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Plücker Coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe -dimensional linear subspaces, or ''flats'', in an -dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control. Geometric intuition A line in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it (a plane-plane intersection). Consider the first case, with points x=(x_1,x_2,x_3) and y=( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Displacement
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position. Displacement is the shift in location when an object in motion changes from one position to another. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity (a vector), whose magnitude is the average speed (a scalar quantity). Formulation A displacement may be formulated as a ''relative position'' (resulting from the motion), that is, as the final position of a point relative to its initial position . The corresponding displacement vector can be defined as the difference be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inscribed Angle Theorem
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle intercepting the same arc. The inscribed angle theorem appears as Proposition 20 in Book 3 of Euclid's ''Elements''. Note that this theorem is not to be confused with the Angle bisector theorem, which also involves angle bisection (but of an angle of a triangle not inscribed in a circle). Theorem Statement The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle that intercepts the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the same arc of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thales's Theorem
In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's '' Elements''. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras. History Babylonian mathematicians knew this for special cases before Greek mathematicians proved it. Thales of Miletus (early 6th century BC) is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) documenting Pamphila's (1st century AD) statement that T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Without Loss Of Generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property ''P''(''x'',''y'') of real numbers is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "Point at infinity, points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translation (geometry), translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane At Infinity
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case. Definition There are two approaches to defining the ''plane at infinity'' which depend on whether one starts with a projective 3-space or an affine 3-space. If a projective 3-space is given, the ''plane at infinity'' is any distinguished projective plane of the space. This point of view emphasizes the fact that this plane is not geometrically different than any other plane. On the other hand, given an affine 3-space, the ''plane at infinity'' is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. Meaning that the points of the ''plane at infinity'' are the points where parallel lines of the affine 3-space will meet, and the lines are the l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line At Infinity
In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. Geometric formulation In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point At Infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point). In the c ... [...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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. 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 Euclidean vector, vectors is widely used. It is often called the inner product (or rarely the 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). It should not be confused with the cross product. Algebraically, the dot product is the sum of the Product (mathematics), products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
