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A basic 3D rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate vectors by an angle θ about the x-, y-, or z-axis, in three dimensions, using the right-hand rule—which codifies their alternating signs.
These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. [7]
When the initial end point is the identity quaternion, slerp gives a segment of a one-parameter subgroup of both the Lie group of 3D rotations, SO(3), and its universal covering group of unit quaternions, S 3. Slerp gives a straightest and shortest path between its quaternion end points, and maps to a rotation through an angle of 2Ω.
The line element for an infinitesimal displacement from (r, θ, φ) to (r + dr, θ + dθ, φ + dφ) is = ^ + ^ + ^, where ^ = ^ + ^ + ^, ^ = ^ + ^ ^, ^ = ^ + ^ are the local orthogonal unit vectors in the directions of increasing r, θ, and φ, respectively, and x̂, ŷ, and ẑ are the unit ...
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3).
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space.
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
Specifying the coordinates (components) of vectors of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors, û, v̂ and ŵ, that form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters.