In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.
If is a fixed vector, known as the translation vector, and is the initial position of some object, then the translation function will work as .
If is a translation, then the image of a subset under the function is the translate of by . The translate of by is often written as .
In classical physics, translational motion is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker: [1]
If a body is moved from one position to another, and if the lines joining the initial and final points of each of the points of the body are a set of parallel straight lines of length ℓ, so that the orientation of the body in space is unaltered, the displacement is called a translation parallel to the direction of the lines, through a distance ℓ.
A translation is the operation changing the positions of all points of an object according to the formula
where is the same vector for each point of the object. The translation vector common to all points of the object describes a particular type of displacement of the object, usually called a linear displacement to distinguish it from displacements involving rotation, called angular displacements.
When considering spacetime, a change of time coordinate is considered to be a translation.
The translation operator turns a function of the original position, , into a function of the final position, . In other words, is defined such that This operator is more abstract than a function, since defines a relationship between two functions, rather than the underlying vectors themselves. The translation operator can act on many kinds of functions, such as when the translation operator acts on a wavefunction, which is studied in the field of quantum mechanics.
The set of all translations forms the translation group, which is isomorphic to the space itself, and a normal subgroup of Euclidean group . The quotient group of by is isomorphic to the group of rigid motions which fix a particular origin point, the orthogonal group :
Because translation is commutative, the translation group is abelian. There are an infinite number of possible translations, so the translation group is an infinite group.
In the theory of relativity, due to the treatment of space and time as a single spacetime, translations can also refer to changes in the time coordinate. For example, the Galilean group and the Poincaré group include translations with respect to time.
One kind of subgroup of the three-dimensional translation group are the lattice groups, which are infinite groups, but unlike the translation groups, are finitely generated. That is, a finite generating set generates the entire group.
A translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector using 4 homogeneous coordinates as . [2]
To translate an object by a vector , each homogeneous vector (written in homogeneous coordinates) can 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:
Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).
While geometric translation is often viewed as an active transformation that changes the position of a geometric object, a similar result can be achieved by a passive transformation that moves the coordinate system itself but leaves the object fixed. The passive version of an active geometric translation is known as a translation of axes .
An object that looks the same before and after translation is said to have translational symmetry. A common example is a periodic function, which is an eigenfunction of a translation operator.
The graph of a real function f, the set of points , is often pictured in the real coordinate plane with x as the horizontal coordinate and as the vertical coordinate.
Starting from the graph of f, a horizontal translation means composing f with a function , for some constant number a, resulting in a graph consisting of points . Each point of the original graph corresponds to the point in the new graph, which pictorially results in a horizontal shift.
A vertical translation means composing the function with f, for some constant b, resulting in a graph consisting of the points . Each point of the original graph corresponds to the point in the new graph, which pictorially results in a vertical shift. [3]
For example, taking the quadratic function , whose graph is a parabola with vertex at , a horizontal translation 5 units to the right would be the new function whose vertex has coordinates . A vertical translation 3 units upward would be the new function whose vertex has coordinates .
The antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other. [4]
For describing vehicle dynamics (or movement of any rigid body), including ship dynamics and aircraft dynamics, it is common to use a mechanical model consisting of six degrees of freedom, which includes translations along three reference axes (as well as rotations about those three axes). These translations are often called surge, sway, and heave.
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them. It may refer to the branch of computer science that comprises such techniques or to the models themselves.
In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
Kinematics is a subfield of physics and mathematics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relative to a rotational axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass & distance from the axis.
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator), or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (p).
In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and the determinant are often referred to simply as the Jacobian in literature. They are named after Carl Gustav Jacob Jacobi.
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph, the discrete Laplace operator is more commonly called the Laplacian matrix.
In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real lineR1, the real coordinate planeR2, and the real coordinate three-dimensional spaceR3. With component-wise addition and scalar multiplication, it is a real vector space.
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.
In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form A + εB, where A and B are ordinary quaternions and ε is the dual unit, which satisfies ε2 = 0 and commutes with every element of the algebra. Unlike quaternions, the dual quaternions do not form a division algebra.
In physics and continuum mechanics, deformation is the change in the shape or size of an object. It has dimension of length with SI unit of metre (m). It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation. A configuration is a set containing the positions of all particles of the body.
In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
There are many ways to derive the Lorentz transformations using a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.
Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.
