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Plane-based Geometric Algebra
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections, Projection (linear algebra), projections, and their angle from one another in 3D space. Originally growing out of research on spin groups, it was developed with applications to robotics in mind. It has since been applied to machine learning, rigid body dynamics, and computer science, especially computer graphics. It is usually combined with a ''duality'' operation into a system known as "Projective Geometric Algebra", see below. Plane-based geometric algebra takes ''planar reflections'' as basic elements, and constructs all other transformations and geometric objects out of them. Formally: it identifies planar reflections with the ''grade-1'' elements of a Clifford Algebra, that is, elements that are written with a single subscript such as "\bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elements Of Plane-based Geometric Algebra
Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of one body around another * DNA element, a functional region of DNA, including genes and cis-regulatory elements. Mathematics * Element (category theory), one of the constituents in general category theory * Element (mathematics), one of the constituents of set theory in mathematics * Differential element, an infinitesimally small change of a quantity in an integral * Euclid's ''Elements'', a mathematical treatise on geometry and number theory * An entry, or element, of a matrix Philosophy and religion * Classical elements, ancient beliefs about the fundamental types of matter (earth, air, fire, water) * The elements, a religious term referring to the bread and wine of the Eucharist * ''Godai'' (Japanese philosophy), the basis of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Screw Theory
Screw theory is the algebraic calculation of pairs of Vector (mathematics and physics), vectors, also known as ''dual vectors'' – such as Angular velocity, angular and linear velocity, or forces and Moment (physics), moments – that arise in the kinematics and Dynamics (mechanics), dynamics of Rigid body, rigid bodies. Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axis, screw axes of spatial movement and the Line of action, lines of action of forces. The pair of vectors that form the Plücker coordinates of a line define a unit screw, and general screws are obtained by multiplication by a pair of real numbers and Vector addition, addition of vectors. Important theorems of screw theory include: the ''transfer principle'' proves that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws; Chasles' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance imaging and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a ''center of rotation''. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientation (geometry), orientations), in contrast to rotation around a fixed axis, rotation around a axis. The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or ''autorotation''). In that case, the surface intersection of the internal ''spin axis'' can be called a ''pole''; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or ''orbit''), e.g. Earth's orbit around the Sun. The en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
