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Circle

Circle

:This article is about the shape and mathematical concept of circle; for other meanings, see Circle (disambiguation). Circle (disambiguation) In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually however, the circumference means the length of the circle, and the interior of the circle is called a disk or disc. An arc is any continuous portion of a circle.

Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that :

\left( x - a \right)^2 + \left( y - b \right)^2=r^2.

If the circle is centered at the origin (0, 0), then this formula can be simplified to :

x^2 + y^2 = r^2.

The circle centered at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, y) can be written as :x = a + r cos(t) :y = b + r sin(t). The slope a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle: :

y' = - \frac.

In the complex plane, a circle with a center at c and radius r has the equation

|z-c|^2 = r^2

. Since

|z-c|^2 = z\overline-\overlinez-c\overline+c\overline

, the slightly generalized equation

pz\overline + gz + \overline = q

for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles. All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words:
- Length of a circle's circumference =

2\pi \times r.

- Area of a circle =

\pi \times r^2.

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r².

Properties

limit limit

Chord properties

- Chords equidistant from the centre of a circle are equal.
- Equal chords are equidistant from the centre.
- A line from the centre, perpendicular to a chord, bisects the chord.
- The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
- The perpendicular bisector of a chord passes through the centre of a circle.

Tangent properties

- The line drawn perpendicular to the end point of a radius is a tangent to the circle.
- A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
- Tangents drawn from a point outside the circle are equal in length.
- Two tangents can always be drawn from a point outside of the circle.

Inscribed angle theorem

- If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
- If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
- An inscribed angle subtended by a semicircle is a right angle.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Secant, tangent, and chord properties

- The chord theorem states that if two chords, CD and EF, intersect at G, then

CG \times DG = EG \times FG

. (Chord Theorem)
- If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then

DC^2 = DG \times DE

. (Tangent Secant Theorem)
- If two secants, DG and DE, also cut the circle at H and F respectively, then

DH \times DG = DF \times DE

. (Corollary of the Tangent Secant Theorem)
- The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent Chord Property)
- If the angle subtended by the chord at the centre is 90 degrees then l = sqrt(2)
- r, where l is the length of the chord and r is the radius of the circle.

External links

- [http://agutie.homestead.com/files/clifford1.htm Clifford's Circle Chain Theorems.] This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression. by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- [http://www.cut-the-knot.org/pythagoras/Munching/circle.shtml Munching on Circles] at cut-the-knot Category:Conic sections ja:円 (数学) simple:Circle

Circle (disambiguation)

The term circle may refer to:
- Circle (dance move), a variety of traditional dance moves where partners circle around each other.
- Circle (Eddie Izzard), a particular tour by the British comedian.
- The Circle, a peer-to-peer distributed file system.
- The Circle (film), a 2000 film by Iranian independent filmmaker Jafar Panahi.
- Circle (band), a Finish rock group.
- The Circle (organization), a Bajorin terrorist organization on Star Trek:Deep Space Nine
- Circle (Wiccan), a "portable temple" or shield cast or drawn around a ritual to raise energy, often a sphere.
- Circle, Alaska, a small village on the Yukon River. An alternate spelling of circle may refer to:
- The Cyrkle, a 1960s U.S. rock and roll band.
- See also ring (diacritic) for the circular diacritic mark.

Euclidean geometry

:For a topic other than geometry whose name includes the word "Euclidean", see Euclidean algorithm. In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. It is important to remember that, in the original and correct conception, geometry is, first of all, a physical science ("the noblest of the physical sciences"); that is, the logical definition of geometry (its fundamental assumptions or axioms) arises directly out of observation. Abstract geometries may be exclusively mathematical, but, whenever so, they are different from physical geometries. Euclidean geometry sometimes means geometry in the plane which is also called plane geometry. Plane geometry is the topic of this article. Euclidean geometry is also based on the Point-Line-Plane postulate. Euclidean geometry in three dimensions is traditionally called solid geometry. For information on higher dimensions see Euclidean space. Plane geometry is the kind of geometry usually taught in secondary school. Euclidean geometry is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry.

Axiomatic approach

The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms. The five postulates of the Elements are as follows: # Any two points can be joined by a straight line. # Any straight line segment can be extended indefinitely in a straight line. # Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. # All right angles are congruent. # If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The fifth postulate is called the parallel postulate, which leads to the same geometry as the following statement (note that it is formulated for two-dimensional geometry): :Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. The parallel postulate seems less obvious than the others and many geometers tried in vain to prove it from them. By 1763 at least 28 different proofs of the fifth postulate had been published, but all were found to be incorrect. In the 19th century it was shown that this could not be done, by constructing hyperbolic geometry where the parallel postulate is false, while the other axioms hold. (If one simply drops the parallel postulate from the list of axioms then the result is the more general geometry called absolute geometry). Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. In Hyperbolic geometry the sum of the three angles are always less than 180 and can approach zero. Only under the umbrella of Euclidean geometry are then angles always portions of 180°. Another thing that was observed was that Euclid's five axioms are actually somewhat incomplete. For instance, one of his theorems is that any line segment is part of a triangle; he constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as third vertex. His axioms, however, do not guarantee that the circles actually intersect. Many revised systems of axioms were constructed, the most standard ones are Hilbert's axioms, Birkhoff's axioms, and Tarski's axioms. Tarski used his axioms to show Euclidean geometry is a complete decidable theory; that is, every proposition of Euclidean geometry can be shown to be either true or false. Euclid also had five "common notions" which can also be taken to be axioms, though he later used other properties of magnitudes. # Things which equal the same thing also equal one another. # If equals are added to equals, then the wholes are equal. # If equals are subtracted from equals, then the remainders are equal. # Things which coincide with one another equal one another. # The whole is greater than the part.

Modern introduction to Euclidean geometry

Today, Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. If one introduces geometry this way, one can then prove the Euclidean (or any other) axioms as theorems in this particular model. This does not have the beauty of the axiomatic approach, but it is extremely concise.

The construction

First let us define the set of points as set of pairs of real numbers

(x,y)

. Then given two points

P=(x,y)

and

Q=(z,t)

one can define distances using the following formula: :

|PQ|=\sqrt

. This is known as the Euclidean metric. All other notions as a straight line, angle, circle can be defined in terms of points as pairs of real numbers and the distances between them. For example straight line through

P

and

Q

can be defined as a set of points

A

such that the triangle

APQ

is degenerate, i.e. :

|PQ| =|PA|+|AQ| \mbox |PQ| =\pm(|PA|-|AQ|)

References

- Gödel, Escher, Bach: an Eternal Golden Braid, by Douglas Hofstadter, ISBN 0465026567 (page 91)

Classical theorems

- Ceva's theorem
- Heron's formula
- Nine-point circle
- Pythagorean theorem
- Tartaglia's formula

External links

- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Euclid's elements]
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
-
- ko:유클리드 기하학 ja:ユークリッド幾何学

Distance

:This article is about distance in the mathematical / physical sense. For other uses, see distance (disambiguation). The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious. Distance as mentioned above is sometimes not symmetric, hence not a metric (see below): this applies to distance by car in the case of one-way streets, and also in the case the distance is expressed in terms of the time to cover it (a road may be more crowded in one direction than in the other, for a ship upstream and downstream makes a difference). As opposed to a position coordinate, a distance can not be negative. Distance is a scalar quantity, containing only a magnitude, whereas displacement is an equivalent vector quantity containing both magnitude and direction. In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

Distance covered

Pythagorean theorem The distance covered by a vehicle (often recorded by a odometer), person, animal, object, etc. should be distinguished from the distance from starting point to end point, even if latter is taken to mean e.g. the shortest distance along the road, because a detour could be made, and the end point can even coincide with the starting point.

Formal definition

A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space. We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B. Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

The distance formula

The (Euclidean) distance, d, between two points expressed in Cartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate. Thus, in a two-dimensional space :

d = \sqrt,

and in a three-dimensional space: :

d = \sqrt.

Here, "Δ" (delta) refers to the change in a variable. Thus, Δx is the change in x, pronounced as such, or as "delta-x". In mathematical terms,

\Delta x = \left|x_1 - x_0\right|

, and so

(\Delta x)^2 = (x_1 - x_2)^2

. This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the arc-length formula.

Generalized distance in arbitrary dimensions: Norms

In the Euclidean space Rⁿ, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead. For a point (x₁, x₂, ...,x_n) and a point (y₁, y₂, ...,y_n), the Minkowski distance of order p (p-norm distance) is defined as: p need not be an integer, but it cannot be less than 1, because then the triangle inequality does not hold. There is no such thing as a negative distance. The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to more than two coordinates. It is what would be obtained if the distance between two points were measured with a ruler: the "intuitive" idea of distance. The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets). The infinity norm distance is also called Chebyshev distance. In 2D it represents the distance kings, queens, and bishops must travel between two squares on a chessboard. Also, if you measure the strength of each of the n links in a chain (where larger numbers mean weaker links), then because a chain is only as strong as its weakest link, the strength of the chain will be the infinity-norm distance from the list of measurements to the origin. The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse. In physical space the Euclidean distance is in a way the most natural one, because in this case the length of a rigid body does not change with rotation.

Distances in other spaces

- Mahalanobis distance is used in statistics.
- Hamming distance is used in coding theory.
- Levenshtein distance

Plane (mathematics)

In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it may be visualized as a flat infinite sheet of paper. There are several definitions of the plane, equivalent in the sense of Euclidean geometry, but which can be extended in different ways to define object in other areas of mathematics. In some areas of mathematics, such as plane geometry or 2D computer graphics, the whole space in which the work is carried out is a single plane. In such situations the definite article is used: the plane. Many fundamental tasks in geometry, trigonometry, and graphing are performed in the two dimensional space, or in other words, in the plane.

Euclidean geometry

A plane is a surface such that, given any two points on the surface, the surface also contains the straight line that passes through the two points. One can introduce a Cartesian coordinate system on a given plane in order to label every point on it uniquely with two numbers, the point's coordinates. Within any Euclidean space, a plane is uniquely determined by any of the following combinations:
- three non-collinear points (not lying on the same line)
- a line and a point not on the line
- two different lines which intersect
- two different lines which are parallel

Planes embedded in R³

This section is specifically concerned with planes embedded in three dimensions: specifically, in R³.

Properties

In three-dimensional space, we may exploit the following facts that do not hold in higher dimensions:
- Two planes are either parallel or they intersect in a line.
- A line is either parallel to a plane or they intersect at a single point.
- Two lines normal to the same plane must be parallel to each other.
- Two planes normal to the same line must be parallel to each other.

Point and a normal vector

In a three-dimensional ambient space, there is another important way of defining a plane:
- a point and a line, which is normal (perpendicular) to the plane We can explicitly describe the resulting plane; let

\vec p

be the point we wish to lie in the plane, and let

\vec n

be a nonzero vector parallel to the line we wish to be normal to the plane. The desired plane is the set of all points

\vec r

such that :

\vec n\cdot(\vec r-\vec p)=0.

If we write

\vec n = (a, b, c)

\vec r = (x, y, z)

, and

\vec n\cdot\vec p=-d

, then the plane is determined by the condition :

ax + by + cz + d = 0

, where a, b, c and d could be any real numbers such that not all of a, b, c are zero. Alternatively, a plane may be described parametrically as the set of all points of the form :

\vec + s\vec + t\vec,

where s and t range over all real numbers, and

\vec

\vec

and

\vec

are given vectors defining the plane.

Plane through three points

The plane passing through three points

\vec p_1 = (x_1,y_1,z_1)

\vec p_2 = (x_2,y_2,z_2)

and

\vec p_3 = (x_3,y_3,z_3)

can be determined by the following determinant equations: :

\begin 
x - x_1 & y - y_1 & z - z_1 \\
x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 
\end =\begin 
x - x_1 & y - y_1 & z - z_1 \\
x - x_2 & y - y_2 & z - z_2 \\
x - x_3 & y - y_3 & z - z_3 
\end = 0.

This plane can also be described by the "point and a normal vector" prescription above. A suitable normal vector is given by the cross product

\vec n = ( \vec p_2 - \vec p_1 ) \times ( \vec p_3 - \vec p_1 ),

and the point

\vec p

can be taken to be

\vec p_1

The distance from a point to a plane

For a plane

ax + by + cz + d = 0

and a point

\vec p_1 = (x_1,y_1,z_1)

not necessarily lying on the plane, the distance from

\vec p_1

to the plane is :

D = \frac.

The line of intersection between two planes

Given intersecting planes described by

\vec n_1\cdot \vec r = h_1

and

\vec n_2\cdot \vec r = h_2

, the line of intersection is perpendicular to both

\vec n_1

and

\vec n_2

and thus parallel to :

\vec n_1 \times \vec n_2

. If we further assume that

\vec n_1

and

\vec n_2

are orthonormal then the closest point on the line of intersection to the origin is :

\vec r_0 = h_1\vec n_1 + h_2\vec n_2

The dihedral angle

Given two intersecting planes described by

a_1 x + b_1 y + c_1 z + d_1 = 0

and

a_2 x + b_2 y + c_2 z + d_2 = 0

, the dihedral angle between them is defined to be the angle

\alpha

between their normal directions: :

\cos\alpha = \hat n_1\cdot \hat n_2 = \frac

External links

- [http://mathworld.wolfram.com/Plane.html Mathworld: Plane]
- Category:Surfaces ja:平面

Simple closed curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

Definitions

In mathematics, a (topological) curve is defined as follows. Let

I

be an interval of real numbers (i.e. a non-empty connected subset of

\mathbb

). Then a curve

\!\,\gamma

is a continuous mapping

\,\!\gamma : I \rightarrow X

, where

X

is a topological space. The curve

\!\,\gamma

is said to be simple if it is injective, i.e. if for all

x

y

I

, we have

\,\!\gamma(x) = \gamma(y) \rightarrow x = y

. If

I

is a closed bounded interval

\,\![a, b]

, we also allow the possibility

\,\!\gamma(a) = \gamma(b)

(this convention makes it possible to talk about closed simple curve). If

\gamma(x)=\gamma(y)

for some

x\ne y

(other than the extremities of

I

), then

\gamma(x)

is called a double (or: multiple) point of the curve. A curve

\!\,\gamma

is said to be closed or a loop if

\,\!I = [a, b]

and if

\!\,\gamma(a) = \gamma(b)

. A closed curve is thus a continuous mapping of the circle

S^1

; a simple closed curve is also called a Jordan curve. A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example.

Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Length of curves

X

is a metric space with metric

d

, then we can define the length of a curve

\!\,\gamma : [a, b] \rightarrow X

by :

\mbox (\gamma)=\sup \left\

A rectifiable curve is a curve with finite length. A parametrization of

\!\,\gamma

is called natural (or unit speed or parametrised by arc length) if for any

t_1

t_2

[a, b]

, we have :

\mbox (\gamma|_)=|t_2-t_1|

\!\,\gamma

is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of

\!\,\gamma

t_0

as :

\mbox(t_0)=\limsup_

and then :

\mbox(\gamma)=\int_a^b \mbox(t) \, dt

In particular, if

X = \mathbb^n

is Euclidean space and

\gamma : [a, b] \rightarrow \mathbb^n

is differentiable then :

\mbox(\gamma)=\int_a^b \left| \,  \, \right| \, dt

Differential geometry

Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If

X

is a differentiable manifold, then we can define the notion of differentiable curve in

X

. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take

X

to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to

X

by means of this notion of curve. If

X

is a smooth manifold, a smooth curve in

X

is a smooth map :

\!\,\gamma : I \rightarrow X.

This is a basic notion. There are less and more restricted ideas, too. If

X

is a

C^k

manifold (i.e., a manifold whose charts are

k

times continuously differentiable), then a

C^k

curve in

X

is such a curve which is only assumed to be

C^k

(i.e.

k

times continuously differentiable). If

X

is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and

\!\,\gamma

is an analytic map, then

\!\,\gamma

is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two

C^k

differentiable curves :

\!\,\gamma_1 :I \rightarrow X

and :

\!\,\gamma_2 : J \rightarrow X

are said to be equivalent if there is a bijective

C^k

map :

\!\,p : J \rightarrow I

such that the inverse map :

\!\,p^ : I \rightarrow J

is also

C^k

, and :

\!\,\gamma_(t) = \gamma_(p(t))

for all

t

. The map

\!\,\gamma_2

is called a reparametrisation of

\!\,\gamma_1

; and this makes an equivalence relation on the set of all

C^k

differentiable curves in

X

. A

C^k

arc is an equivalence class of

C^k

curves under the relation of reparametrisation.

Algebraic curve

Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.

History

A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html List of famous curves] Category:Curves Category:Metric geometry Category:Topology Category:General topology ko:곡선 ja:曲線

Disk (mathematics)

In geometry, a disk is the region in a plane contained inside of a circle. A disk is said to be closed or open according to whether it contains or not the circle which is its boundary. A ball is a disk in a space with more than two dimensions. See ball (mathematics). Sometimes however, one uses the word "disk" synonymously with "ball", allowing it to have other dimensions.

Formula

: For help with typing formulas in Wikipedia markup, see help:formula. In mathematics and in the sciences, a formula (formulas or formulae in plural form) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E=mc² (see special relativity). When one attempts to write a chemical formula, it is important to know whether the substance in question actually exists. For example, one can easily write the formula of carbon nitrate, but no chemist has ever prepared this compound. Here are basic rules for writing chemical formulae: 1. Represent the symbols of the components, placing the positive part first, and then the negative part. 2. Indicate the respective oxidation numbers above and to the right of each symbol. (Enclose radicals in parentheses for the time being.) 3. For each symbol, write a subscript number equal to the oxidation number of the other element or radical. This is the same as the mechanical crisscross method. Since the positive oxidation number shows the number of electrons that may be lost or shared and the negative oxidation number shows the number of electrons that may be gained or shared, you must have just as many electrons lost (or partially lost in sharing) as are gained (or partially gained in sharing). 4. Now rewrite the formulas, omitting the subscript 1, the parentheses of the radicals that have the subscript 1, and the plus and minus numbers. 5. As a general rule, the subscript numbers in the final formula are reduced to their lowest terms. There are, however, certain exception, such as hydrogen peroxide(H₂O₂, and acetylene C₂H₂). For these exceptions, you must have more specific information about the compound. The only way to become proficient at writing formulae is to memorize the oxidations numbers of common elements (or learn to use the periodic chart group numbers) and practice writing formulae.

Other meanings

- Formula auto racing (for example, Formula One)
- Formula Language (Lotus Notes programming language)
- Infant formula (baby feeding substitute for breast milk)
- In a biological/chemical lab, formula can also mean something like a recipe, similar to the term for infant formula ja:公式

Parametric equations

]]In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. When this is done, the equations then tell the values of dependent variables. A simple example is, in kinematics, filling in a time parameter to get the position, velocity, and other facts about a moving object. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of a functions from, say, Rⁿ. It is therefore somewhat more accurately defined as a parametric representation. See also parameter, parametrization, regular parametric representation. For example, the simplest equation for a parabola, :

y = x^2\,

can be parametrized by using a free parameter t, and setting :

x = t\,

y = t^2\,

Although the preceding example appears somewhat trivial, consider the following parametrization of a circle of radius a: :

x = a \cos(t)\,

y = a \sin(t)\,

Finally, there are certain geometric forms that are nearly impossible to describe as a single equation but have very elegant expressions in parametric form: :

x = a \cos(t)\,

y = a \sin(t)\,

z = bt\,

which describe a three-dimensional curve, the helix, which has a radius of a and rises by 2πb units per turn. (Note that the equations are identical in the plane to those for a circle; in fact, a helix is just "a circle whose ends don't have the same z-value".) Such expressions as the one above are commonly written as :

r(t) = (x(t), y(t), z(t)) = (a \cos(t), a \sin(t), b t)\,

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following such a parametrized path as: :

v(t) = r'(t) = (x'(t), y'(t), z'(t)) = (-a \sin(t), a \cos(t), b)\,

and the acceleration as: :

a(t) = r

(t) = (x(t), y(t), z(t)) = (-a \cos(t), -a \sin(t), 0)\, In general, a parametric curve is a function of one independent parameter (usually denoted t). Parametrized surfaces, of great use in such vector calculus applications as Stokes' theorem, are functions of two parameters, most commonly (s, t) or (u,v). An example of a parametrized surface is the (capless) cylinder given by :

r(u, v) = (x(u, v), y(u, v), z(u, v)) = (a \cos(u), a \sin(u), v)\,

Considering the equation as representing a circle in the plane, it is evident that this represents a cylinder. It is then allowed to take on arbitrary values of z. Category:Multivariate calculus Category:Equations

Slope

In mathematics, the slope or the gradient of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of the line relative to the horizontal axis. With an understanding of algebra and geometry, one can calculate the slope of a straight line; with calculus, one can calculate the slope of the tangent to a curve at a point. The concept of slope, and much of this article, applies directly to grades or gradients in geography and civil engineering.

Definition of slope

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: :

m = \frac

(The delta symbol, "Δ", is commonly used in mathematics to mean "difference" or "change".) Given two points (x₁, y₁) and (x₂, y₂), the change in x from one to the other is x₂ - x₁, while the change in y is y₂ - y₁. Substituting both quantities into the above equation obtains the following: :

m = \frac

Since the y-axis is vertical and the x-axis is horizontal by convention, the above equation is often memorized as "rise over run", where Δy is the "rise" and Δx is the "run". Therefore, by convention, m is equal to the change in y, the vertical coordinate, divided by the change in x, the horizontal coordinate; that is, m is the ratio of the changes. This concept is fundamental to algebra, analytic geometry, trigonometry, and calculus. Note that the points chosen and the order in which they are used is irrelevant; the same line will always have the same slope. Other curves have "accelerating" slopes and one can use calculus to determine such slopes.

Example 1

Suppose a line runs through two points: P(13,8) and Q(1,2). By dividing the difference in y-coordinates by the difference in x-coordinates, one can obtain the slope of the line: :

m = \frac = \frac = \frac = \frac = \frac

The slope is 1/2 = 0.5.

Example 2

If a line runs through the points (4, 15) and (3, 21) then: :

m = \frac = \frac = -6

Geometry

The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. The slope of a vertical line is not defined (it does not make sense to define it as +∞, because it might just as well be defined as -∞). The angle θ a line makes with the positive x axis is closely related to the slope m via the tangent function: :

m = \tan\,\theta

and :

\theta = \arctan\,m

(see trigonometry). Two lines are parallel if and only if their slopes are equal or if they both are vertical and therefore undefined; they are perpendicular (i.e. they form a right angle) if and only if the product of their slopes is -1 or one has a slope of 0 and the other is vertical and undefined.

Slope of a road, etc.

perpendicular There are two common ways to describe how steep a road is. One is by the angle in degrees, and the other is by the slope in a percentage. See also mountain railway. The formula for converting a slope in percentage to degrees is: :

\theta = \arctan\frac.

Algebra

If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form :

y = mx + b \,

then m is the slope. This form of a line's equation is called the slope-intercept form, because b can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the slope m of a line and a point (x₀, y₀) on the line are both known, then the equation of the line can be found using the point-slope formula: :

y - y_0 = m(x - x_0) \,

For example, consider a line running through the points (2, 8) and (3, 20). This line has a slope, m, of (20 - 8) / (3 - 2) = 12. One can then write the line's equation, in point-slope form: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16. The slope of a linear equation in the general form: :

Ax + By + C = 0 \,

is given by the formula: −A/B.

Calculus

The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point.

Why calculus is necessary

tangent If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, :

m = \frac

, is the slope of a secant line to a curve. For a line, the secant between any two points is identical to the line itself; however, this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x² at (0,0) and (3,9) is m = (9 - 0) / (3 - 0) = 3 (which happens to be the slope of the tangent at, and only at, x = 1.5). However, by moving the points used in the above formula closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve. It follows that the secant line is identical to the tangent line when Δy and Δx equal zero; however, this results in a slope of 0/0, which is an indeterminate form (see also division by zero). The concept of a limit is necessary to calculate this slope; the slope is the limit of Δy / Δx as Δy and Δx approach zero. However, Δx and Δy are interrelated such that it is sufficient to take the limit where only Δx approaches zero. This limit is the derivative of y with respect to x. It may be written (in calculus notation) as dy/dx.

Similarity (mathematics)

Several equivalence relations in mathematics are called similarity. The first one discussed below is about similar objects, the second one about similar matrices. The term "similarity transformation" has two different meanings, each related to one of the meanings of "similar". For similarity between people, see similarity (psychology).

Geometry

Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition. One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have :

d(f(x),f(y)) = r d(x,y), \,

where "d(x,y)" is the Euclidean distance from x to y. Two sets are called similar if one is the image of the other under such a similarity. A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image. A similarity is a composition of a homothety and an isometry. The 2D similarity transformations expressed in terms of the complex plane are

f(z)=az+b

and

f(z)=a\overline z+b

, and all affine transformations are of the form

f(z)=az+b\overline z+c

(a, b, and c complex). (This paragraph views the complex plane as a 2-dimensional space over the reals; over the complex field, it is 1-dimensional.) Self-similarity means that a pattern is non-trivially similar to itself, e.g., the set . When this set is plotted on a logarithmic scale is has translational symmetry.

Similar triangles

If triangle ABC is similar to triangle DEF, then this relation can be denoted as :

\triangle ABC \sim \triangle DEF

. In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°. Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following: :

= ,

= ,

= ,

=  = .

Angle/side similarities

A concept commonly taught in high school mathematics is that of proving the angle and side theorems, which can be used to define two triangles as similar (or indeed, congruent). In each of these three-letter acronyms, A stands for equal angles, and S for sides. For example, ASA denominates an angle, side and angle that are all equal and adjacent, in that order.
- AAA - Angle-Angle-Angle. If two triangles share three common angles, they are similar. (Obviously, this means that the side lengths are locked in a common ratio, but can vary proportionally, making the triangles similar.) . Additionally, knowing that that the interior angles of a triangle have a sum of 180 degrees, having two triangles with only two common angles (sometimes known as AA) implies similarity as well. See also Congruence (geometry).

Linear algebra

In linear algebra, two n-by-n matrices A and B over the field K are called similar if there exists an invertible n-by-n matrix P over K such that :P⁻¹AP = B. One of the meanings of the term "similarity transformation" is such a transformation of a matrix A into a matrix B. In group theory similarity is called conjugacy. Similar matrices share many properties: they have the same rank, the same determinant, the same trace, the same eigenvalues (but not necessarily the same eigenvectors), the same characteristic polynomial and the same minimal polynomial. There are two reasons for these facts:
- two similar matrices can be thought of as describing the same linear map, but with respect to different bases
- the map X

\mapsto

P⁻¹XP is an automorphism of the associative algebra of all n-by-n matrices Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A -- the study of A then reduces to the study of the simpler matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form. Another normal form, the rational canonical form, works over any field. By looking at the Jordan forms or rational canonical forms of A and B, one can immediately decide whether A and B are similar. Similarity of matrices does not depend on the base field: if L is a field containing K as a subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices over L. This is quite useful: one may safely enlarge the field K, for instance to get an algebraically closed field; Jordan forms can then be computed over the large field and can be used to determine whether the given matrices are similar over the small field. This approach can be used, for example, to show that every matrix is similar to its transpose. If in the definition of similarity, the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Another important equivalence relation for real matrices is congruency. Two real matrices A and B are called congruent if there is a regular real matrix P such that :P^TAP = B.

Topology

In topology, a metric space can be constructed by defining a similarity instead of a distance. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of disimilarity: the closer the points, the lesser the distance). The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are # Positive defined:

\forall (a,b), S(a,b)\geq 0

# Majored by the similarity of one element on itself (auto-similarity):

S (a,b) \leq S (a,a)

and

\forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b

More properties can be invoked, such as reflectivity (

\forall (a,b)\ S (a,b) = S (b,a)

) or finiteness (

\forall (a,b)\ S(a,b) < \infty

). The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

External links

- [http://www.cut-the-knot.org/Curriculum/Geometry/SimilarTriangles.shtml Three Similar Triangles] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/DirectlySimilar2.shtml Directly Similar Figures] at cut-the-knot
- [http://www.cut-the-knot.org/ctk/NapoleonPropeller.shtml Napoleon's Propeller] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/SimTri.shtml Two Triples of Similar Triangles] at cut-the-knot
- [http://www.geocities.com/bwakeel/similitudes.htm About similarities of Euclidean spaces] Category:Euclidean geometry Category:Linear algebra ja:相似

Area (geometry)

Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area. Depending on the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume.

How to define area

Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition: :The area of a polygon in the Euclidean plane is a positive number such that: #The area of the unit square is equal to one. #Congruent polygons have equal areas. #(additivity) If a polygon is a union of two polygons which do not have common interior points, then its area is the sum of the areas of these polygons. But before using this definition one has to prove that such an area indeed exists. In other words, one can also give a formula for the area of an arbitrary triangle, and then define the area of an arbitrary polygon using the idea that the area of a union of polygons (without common interior points) is the sum of the areas of its pieces. But then it is not easy to show that such area does not depend on the way you break the polygon into pieces. Nowadays, the most standard (correct) way to introduce area is through the more advanced notion of Lebesgue measure, but one should note that in general, if one adopts the axiom of choice then it is possible to prove that there are some shapes whose Lebesgue measure cannot be meaningfully defined. Such 'shapes' (they cannot a fortiori be simply visualised) enter into Tarski's circle-squaring problem (and, moving to three dimensions, in the Banach-Tarski paradox). The sets involved do not arise in practical matters. In three dimensions, the analog of area is called volume. The n dimensional analog is defined by means of a measure or as a Lebesgue integral; this is sometimes referred to as content.

Formula

content

Areas of 2-dimensional figures

- square or rectangle:

\ell w

(where l is the length and w is the width; in the case of a square, l = w).
- circle:

\pi r^2

(where r is the radius)
- ellipse:

\pi ab

(where a and b are the semi-major and semi-minor axes)
- any regular polygon:

\frac

(where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side])
- a parallelogram:

Bh

(where the base B is any side, and the height h is the distance between the lines that the sides of length B lie on)
- a trapezoid:

\frac

(B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
- a triangle:

\frac

(where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known. If the lengths of the three sides are known then Heron's formula can be used:

\sqrt

(where a, b, c are the sides of the triangle, and

s = \frac

is half of its perimeter)
- the area between the graphs of two functions is equal to the integral of one function, f(x), minus the integral of the other function, g(x).
- an area bounded by a function r = r(θ) expressed in polar coordinates is

\int_0^ r^2 \, d\theta

.
- the area enclosed by a parametric curve

\vec u(t) = (x(t), y(t))

with endpoints

\vec u(t_0) = \vec u(t_1)

is given by the path integrals ::

\oint_^ x \dot y \, dt  = - \oint_^ y \dot x \, dt  =   \oint_^ (x \dot y - y \dot x) \, dt

(see Green's theorem) :or the z-component of ::

\oint_^ \vec u \times \dot \vec u \, dt

Surface area of 3-dimensional figures

- cube:

6s^2

, where s is the length of any side
- rectangular box:

2 (\ell w + \ell  h + w h)

, where l, w, and h are the length, width, and height of the box
- sphere:

4 \pi  r^2

, where π is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere
- ellipsoid: see the article
- cylinder:

2\pi r (h + r)

, where r is the radius of the circular base, and h is the height
- cone:

\pi  r (r + \sqrt)

, where r is the radius of the circular base, and h is the height. Category:Euclidean plane geometry Category:Area

Mathematical constant

A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Unlike physical constants, mathematical constants are defined independently of any physical measurement. Apart from the obvious cases of zero and unity, many particular numbers have special significance in mathematics, and arise in many different contexts. For example, up to multiplication with nonzero complex numbers, there is a unique holomorphic function f with f = f. Therefore, f(1)/f(0) is a mathematical constant, the constant e. f is also a periodic function, and the absolute value of its period is another mathematical constant, 2π. Mathematical constants are typically elements of the field of real numbers or complex numbers. Mathematical constants that one can talk about are definable numbers (and almost always also computable). However, there are still some mathematical constants for which only very rough estimates are known. An alternate sorting may be found at Mathematical constant (sorted by continued fraction representation)
Table of selected mathematical constants
Abbreviations used: : I - irrational number, A - algebraic number, T - transcendental number, ? - unknown : Gen - General, NuT - Number theory, ChT - Chaos theory, Com - Combinatorics, Inf - Information theory, Ana - Mathematical analysis
See also

- An alternative sorting based on the continued fraction representations
- Physical constants
External links

- Steven Finch's page of mathematical constants: http://pauillac.inria.fr/algo/bsolve/constant/constant.html
- Steven Finch's alternative indexing: http://pauillac.inria.fr/algo/bsolve/constant/table.html
- Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms: http://numbers.computation.free.fr/Constants/constants.html
- Simon Plouffe's inverter: http://pi.lacim.uqam.ca/eng/
- CECM's Inverse symbolic calculator (ISC) (tells you how a given number can be constructed from mathematical constants): http://www.cecm.sfu.ca/projects/ISC/
- ko:수학 상수 ja:数学定数

Triangle (geometry)

:For alternative meanings, such as the musical instrument, see triangle (disambiguation). 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.

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 sides are of equal length. An isosceles triangle also has two equal internal angles.
- In a scalene triangle all sides have different lengths. The internal angles in a scalene triangle are all different.

Equilateral triangle	Isosceles triangle	Scalene triangle
Equilateral	Isosceles	Scalene

Triangles can also be classified according to the size of their largest internal angle, described below using degrees of arc.
- A right triangle (or right-angled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs of the triangle.
- An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
- An acute triangle has internal angles that are all smaller than 90° (three acute angles).

Right triangle	Obtuse triangle	Acute triangle
Right	Obtuse	Acute

Basic facts

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 which are investigated in trigonometry. In the remainder we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides. trigonometry In Euclidean geometry, the sum of the angles α + β + γ is equal to two right angles (180° or π radians). This allows determination of the third angle of any triangle as soon as two angles are known. trigonometry A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. If side C is the hypotenuse, we can write this as :

c^2 = a^2 + b^2 \,

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third—something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines: :

c^2 = a^2 + b^2 - 2ab \cos\gamma \,

which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known. The law of sines states :

\fraca=\fracb=\fracc=\frac1d

where d is the diameter of the circumcircle (the circle which passes through all three points of the triangle). The law of sines can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions. There are two special right triangles that appear commonly in geometry. The so-called "45-45-90 triangle" has angles with those angle measures and the ratio of its sides is :

;1:1:\sqrt

. The "30-60-90 triangle" has sides in the ratio of

1:\sqrt:2

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 triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. Menelaus' theorem A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above. Thales' theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. Thales' theorem 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. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse. The three vertices together with the orthocenter are said to form an orthocentric system. orthocentric system An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
orthocentric system A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. This is also the triangle's center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side. center of gravity The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.
excircle The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

Computing the area of a triangle

Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.

Using geometry

The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction. area To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is bh, the area of the given triangle must be ½bh. parallelogram

Using vectors

The area of a parallelogram can also be calculated by the use of vectors. If AB and AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |AB × AC|, the magnitude of the cross product of vectors AB and AC. |AB × AC| is also equal to |h × AC|, where h represents the altitude h as a vector. The area of triangle ABC is half of this, or S = ½|AB × AC|. cross product

Using trigonometry

The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ. It is of course no coincidence that the area of a parallelogram is ab sin γ.

Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x₁, y₁) and C = (x₂, y₂), then the area S can be computed as 1/2 times the absolute value of the determinant :

\beginx_1 & x_2 \\ y_1 & y_2 \end

or S = ½ |x₁y₂ − x₂y₁|.

Using Heron's formula

Yet another way to compute S is Heron's formula: :

S = \sqrt

where s = ½ (a + b + c) is the semiperimeter, or one half of the triangle's perimeter.

Using the side lengths and a numerically stable formula

Heron's formula is numerically unstable for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that: :a ≥ b ≥ c and computing :

S = \frac\sqrt

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Non-planar triangles

If any four of a triangle's elements (vertices, and/or elements of its sides) are plane to each other, the triangle is called plane. Geometers also study non-planar triangles in noneuclidean geometries, such as spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, he would find that its angles would be greater than 180°.

External links

- [http://ostermiller.org/calc/triangle.html Triangle Calculator] - solves for remaining sides and angles when given three sides or angles, supports degrees and radians.
- [http://agutie.homestead.com/files/Napoleon0.htm Napoleon's theorem] A triangle with three equilateral triangles. A purely geometric proof. It uses the Fermat point to prove Napoleon's theorem without transformations by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
- William Kahan: [http://http.cs.berkeley.edu/~wkahan/Triangle.pdf Miscalculating Area and Angles of a Needle-like Triangle].
- Clark Kimberling: [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]. Lists some 1600 interesting points associated with any triangle.
- Christian Obrecht: [http://perso.wanadoo.fr/obrecht/ Eukleides]. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry.
- [http://www.cut-the-knot.org/triangle Triangle constructions, remarkable points and lines, and metric relations in a triangle] at cut-the-knot
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1469&CurriculumID=4 Printable Worksheet on Types of Triangles]
- [http://www.vias.org/comp_geometry/geom_triangle.html Compendium Geometry] Analytical Geometry of Triangles Category:Polygons ko:삼각형 ja:三角形 th:รูปสามเหลี่ยม

Approximation

An approximation is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. Approximations may be used because incomplete information prevents use of exact representations. Alternately, even when the exact representation is known, it may be preferable to use an approximation that simplifies analysis without too great a cost in accuracy. For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behaviours — e.g. gravity — are much easier to calculate for a sphere than for less regular shapes. The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.

Science

The scientific method is carried out with a constant interaction between scientific laws (theory) and empirical measurements, which are constantly compared to one another. Approximation also refers to using a simpler process. This model is used to make predictions easier. The most common versions of philosophy of science accept that empirical measurements are always approximations — they do not perfectly represent what is being measured. The history of science indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements. This is the correspondence principle.

Mathematics

Numerical approximations sometimes result from using a small number of significant digits. Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximation to real numbers by rational numbers. The symbol "≈" means "approximately equal to".

Octagon

An octagon is a polygon that has eight sides.

Regular octagons

A Regular Octagon is an octagon with equal sides. The internal angle at each vertex of a regular octagon is 135°. The area of a regular octagon of side length a is given by :

A = 2a^2 \cot \frac = 2(1+\sqrt)a^2 \simeq 4.82843 a^2.

Construction

A regular octagon is constructible with straightedge and compass. To do so, follow steps 1 through 18 of the animation, noting that the compass radius is not altered during steps 7 though 10. straightedge and compass

Octagons in general use

The most common use of Octagons are for the shape of stop signs. In architecture, an octagon house is a house built in an octagonal shape. Examples include Thomas Jefferson's Poplar Forest, near Lynchburg, Virginia and the Tayloe House (now simply called the "Octagon House") in Washington, DC, designed by William Thornton for Colonel John Tayloe. In mixed martial arts, The Octagon is the name of the octagonal ring used in the Ultimate Fighting Championship, and by metonymy a term for the UFC itself as well. The Octagon is the name of the city centres of Bristol, England, and Dunedin, New Zealand. Oktogon is a major square in downtown Budapest, the crossing point of Andrássy Avenue and the Grand Boulevard.

External links

[http://mathcentral.uregina.ca/QQ/database/QQ.09.01/laurie2.html How to find the area of an octagon] Category:Polygons Category:Polygons ja:八角形 th:รูปแปดเหลี่ยม

Descartes' theorem

In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.

History

Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. Unfortunately the book, which was called On Tangencies, is not among his surviving work. René Descartes touched on the problem briefly in 1643, in a letter to Princess Elizabeth of Bohemia (as such things might go in those times). He came up with essentially the same solution as given in equation (1) below, and thus attached his name to the theorem. Frederick Soddy rediscovered the equation in 1936. The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature (June 20, 1936). Soddy also extended the theorem to spheres.

Definition of curvature

Nature Descartes' theorem is most easily stated in terms of the circles' curvature. The curvature of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa. The plus sign in k = ±1/r applies to a circle that is externally tangent to the other circles, like the three black circles in the image. For an internally tangent circle like the big red circle, that circumscribes the other circles, the minus sign applies. If a straight line is considered a degenerate circle with curvature k = 0, Descartes' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line.

Descartes' theorem

If four mutually tangent circles have curvature k_i (for i = 1…4), Descartes' theorem says:

(1)

(k_1+k_2+k_3+k_4)^2=2\,(k_1^2+k_2^2+k_3^2+k_4^2).

When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:

(2)

Circle

Mathematical definitions

Properties

Chord properties

Tangent properties

Inscribed angle theorem

Secant, tangent, and chord properties

See also

External links

Circle (disambiguation)

Euclidean geometry

Axiomatic approach

Modern introduction to Euclidean geometry

The construction

References

Classical theorems

See also

External links

Distance

Distance covered

Formal definition

The distance formula

Generalized distance in arbitrary dimensions: Norms

Distances in other spaces

See also

Plane (mathematics)

Euclidean geometry

Planes embedded in R3

Properties

Point and a normal vector

Plane through three points

The distance from a point to a plane

The line of intersection between two planes

The dihedral angle

See also

External links

Simple closed curve

Definitions

Conventions and terminology

Length of curves

Differential geometry

Algebraic curve

History

See also

External links

Disk (mathematics)

See also

Formula

Other meanings

Parametric equations

Slope

Definition of slope

Example 1

Example 2

Geometry

Slope of a road, etc.

Algebra

Calculus

Why calculus is necessary

See also

Similarity (mathematics)

Geometry

Similar triangles

Angle/side similarities

Linear algebra

Topology

See also

External links

Area (geometry)

How to define area

Formula

Areas of 2-dimensional figures

Surface area of 3-dimensional figures

Mathematical constant

Table of selected mathematical constants

See also

External links

Triangle (geometry)

Types of triangles

Basic facts

Planes embedded in R³