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

Read more here: » Triangle: Encyclopedia - Triangle

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 - Computing the area of a triangle

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). All triangles are two-dimensional. 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 si ...

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 - Basic facts

In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. It is also known as the vector product or outer product. It differs from the dot product in that it results in a vector rather than in a scalar. Its main use lies in the fact that the cross product of two vectors is orthogonal to both of them. Cross product - Definition. The cross product of the two vectors a and b is denoted by a × b (in longhand some mathema ...

Including:

• Cross product - Definition
• Cross product - Properties
• Cross product - Geometric meaning
• Cross product - Algebraic properties
• Cross product - Associativity
• Cross product - Matrix notation
• Cross product - Lagrange's formula
• Cross product - Applications
• Cross product - Higher dimensions
• Cross product - Symbol

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In mathematics, the Cauchy-Schwarz inequality, also known as the Schwarz inequality, or the Cauchy-Bunyakovski-Schwarz inequality, named after Augustin Louis Cauchy, Viktor Yakovlevich Bunyakovsky and Hermann Amandus Schwarz, is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The inequality states that if x and y are elements o ...

Including:

• Cauchy-Schwarz inequality - Proof
• Cauchy-Schwarz inequality - Notable special cases
• Cauchy-Schwarz inequality - Usage

Read more here: » Cauchy-Schwarz inequality: Encyclopedia - Cauchy-Schwarz inequality

In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In graph theory, a graph describes a set of connections between objects. Each object is called a node or vertex. The connections themselves are called edges or arcs. In nuclear and particle physics, a vertex is the interaction point, where some subnuclear process occurs, changing the number and/o ...

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In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3. In computers, the mathematical function used to perform this calculation is usually given the name abs(). Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the ...

Including:

• Absolute value - Real numbers
• Absolute value - Complex numbers
• Absolute value - Absolute value functions
• Absolute value - Ordered rings
• Absolute value - Distance
• Absolute value - Fields
• Absolute value - Vector spaces
• Absolute value - Algorithms
• Absolute value - Notes

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In mathematics, a unit circle is a circle with unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit ball. If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and ...

Including:

• Unit circle - Trigonometric functions on the unit circle
• Unit circle - Circle group

Read more here: » Unit circle: Encyclopedia - Unit circle

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to comp ...

Including:

• Trigonometric function - List of trigonometric functions
• Trigonometric function - History
• Trigonometric function - Right triangle definitions
• Trigonometric function - Mnemonics
• Trigonometric function - Slope definitions
• Trigonometric function - Unit-circle definitions
• Trigonometric function - Series definitions
• Trigonometric function - Relationship to exponential function
• Trigonometric function - Definitions via differential equations
• Trigonometric function - The significance of radians
• Trigonometric function - Other definitions
• Trigonometric function - Computation
• Trigonometric function - Inverse functions
• Trigonometric function - Identities
• Trigonometric function - Properties and applications
• Trigonometric function - Law of sines
• Trigonometric function - Law of cosines
• Trigonometric function - Law of tangents

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A 3D projection is a mathematical transformation used to project three dimensional points onto a two dimensional plane. Often this is done to simulate the relationship of the camera to subject. 3D projection is often the first step in the process of representing three dimensional shapes two dimensionally in computer graphics, a process known as rendering. The following algorithm was a standard on early computer simulations and videogames, and it is still in use with heavy modifications for each particular case. This article des ...

Including:

• 3D projection - Data necessary for projection
• 3D projection - First step: world transform
• 3D projection - Second step: camera transform
• 3D projection - Third step: perspective transform
• 3D projection - Simple Version

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In physical simulations, video games and computational geometry, collision detection includes algorithms from checking for collision, i.e. intersection, of two given solids, to calculating trajectories, impact times and impact points in a physical simulation. Collision detection - Overview. In physical simulation, we wish to conduct experiments, such as playing billiards. The physics of bouncing billiard balls are well understood, under the umbrella of rigid body motion and elastic collisions. An initial de ...

Including:

• Collision detection - Overview
• Collision detection - Collision detection in physical simulation
• Collision detection - A posteriori vs a priori
• Collision detection - n-body pruning
• Collision detection - Temporal coherence
• Collision detection - Physically based temporal coherence
• Collision detection - Pairwise pruning
• Collision detection - Exact pairwise collision detection
• Collision detection - A priori collision detection
• Collision detection - Spatial partitioning miscellanea
• Collision detection - Collision detection in video games
• Collision detection - Collision detection in computational geometry

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Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, ...

Including:

• Symmetry - Mathematical model for symmetry
• Symmetry - Non-isometric symmetry
• Symmetry - Reflection symmetry
• Symmetry - Rotational symmetry
• Symmetry - Translational symmetry
• Symmetry - Glide reflection symmetry
• Symmetry - Rotoreflection symmetry
• Symmetry - Screw axis symmetry
• Symmetry - Symmetry combinations
• Symmetry - Color
• Symmetry - Similarity vs. sameness
• Symmetry - More on symmetry in geometry
• Symmetry - Symmetry in mathematics
• Symmetry - Symmetry in logic
• Symmetry - Generalization of symmetry
• Symmetry - Symmetry in physics
• Symmetry - Symmetry in biology
• Symmetry - Symmetry in chemistry
• Symmetry - Symmetry in the arts and crafts
• Symmetry - Architecture
• Symmetry - Pottery
• Symmetry - Quilts
• Symmetry - Carpets rugs
• Symmetry - Music
• Symmetry - Other arts and crafts
• Symmetry - Aesthetics
• Symmetry - Symmetry in games and puzzles
• Symmetry - Symmetry in literature
• Symmetry - Symmetry in telecommunications
• Symmetry - Moral symmetry
• Symmetry - Related topics

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A cone is a basic geometrical shape: see cone (solid). Several things have also been called "cones" on account of their shape: A volcanic cone is a mountain formed by material ejected from a volcanic vent. In relativity, the light cone of an event consists of all spacetime events that can interact with it. The scaly fruit-like reproductive bodies of certain plants, especially conifers and cycads, are called cones: see conifer cone. In vertebrate anatomy, a cone cel ...

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An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek ἀγκύλος (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference ...

Including:

• Angle - Units of measure for angles
• Angle - Conventions on measurement
• Angle - Types of angles
• Angle - Some facts
• Angle - A formal definition
• Angle - Angles in different contexts
• Angle - Angles in Riemannian geometry
• Angle - Angles in astronomy
• Angle - Angles in maritime navigation

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The following consequences of the triangle inequalities are often useful; they give lower bounds instead of upper bounds: | ||x|| - ||y|| | ≤ ||x - y|| or for metric | d(x, y) - d(x, z) | ≤ d(y, z) this implies that the norm ||-|| as well distance function d(x, -) are 1-Lipschitz and therefore co ...

See also:

Triangle inequality, Triangle inequality - Normed vector space, Triangle inequality - Metric space, Triangle inequality - Consequences, Triangle inequality - Reversal in Minkowski space

Read more here: » Triangle inequality: Encyclopedia II - Triangle inequality - Consequences

Using vectors and vector dot products, we can easily prove the law of cosines. If we have a triangle with vertices A, B, and C whose sides are the vectors a, b, and c, we know that: and Using the dot product, we simplify the above into ...

See also:

Law of cosines, Law of cosines - Proof, Law of cosines - Alternative proof for acute angles, Law of cosines - Finding the angles when the sides are known, Law of cosines - Isosceles case, Law of cosines - External link

Read more here: » Law of cosines: Encyclopedia II - Law of cosines - Proof

The cross product occurs in the formula for the vector operator curl. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product. The cross product can also be used to calculate the normal for a triangle or polygon. Given a point p and a line through a and b in a plane, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negat ...

See also:

Cross product, Cross product - Definition, Cross product - Properties, Cross product - Geometric meaning, Cross product - Algebraic properties, Cross product - Associativity, Cross product - Matrix notation, Cross product - Lagrange's formula, Cross product - Applications, Cross product - Higher dimensions, Cross product - Symbol

Read more here: » Cross product: Encyclopedia II - Cross product - Applications

For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two edges of the polygon. For a plane given by the equation ax + by + cz = d, the vector (a,b,c) is a normal. If a (possibly non-flat) surface S is parametrized by a system of curvilinear coordinates x(s, t), with s and t real variables, then a normal is given by th ...

See also:

Surface normal, Surface normal - Calculating a surface normal, Surface normal - Uniqueness of the normal, Surface normal - Uses, Surface normal - External link

Read more here: » Surface normal: Encyclopedia II - Surface normal - Calculating a surface normal

A semi normed vector space is a 2-tuple (V,p) where V is a vector space and p a semi norm on V. A normed vector space is a 2-tuple (V,||·||) where V is a vector space and ||·|| a norm on V. We often omit p or ||·|| and just write V for a space if it is clear from the context what (semi) norm we are using. ...

See also:

Normed vector space, Normed vector space - Definition, Normed vector space - Topological structure, Normed vector space - Linear maps and dual spaces, Normed vector space - Normed spaces as quotient spaces of semi normed spaces, Normed vector space - Finite product spaces

Read more here: » Normed vector space: Encyclopedia II - Normed vector space - Definition

We use the following facts: the sum of the angles in a triangle is equal to two right angles and that the base angles of an isosceles triangle are equal. Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let γ = BAO and δ = OBC. Since the sum of the angles of a triangle is equal to two right angles, we have 2γ + γ ′ = 180° and 2δ + δ ′ = 180° We also know that

Thales' theorem, Thales' theorem - Proof, Thales' theorem - Converse, Thales' theorem - Proof of the converse, Thales' theorem - Generalization, Thales' theorem - History, Thales' theorem - External link

Read more here: » Thales' theorem: Encyclopedia II - Thales' theorem - Proof