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Learn how to rotate a Cartesian coordinate system by an angle and transform the coordinates of points and curves. See examples, formulas, and applications of rotation of axes in two dimensions.
Learn about the concept of rotation in geometry and algebra, and how it can be represented in different dimensions and formalisms. Find out the definitions, properties, and examples of rotations, as well as their applications in physics and mechanics.
A rotation matrix is a transformation matrix that performs a rotation in Euclidean space. Learn about the definition, properties, examples, and applications of rotation matrices in two and three dimensions.
Learn how quaternions, a type of four-dimensional number, can represent and apply rotations and orientations in three-dimensional space. Quaternions are compact, efficient, and numerically stable compared to other methods, but have some limitations and challenges.
Learn about the spherical coordinate system for three-dimensional space, where the position of a point is specified by radial distance, polar angle, and azimuthal angle. Compare different conventions and terminologies for these angles, such as colatitude, elevation, and longitude.
Learn about the angle between magnetic north and true north at a particular location on the Earth's surface, and how it varies over time and place. Find out how to measure, determine and model magnetic declination from maps, charts and instruments.
Optical rotation is the rotation of the plane of polarization of light by chiral materials, such as sugars, proteins, or liquid crystals. Dextrorotation is the clockwise rotation of light by a dextrorotary compound, such as sucrose or camphor, while laevorotation is the counterclockwise rotation by a laevorotary compound, such as cholesterol.
Learn how to express a rotation in three dimensions as a mathematical transformation using different methods, such as rotation matrix, Euler axis and angle, and Euler rotations. Compare the properties, advantages, and disadvantages of each formalism.