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Rodrigues' Rotation Formula
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in , the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from , the Lie algebra of , to without actually computing the full matrix exponential. This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to Euler, and recommended calling it "Euler's finite rotation formula." This proposal has received notable support, but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both. State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted simply as \sin \theta and \cos \theta. More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and lig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Triple Product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product : \mathbf\cdot(\mathbf\times \mathbf) is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined. Properties * The scalar triple product is unchanged under a circular shift of its three operands (a, b, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Rejection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: \mathbf_1 = a_1\mathbf where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and . Which finally gives: \mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ . The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Projection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: \mathbf_1 = a_1\mathbf where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and . Which finally gives: \mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ . The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Hand Rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of the various left-hand and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. One can see this by holding one's hands outward and together, palms up, with the thumbs out-stretched to the right and left, and the fingers making a curling motion from straight outward to pointing upward. (Note the picture to right is not an illustration of this.) The curling motion of the fingers represents a movement from the first (''x'' axis) to the second (''y'' axis); the third (''z'' axis) can point along either thumb. Left-hand and right-hand rules arise when dealing with coordinate axes. The rule can be used to find the direction of the magnetic field, rotation, spirals, electromagnetic field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Decomposition Unit Vector Rodrigues Rotation Formula
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
