There are an infinite number of moments of inertia for any object, one for every possible axis of rotation through the object's centroid. For convenience, the three moments of inertia typically used for an object are about axes parallel to the three Cartesian axes (X, Y, and Z): moment of inertia about the current axis of rotation moment of inertia about the line through the centroid, parallel to the X-axis moment of inertia about the line through the centroid, parallel to the Y-axis moment of inertia ab ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

Read more here: » Moment of inertia: Encyclopedia II - Moment of inertia - Types of moment of inertia

Euler-Bernoulli beam equation - Definitions. x = location along the beam axis y = location perpendicular to beam and to loading z = location perpendicluar to beam, in load plane, with the axis origin at the centroid of the area of the cross-section ux = deflection along beam axis uSee also:

Euler-Bernoulli beam equation, Euler-Bernoulli beam equation - History, Euler-Bernoulli beam equation - Assumptions, Euler-Bernoulli beam equation - Predictions, Euler-Bernoulli beam equation - Definitions, Euler-Bernoulli beam equation - Final equations, Euler-Bernoulli beam equation - Derivation, Euler-Bernoulli beam equation - Practical simplifications, Euler-Bernoulli beam equation - Extensions

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There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangl ...

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Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Using the side lengths and a numerically stable formula, Triangle - Non-planar triangles

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In geometry, there are several theorems that are known by the generic name Pappus's Theorem, attributing them to Pappus of Alexandria. They include: Pappus's centroid theorem, the Pappus chain, Pappus's harmonic theorem, and Pappus's hexagon theorem. ...

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Pappus of Alexandria, Pappus of Alexandria - His Work, Pappus of Alexandria - Theorems, Pappus of Alexandria - Reference

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There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangl ...

See also:

Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points, lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Using the side lengths and a numerically stable formula, Triangle - Non-planar triangles

Read more here: » Triangle: Encyclopedia II - Triangle - Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangl ...

See also:

Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Non-planar triangles

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The UNECE Regulations formally define selective yellow in terms of the CIE 1931 colour space as follows: For front fog lamps, the limit towards white is extended: There are currently competing proposals before UNECE to redefine selective yellow to include this extended range, and to eliminate selective yellow altogether from all lighting regulations. The entirety of the basic selective yellow definition lies outside the gamut of the sRGB colour space — such a pure yellow cannot be represented using RGB primaries. The c ...

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Selective yellow, Selective yellow - Formal definition, Selective yellow - External link

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REALITY ANOMALY

From David Brinn's The Practice Effect: "A psychosomatic reality anomaly has its start when we surround [the centroid mass] by a field of improbability..." But, yes. Seth aside, we might add to Brinn's observation that this "improbability" is the key to ex nihilo manifestation.

(See also: REALITY ANOMALY, Magick, Mysticism, Mysticism Dictionary, Body Mind and Soul, )

Sometimes the term "barycentric subdivision" is improperly used for any subdivision of a polytope P into simplices that have one vertex at the centroid of P, and the opposite facet on the boundary of P. While this property hold for the true barycentric subdivision, it also holds for other subdivisions which are not BCS. For example, if one makes a straight cut from the barycenter of a triangle to each of its three corners, one o ...

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Barycentric subdivision, Barycentric subdivision - Barycentric subdivision of a simplex, Barycentric subdivision - Barycentric subdivision of a convex polytope, Barycentric subdivision - Barycentric subdivision in topology, Barycentric subdivision - Applications, Barycentric subdivision - Repeated barycentric subdivision, Barycentric subdivision - False barycentric subdivision

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The circumcircle of a triangle is the unique circle on which all its three vertices lie. (This is not the same as the first definition for "thin" triangles where only two points would lie on the first definition's circumcircle.) The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular b ...

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Circumcircle, Circumcircle - Cyclic polygons, Circumcircle - Circumcircles of triangles, Circumcircle - Circumcircle equation, Circumcircle - Circumcircle of circles

Read more here: » Circumcircle: Encyclopedia II - Circumcircle - Circumcircles of triangles

Data clustering - k-means and derivatives. The k-means algorithm assigns each point to the cluster whose center (also called centroid) is nearest. The center is the average of all the points in the cluster, ie its coordinates is the arithmetic mean for each dimension separately for all the points in the cluster. Example: The data set has three dimensions and the cluster has two points: X = (x1, x2, x3) and Y< ...

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Data clustering, Data clustering - Types of clustering, Data clustering - Hierarchical clustering, Data clustering - Introduction, Data clustering - Agglomerative hierarchical clustering, Data clustering - Partitional clustering, Data clustering - k-means and derivatives, Data clustering - The elbow criterion, Data clustering - Spectral clustering, Data clustering - Applications, Data clustering - Biology, Data clustering - Marketing research, Data clustering - Other applications, Data clustering - Comparisons between data clusterings, Data clustering - Bibliography, Data clustering - Software implementations, Data clustering - Free, Data clustering - Non-free

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Say we wanted to develop a formula for rotational kinetic energy that is analogous to the linear kinetic energy formula for a point mass. An object with mass m that is moving with velocity v has a kinetic energy (KE) given as: If an object is rotating at an angular velocity ω in a circle of radius r, its linear velocity, v is equal to ωr. Substituting v = ωr into t ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

Read more here: » Moment of inertia: Encyclopedia II - Moment of inertia - Derivation for point mass

For a small (pointlike) mass m, located at distance r from the axis of rotation, moment of inertia (versus that axis) is defined as: For a system with N particles, each with mass mi and distance ri, moment of inertia is defined as the ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

Read more here: » Moment of inertia: Encyclopedia II - Moment of inertia - Mathematical definition

There are several types of clustering methods: Non-Hierarchical clustering (also called k-means clustering) first determine a cluster center, then group all objects that are within a certain distance examples: Sequential Threshold method - first determine a cluster center, then group all objects that are within a predetermined threshold from the center - one cluster is created at a time Parallel Threshold method - simultaneously several cluster centers are determined, then ...

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Cluster analysis in marketing, Cluster analysis in marketing - In marketing cluster analysis is used for:, Cluster analysis in marketing - The basic procedure is:, Cluster analysis in marketing - Clustering procedures

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A common equation which describes the relationship between the linear force applied to an object, the object's mass, and the object's linear acceleration, in a frictionless setting, is: A similar equation can be used to describes the relationship between the rotational force (torque) applied to an object, the object's rotational mass (moment of inertia), and the object's rotational (angular) acceleration, in a frictionless setting: Where: torque moment of ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

Read more here: » Moment of inertia: Encyclopedia II - Moment of inertia - Application of moment of inertia

The input variables in a fuzzy control system are in general mapped into by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". A control system may also have various types of switch, or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fu ...

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Fuzzy control system, Fuzzy control system - Antilock brakes, Fuzzy control system - Fuzzy sets, Fuzzy control system - Fuzzy control in detail, Fuzzy control system - Building a fuzzy controller, Fuzzy control system - History & applications, Fuzzy control system - Reference

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Triangles can be classified according to the relative lengths of their sides: In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles (namely, the angles where each of the equal sides meets the third side). In a scalene triangle all sides have different lengths. The internal angles ...

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Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Using the side lengths and a numerically stable formula, Triangle - Non-planar triangles

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The moment of inertia can be used to describe the amount of angular momentum a rigid body possesses, via the relation: For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar. However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity. The definition of the moment of inertia tensor is very similar to that above, except that i ...

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Moment of inertia, Moment of inertia - Explanation, Moment of inertia - Confusion with second moment of area, Moment of inertia - Derivation for point mass, Moment of inertia - Mathematical definition, Moment of inertia - Types of moment of inertia, Moment of inertia - Application of moment of inertia, Moment of inertia - Inertia tensor

Read more here: » Moment of inertia: Encyclopedia II - Moment of inertia - Inertia tensor

Triangles can be classified according to the relative lengths of their sides: In an equilateral triangle all sides are of equal length. An equilateral triangle is also equiangular, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon In an isosceles triangle two sides are of equal length. An isosceles triangle also has two equal internal angles (namely, the angles where each of the equal sides meets the third side). In a scalene triangle all sides have different lengths. The internal angles ...

See also:

Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Non-planar triangles

Read more here: » Triangle: Encyclopedia II - Triangle - Types of triangles

Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle ...

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Triangle, Triangle - Types of triangles, Triangle - Basic facts, Triangle - Points lines and circles associated with a triangle, Triangle - Computing the area of a triangle, Triangle - Using geometry, Triangle - Using vectors, Triangle - Using trigonometry, Triangle - Using coordinates, Triangle - Using Heron's formula, Triangle - Using the side lengths and a numerically stable formula, Triangle - Non-planar triangles

Read more here: » Triangle: Encyclopedia II - Triangle - Basic facts

A solid body immersed in a fluid will have an upward buoyant force acting on it equal to the weight of displaced fluid. This is due to the hydrostatic pressure in the fluid. In the case of a container ship, for instance, its weight force is balanced by a buoyant force from the displaced water, allowing it to float. If more cargo is loaded onto the ship, it would sit lower in the water - displacing more water and thus receive a higher buoyant force to balance the increased weight force. Discovery of the principle of buoyancy is attributed to Archimedes. < ...

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Fluid statics, Fluid statics - Static pressure in fluids, Fluid statics - Hydrostatic pressure, Fluid statics - Atmospheric pressure, Fluid statics - Buoyancy, Fluid statics - Stability, Fluid statics - Liquids-fluids with free surfaces, Fluid statics - Surface tension effects

Read more here: » Fluid statics: Encyclopedia II - Fluid statics - Buoyancy