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The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) [3] [4] can be expressed in terms of the axis and angle of the rotation as follows: = ^ This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure ...
The vector cross product, used to define the axis–angle representation, does confer an orientation ("handedness") to space: in a three-dimensional vector space, the three vectors in the equation a × b = c will always form a right-handed set (or a left-handed set, depending on how the cross product is defined), thus fixing an orientation in ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
The derivative of slerp(q 0, q 1; t) with respect to t, assuming the ends are fixed, is log(q 1 q 0 −1) times the function value, where the quaternion natural logarithm in this case yields half the 3D angular velocity vector. The initial tangent vector is parallel transported to each tangent along the curve; thus the curve is, indeed, a geodesic.
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the ...
Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer.
UV texturing permits polygons that make up a 3D object to be painted with color (and other surface attributes) from an ordinary image. The image is called a UV texture map. [ 1 ] The UV mapping process involves assigning pixels in the image to surface mappings on the polygon, usually done by "programmatically" copying a triangular piece of the ...
A closure-constructing operator creates a function object from a part of the program: the part of code given as an argument to the operator is part of the function, and so is the lexical environment: the bindings of the lexically visible variables are captured and stored in the function object, which is more commonly called a closure.