Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. 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. In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. In Euclidean geometry a transformation is a one-to-one correspondence between two sets of points or a mapping from one plane to another. A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections. 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. A translation operator is an operator T δ {displaystyle T_{mathbf {delta } }} such that T δ f ( v ) = f ( v + δ ) . {displaystyle T_{mathbf {delta } }f(mathbf {v} )=f(mathbf {v} +mathbf {delta } ).} 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 a Euclidean space, any translation is an isometry. The set of all translations forms 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): 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 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) can be multiplied by this translation matrix: