Information about Translation (geometry)
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A translation moves every point of a figure or a space by the same amount in a given direction.
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A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion which is a translation.
If v is a fixed vector, then the translation Tv will work as Tv(p) = p + v.
If T is a translation, then the image of a subset A under the function T is the translate of A by T. The translate of A by Tv is often written A + v.
In an Euclidean space, any translation is an isometry. The set of all translations form the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E(n ). The quotient group of E(n ) by T is isomorphic to the orthogonal group O(n ):
- E(n ) / T ≅ O(n ).
Matrix representation
Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 1).To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix:
As shown below, the multiplication will give the expected result:
The inverse of a translation matrix can be obtained by reversing the direction of the vector:
Similarly, the product of translation matrices is given by adding the vectors:
See also
External links
- Translation Transform at cut-the-knot
- Geometric Translation (Interactive Animation) at Math Is Fun
Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves.
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.
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coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
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image of a function is the set of all possible values (i.e. outputs) of the function.
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Definition
Let X and Y be sets, f be the function f : X → Y, and x be some member of X...... Click the link for more information.
function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.
An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent.
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In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group.
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In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves.
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quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element. The quotient group is written G/N and is usually spoken in English as G mod N
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In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
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In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. Also they make calculations possible in projective space just as Cartesian coordinates do in Euclidean space.
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matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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In physics, translation is movement that changes the position of an object, as opposed to rotation.
A translation is the operation changing the positions of all objects according to the formula
where is a constant vector.
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A translation is the operation changing the positions of all objects according to the formula
where is a constant vector.
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a: Ta(p) = p + a.
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation.
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In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation.
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cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. The site has won more than 20 awards from scientific and educational publications [1]
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