In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X. In physics, the words centroid and barycenter may mean either the center of mass or the center of gravity of an object, depending on ...

Including:

• Centroid - Centroid of triangle and tetrahedon
• Centroid - Centroids of cones and pyramids
• Centroid - Centroid and convexity
• Centroid - Integral formula
• Centroid - Center of symmetry
• Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia - Centroid

See also:

Centroid, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Centroid and convexity, Centroid - Integral formula, Centroid - Center of symmetry, Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia II - Centroid - Centroid of triangle and tetrahedon

The abscissa of the centroid of a plane figure can be given as the integral , where f(x) is the vertical extent of the object at abscissa x. The same formula yields the first coordinate of the centroid of an object in , for any dimension n, provided that f(x) is the (n − 1)-dimensional measure of the object's cross-section at coordinate See also:

Centroid, Centroid - Centroid of triangle and tetrahedon, Centroid - Centroids of cones and pyramids, Centroid - Centroid and convexity, Centroid - Integral formula, Centroid - Center of symmetry, Centroid - Physical centroids

Read more here: » Centroid: Encyclopedia II - Centroid - Integral formula

The center of mass of an object is a point at which the object's mass can be assumed, for many purposes, to be concentrated. Center of mass - Example. For example, an object can balance on a point only if its center of mass is directly above the point. Alternatively, if you hang an object from a string, the object's center of mass will be directly below the string. Center of gravity, Centroid, Pappus's centroid theorem, Center of pressure Center ...

Including:

• Center of mass - Example
• Center of mass - Comparison with center of gravity
• Center of mass - Definition
• Center of mass - More formulas
• Center of mass - Aeronautical significance
• Center of mass - Motion of the center of mass
• Center of mass - Examples
• Center of mass - Barycenter
• Center of mass - Animations

Read more here: » Center of mass: Encyclopedia - Center of mass

In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side or an extension of the opposite side. The intersection between the (extended) side and the altitude is called the foot of the altitude. This opposite side is called the base of the altitude. The length of the altitude is the distance between the base and the vertex. In an isosceles triangle (a triangle with two congruent sides), the altitude having the incongruent side as ...

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A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. Triangle - Types of triangles. 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 sid ...

Including:

• 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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A Contact patch is the term applied to the portion of a vehicle's tire that is in actual contact with the road surface. The shape of a tire's contact patch can have a great effect on the handling of the vehicle to which it is fitted. Specifically, for the type of wide tire fitted to many modern performance cars, a contact patch that is wider than it is long will increase the tendency for the vehicle to 'tramline' or follow uneven road contours. Furthermore in front wheel drive cars, the offset between the centroid of the contact patch

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In geometry, the circumcircle of a given two-dimensional geometric shape is a circle which contains the shape completely within it. For a triangle, it is the unique circle containing all three vertices. The center of this circumcircle is known as the shape's circumcenter. Note that although the circumcircle of an acute triangle is indeed the smallest circle containing this triangle, this is not true of obtuse triangles. Circumcircle - Cyclic polygons. At least three ver ...

Including:

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

Read more here: » Circumcircle: Encyclopedia - Circumcircle

Data clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. Clustering is the classification of similar objects into different groups, or more precisely, the partitioning of a data set into subsets (clusters), so that the data in each subset (ideally) share some common trait - often proximity according to some defined distance measure. Machine learning typically regar ...

Including:

• 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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The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid. For example, the surface area of the torus with minor radius r and major radius R is ...

See also:

Pappus's centroid theorem, Pappus's centroid theorem - The first theorem, Pappus's centroid theorem - The second theorem

Read more here: » Pappus's centroid theorem: Encyclopedia II - Pappus's centroid theorem - The first theorem

In the discrete case: where n is the number of mass particles. Or in the continuous case: where ρ(s) is the scalar mass density as a function of the position vector. If an object has uniform density then the center of mass is the same thing as the centroid. ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - More formulas

The center of mass is defined as the weighted average of position, with in the discrete case the masses as weights, and in the continuous case the density function as the weight function. Thus the center of mass of an object is the position vector given by: . ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Definition

The center of mass is an important point on an aircraft, as it defines the amount of mass forward or behind the center of gravity that needs to be moved in order to pitch the plane up or down without applying any external forces. In conventional designs the center of mass is often located very near the line 1/3rd back from the front of the wing. That is the line where most wings generate their lift, known as the center of pressure, so by balancing the plane at that point, the lift and weight balance out with no net torque. The center ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Aeronautical significance

The following equations of motion assume that there is a system of particles governed by internal and external forces. An internal force is a force caused by the interaction of the particles within the system. An external force is a force that originates from outside the system, and acts on one or more particles within the system. The external force need not be due to a uniform field. For any system with no external forces, the center of mass moves with constant velocity. This applies for all systems with classical internal forces, in ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Motion of the center of mass

If two bodies are rigidly fixed to each other, then the center of gravity of the combination lies on the line joining the centers of gravity of the individual bodies. In symmetric bodies, the CG lies on the line of symmetry. If there are two or more lines of symmetry, then the CG is at the point of intersection of these lines. The CG of a rectangle is at the intersection of the two diagonals. This principle is used in the example given below. The CG of a triangle lies on the median (line joining the vertex to the mid-point of the opposite side ...

See also:

Center of gravity, Center of gravity - Centers of gravity of simple objects, Center of gravity - Locating center of gravity 1, Center of gravity - Locating center of gravity 2, Center of gravity - Similarities between center of mass and center of inertia, Center of gravity - Differences between center of mass and center of inertia

Read more here: » Center of gravity: Encyclopedia II - Center of gravity - Centers of gravity of simple objects

When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planet, or a planet orbits around a star, both of them are actually orbiting around their center of mass, called the barycenter, see two-body problem. The barycenter (from the Greek βαρύκεντρον) is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. It is an important concept in the fie ...

See also:

Center of mass, Center of mass - Example, Center of mass - Comparison with center of gravity, Center of mass - Definition, Center of mass - More formulas, Center of mass - Aeronautical significance, Center of mass - Motion of the center of mass, Center of mass - Examples, Center of mass - Barycenter, Center of mass - Animations

Read more here: » Center of mass: Encyclopedia II - Center of mass - Barycenter

In a uniform gravitational field (in other words, when the tidal force is insignificant), the center of mass and the center of gravity are at the same location. In a radially uniform gravitational field (such as the one formed by a typical star), the center of mass and the center of gravity of a radially symmetric object (such as planets and stars) are both at the center of the sphere. ...

See also:

Center of gravity, Center of gravity - Centers of gravity of simple objects, Center of gravity - Locating center of gravity 1, Center of gravity - Locating center of gravity 2, Center of gravity - Similarities between center of mass and center of inertia, Center of gravity - Differences between center of mass and center of inertia

Read more here: » Center of gravity: Encyclopedia II - Center of gravity - Similarities between center of mass and center of inertia

The center of mass of a long, uniform beam (of rectangular or circular cross section) is always at the center of the beam. In locations where Earth's gravity dominates, the center of gravity of a vertical, long, uniform beam is closer to Earth than the center of the beam (although it is still inside the beam). In locations where Earth's gravity dominates, the center of gravity of a horizontal, long, uniform beam is further away from Earth than the center of the beam. (The CG may be outside the beam, if the beam is long enough a ...

See also:

Center of gravity, Center of gravity - Centers of gravity of simple objects, Center of gravity - Locating center of gravity 1, Center of gravity - Locating center of gravity 2, Center of gravity - Similarities between center of mass and center of inertia, Center of gravity - Differences between center of mass and center of inertia

Read more here: » Center of gravity: Encyclopedia II - Center of gravity - Differences between center of mass and center of inertia

In general, an object's moment of inertia depends on its shape and the distribution of mass within that shape: the greater the concentration of material away from the object's centroid, the larger the moment of inertia. It also varies depending upon the axis of rotation specified; values relative to the object's centroid are typically taken as baseline values. See the list of moments of inertia for specific examples. The parallel axes theorem can be used to determine moments o ...

See also:

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 - Explanation

Here is an interesting way of determining the CG of an 'L' shaped 2-D object as given in fig 1: 1) Divide the shape into two rectangles, as shown in fig 2. Find the CGs of these two rectangles by drawing the diagonals. Draw a line joining the CGs. The CG of the 'L' shape must lie on this line AB. 2) Divide the shape into two other rectangles, as shown in fig 3. Find the CGs of these two rectangles by drawing the diagonals. Draw a line ...

See also:

Center of gravity, Center of gravity - Centers of gravity of simple objects, Center of gravity - Locating center of gravity 1, Center of gravity - Locating center of gravity 2, Center of gravity - Similarities between center of mass and center of inertia, Center of gravity - Differences between center of mass and center of inertia

Read more here: » Center of gravity: Encyclopedia II - Center of gravity - Locating center of gravity 2