorphism of a region of the complex planeis a conformal self-map (Krantz 1999, p. 81).

A transformation in which all points along a given line remain fixed while other points are shifted parallel to by a distance proportional to their perpendicular distance from . Shearing a plane figure does not change its area. The shear can also be generalized to three dimensions, in which planes are translated instead of lines.

A homographic transformation(1)(2)with substituted for according to(3)

Let be a map. Then is expansive if the statement that the distance for all implies that . Equivalently, is expansive if the orbits of two points and are never very close.

A transformation is unimodular if the determinant of the matrix satisfiesA necessary and sufficient condition that a linear transformation transform a lattice to itself is that the transformation be unimodular.If is a complex number, then the transformationis called a unimodular if , , , and are integers with . The set of all unimodular transformations forms a group called the modular group.

A projective correlation of period two. In a polarity, is called the polar of , and the inversion pole .

A perspective collineation with center and axis is a collineation which leaves all lines through and points of invariant. Every perspective collineation is a projective collineation.

A perspective collineation with center and axis not incident is called a geometric homology. A geometric homology is said to be harmonic if the points and on a line through are harmonic conjugates with respect to and . Every perspective collineation of period two is a harmonic homology.

A point-to-line and line-to-point transformation which transforms points into lines and lines into points such that passes through iff lies on .

Expansion is an affine transformation (sometimes called an enlargement or dilation) in which the scale is increased. It is the opposite of a geometric contraction, and is also sometimes called an enlargement. A central dilation corresponds to an expansion plus a translation.Another type of expansion is the process of radially displacing the edges or faces of a polyhedron (while keeping their orientations and sizes constant) while filling in the gaps with new faces (Ball and Coxeter 1987, pp. 139-140). This procedure was devised by Stott (1910), and can be used to construct all 11 amphichiral (out of 13 total) Archimedean solids. The opposite operation of expansion (i.e., inward expansion) is called contraction. Expansion is a special case of snubification in which no twist occurs.The following table summarizes some expansions of some unit edge length Platonic and Archimedean solids, where is the displacement and is the golden ratio.base..

A similarity transformation which transforms each line to a parallel line whose length is a fixed multiple of the length of the original line. The simplest dilation is therefore a translation, and any dilation that is not merely a translation is called a central dilation. Two triangles related by a central dilation are said to be perspective triangles because the lines joining corresponding vertices concur. A dilation corresponds to an expansion plus a translation.

A map is called bijective if it is both injective and surjective. A bijective map is also called a bijection. A function admits an inverse (i.e., " is invertible") iff it is bijective.Two sets and are called bijective if there is a bijective map from to . In this sense, "bijective" is a synonym for "equipollent" (or "equipotent"). Bijectivity is an equivalence relation on the class of sets.

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp() is defined to be , where is the unique Lie group homeomorphism from the real numbers to the Lie group such that its velocity at time 0 is .On a Riemannian manifold, exp is a map from the tangent bundle of the manifold to the manifold, and exp() is defined to be , where is the unique geodesic traveling through the base-point of such that its velocity at time 0 is .The three notions of exp (exp from complex analysis, exp from Lie groups, and exp from Riemannian geometry) are all linked together, the strongest link being between the Lie groups and Riemannian geometry definition. If is a compact Lie group, it admits a left and right invariant Riemannian metric. With respect to that metric, the two exp maps agree on their common domain. In other words, one-parameter subgroups are geodesics. In the case of the manifold..

A formula which transforms a given coordinate system by rotating it through a counterclockwise angle about an axis . Referring to the above figure (Goldstein 1980), the equation for the "fixed" vector in the transformed coordinate system (i.e., the above figure corresponds to an alias transformation), is(1)(2)(3)(Goldstein 1980; Varshalovich et al. 1988, p. 24). The angle and unit normal may also be expressed as Euler angles. In terms of the Euler parameters,(4)The rotation matrix can be calculated in the Wolfram Language as follows: With[{n = {nx, ny, nz}}, Cos[phi] IdentityMatrix[3] + (1 - Cos[p]) Outer[Times, n, n] + Sin[p] {{0, n[[3]], -n[[2]]}, {-n[[3]], 0, n[[1]]}, {n[[2]], -n[[1]], 0}} ]

When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.In algebraic topology, a free abstract mathematical object is generated by elements in a "free manner" ("freely"), i.e., such that the elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget is freely generated by a subset if, for any function where is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) such that restricted to is .If the algebraic gadgets are vector spaces, then freely generates iff is a basis for . If the algebraic gadgets are..

The map(1)where is computed modulo 1. A generalized Baker's map can be defined as(2)(3)where , , and and are computed mod 1. The q-dimension is(4)If , then the general q-dimension is(5)

The term "similarity transformation" is used either to refer to a geometric similarity, or to a matrix transformation that results in a similarity.A similarity transformation is a conformal mapping whose transformation matrix can be written in the form(1)where and are called similar matrices (Golub and Van Loan 1996, p. 311). Similarity transformations transform objects in space to similar objects. Similarity transformations and the concept of self-similarity are important foundations of fractals and iterated function systems.The determinant of the similarity transformation of a matrix is equal to the determinant of the original matrix(2)(3)(4)The determinant of a similarity transformation minus a multiple of the unit matrixis given by(5)(6)(7)(8)If is an antisymmetric matrix () and is an orthogonal matrix (), then the matrix for the similarity transformation(9)is itself antisymmetric, i.e., . This follows..

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