Mathematics

Roto-translation (rigid transformation)

The rigid transformation (or roto-translation motion) is the composition between reflection, translation, and rotation, and therefore it is an isometry, that is, a geometric transformation that leaves the distances unchanged. In other words, we can think of roto-translation as a rigid movement in which a geometric figure first rotates and then translates. The rototranslation motion of a rigid body …

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Rotation

Rotation is defined as rigid movement having as fixed points a point called center (in two dimensions) or a straight line called axis (in three dimensions) of rotation. This movement shifts all points around the center, or axis, by a fixed angle. In other words a rotation is the movement of a body following a circular …

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Translation

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or space by the same distance in a given direction. It can also be interpreted as the addition of a constant vector at each point, or as a displacement of the origin of the coordinate system. In Physics, translation is defined …

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Coordinate systems

A coordinate system is a system that uses one or more numbers, called coordinates, to uniquely determine the position of a point or other geometric elements on 1D, 2D, and 3D dimensions. Each of these numbers indicates the distance between the point and some fixed reference point, called the origin. The first number, known as the \(x\) value, indicates how far left …

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Vector

Vectors are indicated in the scientific literature with a letter, generally lowercase, with an arrow above it: \(\vec{v}\). Vectors are essential to physics and engineering. Many fundamental physical quantities are vectors, including displacement, velocity, force, and electric and magnetic vector fields. In this context, the following fundamental entities are assigned: free vectors: characterized by magnitude, direction, …

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