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Orientation representation is like telling someone which way you're facing or pointing without using words. It uses special codes or symbols to show direction, like arrows on a map or numbers in a game.
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Euler angles are a method for representing the orientation of a rigid body in three-dimensional space using three angles, which describe rotations about the axes of a coordinate system. They are widely used in fields like robotics, aerospace, and computer graphics, but can suffer from gimbal lock, a situation where the representation loses a degree of freedom.
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Quaternions extend complex numbers to four dimensions, providing a mathematical framework for representing rotations in three-dimensional space. They are widely used in computer graphics, robotics, and physics due to their ability to avoid gimbal lock and provide smooth interpolation of orientations.
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Rotation matrices are orthogonal matrices used to perform rotations in Euclidean space, preserving the length of vectors and the angles between them. They are fundamental in computer graphics, robotics, and physics for transforming coordinates while maintaining geometric properties.
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Axis-Angle Representation is a mathematical model used to describe the orientation of a rigid body in three-dimensional space, where a rotation is defined by an axis of rotation and an angle of rotation around that axis. This representation is particularly useful in robotics and computer graphics for interpolating rotations and converting between different rotation representations like quaternions and rotation matrices.
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Polar coordinates represent points in a plane using a distance from a reference point and an angle from a reference direction, offering an alternative to Cartesian coordinates for problems involving circular or rotational symmetry. This system is particularly useful in fields such as physics and engineering where it simplifies the analysis of systems with radial symmetry.
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Cartesian coordinates provide a system for specifying positions in a plane or space using ordered pairs or tuples of numbers, typically represented as ((x, y)) in two dimensions or ((x, y), z) in three dimensions. This system, developed by René Descartes, allows for the precise plotting and analysis of geometric shapes, algebraic equations, and spatial relationships.
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Spherical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a radial distance from a fixed origin, an azimuthal angle from a reference direction on a reference plane, and a polar angle from the reference plane. They are particularly useful in fields such as physics and engineering for solving problems involving spherical symmetry, such as gravitational fields and electromagnetic waves.
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Gimbal lock occurs when using Euler angles to represent 3D rotations, causing a loss of one degree of freedom as two of the rotational axes align. This can lead to unexpected behavior in systems relying on these rotations, such as in animations or aircraft control, necessitating alternative representations like quaternions to avoid the issue.
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