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Conic Section

Conic section

In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Types of conics

Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.) The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line, or a pair of intersecting lines, are often excluded from the list of conic sections. In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form :

ax^2 + 2hxy + by^2 +2gx + 2fy + c = 0\;

quadratic equation then:
- if h² = ab, the equation represents a parabola;
- if h² < ab and a

\ne

b and/or h

\ne

0 , the equation represents an ellipse;
- if h² > ab, the equation represents a hyperbola;
- if h² < ab and a = b and h = 0, the equation represents a circle;
- if a + b = 0, the equation represents a rectangular hyperbola.

Eccentricity

An alternative definition of conic sections starts with a point F (the focus), a line L (the directrix) not containing F and a positive number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is

\over

, where

a

is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is

ae

. In the case of a circle e = 0 and one imagines the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity. The eccentricity of a conic section is thus a measure of how far it deviates from being circular. For a given

a

, the closer

e

is to 1, the smaller is the semi-minor axis.

Semi-latus rectum and polar coordinates

semi-minor axis The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to a and b by the formula

al=b^2\,\!

, or

l=a(1-e^2)\,\!

. In polar coordinates, a conic section with one focus at the origin and, if any, the other on the positive x-axis, is given by the equation :

r (1 - e \cos \theta) = l\,\!

Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow/prevent turbulence.

Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola.

Dandelin spheres

See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix.

Derivation

Let there be a cone whose axis is the z-axis. Let its vertex be the origin. The equation for the cone is :

x^2 + y^2 - a^2 z^2 = 0 \qquad \qquad (1)

where :

a = \tan \theta > 0 \;

and

\theta

is the angle which the generators of the cone make with respect to the axis. Notice that this cone is actually a pair of cones: one cone standing upside down on the vertex of the other cone—or, as mathematicians say, this cone consists of two "nappes." Let there be a plane with a slope running along the x direction but which is level along the y direction. Its equation is :

z = mx + b \qquad \qquad (2)

where :

m = \tan \phi > 0 \;

and

\phi

is the angle of the plane with respect to the x-y plane. We are interested in finding the intersection of the cone and the plane, which means that equations (1) and (2) shall be combined. Both equations can be solved for z and then equate the two values of z. Solving equation (1) for z yields :

z = \sqrt

therefore :

\sqrt = m x + b.

Square both sides and expand the squared binomial on the right side, :

= m^2 x^2 + 2 m b x + b^2. \;

Grouping by variables yields :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0. \qquad \qquad (3)

Note that this is the equation of the projection of the conic section on the xy-plane, hence contracted in the x-direction compared with the shape of the conic section itself.

Derivation of the parabola

The parabola is obtained when the slope of the plane is equal to the slope of the generators of the cone. When these two slopes are equal, then the angles

\theta

and

\phi

become complementary. This implies that :

\tan \theta = \cot \phi \;

therefore :

m = . \qquad \qquad (4)

Substituting equation (4) into equation (3) makes the first term in equation (3) vanish, and the remaining equation is :

-  b x - b^2 = 0.

Multiply both sides by a², :

y^2 - 2 a b x - a^2 b^2 = 0 \;

then solve for x, :

x =  y^2 - . \qquad \qquad (5)

Equation (5) describes a parabola whose axis is parallel to the x-axis. Other versions of equation (5) can be obtained by rotating the plane around the z-axis.

Derivation of the ellipse

An ellipse happens when the angles

\theta

and

\phi

, when added together, do not measure up to a right angle: :

\theta + \phi <  \qquad \qquad \mbox

which implies that the tangent of the sum of these two angles is positive. :

\tan (\theta + \phi) > 0. \;

But a trigonometric identity states that :

\tan (\theta + \phi) =

therefore :

\tan (\theta + \phi) =  > 0 \qquad \qquad (6)

but m + a is positive, since the summands are given to be positive, so inequality (6) is positive if the denominator is also positive: :

1 - m a > 0. \qquad \qquad (7)

From inequality (7) we can deduce :

m a < 1, \;

m^2 a^2 < 1, \;

1 - m^2 a^2 > 0, \;

> 1,

- 1 > 0,

- m^2 > 0 \qquad \qquad \mbox.

Let us start out again from equation (3), :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0, \qquad \qquad (3)

but this time the coefficient of the x² term does not vanish but is instead positive. Solve for y, :

y = a \sqrt. \qquad \qquad (8)

This would clearly describe an ellipse were it not for the second term under the radical, the 2 m b x: it would be the equation of a circle which has been stretched proportionally along the directions of the x-axis and the y-axis. Equation (8) is an ellipse but it is not obvious, so it will be rearranged further until it is obvious. Complete the square under the radical, :

y = a \sqrt.

Group together the b² terms, :

y = a \sqrt.

Divide by a then square both sides, :

+ \left( x \sqrt -  \right)^2 = b^2 \left( 1 +  \right).

The x has a coefficient. It is desired to pull this coefficient out by factoring it out of the second term which is a square, :

+ \left(  - m^2 \right) \left( x -  \right)^2 = b^2 \left( 1 +  \right).

Further rearrangements of constants finally leads to :

+ \left( x -  \right)^2 = .

The coefficient of the y term is positive (for an ellipse). Renaming of coefficients and constants leads to :

+ (x - C)^2 = R^2 \qquad \qquad (9)

which is clearly the equation of an ellipse. That is, equation (9) describes a circle of radius R and center (C,0) which is then stretched vertically by a factor of

\sqrt

. The second term on the left side (the x term) has no coefficient but is a square, so that it must be positive. The radius is a product of squares, so it must also be positive. The first term on the left side(the y term) has a coefficient which is positive (one of the inequalities derived earlier), so the equation describes an ellipse.

Derivation of the hyperbola

The hyperbola happens when the angles

\theta

and

\phi

add up to an obtuse angle, which is greater than a right angle. The tangent of an obtuse angle is negative. All the inequalities which were valid for the ellipse become reversed. Therefore :

1 - a^2 m^2 < 0 \qquad \qquad \mbox.

Otherwise the equation for the hyperbola is the same as equation (9) for the ellipse, except that the coefficient A of the y term is negative. The sign change is enough to convert an ellipse into a hyperbola. This is because the equation of a real ellipse contains an imaginary hyperbola, and the equation of a real hyperbola contains an imaginary ellipse (see imaginary number). The sign change of coefficient A causes real and imaginary values of the function y=f(x) equivalent to equation (9) to swap.

External links

- [http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html Special plane curves: Conic sections]
- http://mathworld.wolfram.com/Focus.html
- [http://ccins.camosun.bc.ca/~jbritton/jbconics.htm Occurrence of the conics] in nature and elsewhere
- [http://fishrock.com/conics/default.htm Conic structures] in architecture
- Category:Euclidean solid geometry ja:円錐曲線

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

Point (geometry)

A spatial point is an entity with a location in space but no extent (volume, area or length). In geometry, a point therefore captures the notion of location; no further information is captured. Points are used in the basic language of geometry, physics, vector graphics (both 2d and 3d), and many other fields. In mathematics generally, particularly in topology, any form of space is considered as made up of points as basic elements.

Points in Euclidean geometry

A point in Euclidean geometry has no size, orientation, or any other feature except position. Euclid's axioms or postulates assert in some cases that points exist: for example, they assert that if two lines on a plane are not parallel, there is exactly one point that lies on both of them. Euclid sometimes implicitly assumed facts that did not follow from the axioms (for example about the ordering of points on lines, and occasionally about the existence of points distinct from a finite list of points). Therefore the traditional axiomatization of point was not entirely complete and definitive.

Points in Cartesian geometry

Intuitively one can understand a location in the Cartesian 3D space. This location could be described with three real number coordinates: for instance :P = (2, 6, 9). But one can also describe points in 1, 2 or more than 3 dimensions. The description of a point is quite similar to the description of a spatial vector, which also can exist in space with dimensions from one to many. The conceptual difference between these notions is significant, though: a point indicates a location, while a vector indicates a direction and length. If a distinguished point (the origin) is given, one can describe a location by giving the direction and distance from the origin to that point. One could argue that in this world it makes no sense to say that a point is in a one or two dimensional space, because we experience space in 3 dimensions, where one or two dimensions exists within this space, thus forcing 1d and 2d points to actually be 3d points. This way one could say that the only real spatial points are 3d points. And one could also argue that by giving more than 3 coordinates one starts to describe features which are not related to space (how would you describe the fourth dimension in spatial terms?) This is really a question about what we mean by space.

Points in differential geometry

:to be written Here is where the difference between points and vector becomes obvious; here is where the atomic nature of points becomes clear. Category:Elementary geometry ko:점 (기하) ja:点

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

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

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix

A = PDP^T

, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :

\frac + \frac = 1

The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted

l\,\!

(lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to

a\,\!

and

b\,\!

(the ellipse's semi-axes) by the formula

al=b^2\,\!

or, if using the eccentricity,

l=a(1-e^2)\,\!

. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation :

r (1 + e \cos \theta) = l \,\!

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

Closed 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

Conventions and terminology

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

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

History

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

Hyperbola

:For hyperbole, the figure of speech, see hyperbole. hyperbole In mathematics, a hyperbola is a type of conic section (literally: 'exaggeration' from the Greek ) defined as the intersection between a cone and a plane which cuts through both halves of the cone. It may also be defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant. For a simple geometric proof that the two characterizations above are equivalent to each other, see Dandelin spheres. Algebraically, a hyperbola is a curve in the Cartesian plane defined by an equation of the form :

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 > 4 AC

, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the hyperbola, exists.

Definitions

- It can also be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola. These foci lie on the transverse axis and their midpoint is called the center. A hyperbola comprises two disconnected curves called its arms which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes. A hyperbola has the property that a ray originating at one of the foci is reflected in such a way as to appear to have originated at the other focus. reflectedA special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant. Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola. If on the hyperbola equation one switches x and y, the conjugate hyperbola is obtained. A hyperbola and its conjugate have the same asymptotes.

Equations

Cartesian

(center (h, k) ) :

\frac - \frac = 1

\frac - \frac = 1

In both formulas a is called the semi-major axis; it is half the distance between the two branches; b is called the semi-minor axis. Note that b can be larger than a! The reason for this is because the values of a and b do not dictate which way the hyperbola opens. It is the order in which y and x are subtracted. If y is positive, then the hyperbola opens up and down. If x is positive, then the hyperbola opens left and right. The eccentricity is given by :

e = \sqrt

For rectangular hyperbolas with the coordinate axes parallel to their asymptotes: :

(x-h)(y-k) = c \,

Polar

r^2 =\ \ \,  a\,\sec 2t

r^2 =    -a\,\sec 2t

r^2 =\ \ \, a\,\csc 2t

r^2 =    -a\,\csc 2t

Parametric

x = a\,\cosh \theta;\; y = b\,\sinh \theta

x = a\,\tan \theta;\ \ y = b\,\sec \theta

External links

-
-
-
- [http://mathworld.wolfram.com/Hyperbola.html Mathworld - Hyperbola] Category:Conic sections ja:双曲線

Straight line

A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve is not always a line. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection between the points. Three or more points that lie on the same line are called collinear. Two different lines can intersect in at most one point; two different planes can intersect in at most one line. This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. "Everything that satisfies the axioms for a line is a line." While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development. In Euclidean space Rⁿ (and analogously in all other vector spaces), we define a line L as a subset of the form :

L = \

where a and b are given vectors in Rⁿ with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line. In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines. In R², every line L is described by a linear equation of the form :

L=\

with fixed real coefficients a, b and c such that a and b are not both zero (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity. More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology. The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.

Line segment

In mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end points are both vertices of a polygon, the line segment is either an edge (of that polygon) if they are adjacent vertices, or otherwise a diagonal. The midpoint of a line segment is its 'middle' point: the unique point at an equal distance from the two end points.

Ray

In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----
- ---> A B C In geometric optics a ray or a (light) beam is a line or curve that describes the direction in which light or other electromagnetic radiation is propagated. The ray is perpendicular to the wavefront in wave optics. In most media, light rays are straight lines. Light passing from one medium to another undergoes refraction or total internal reflection following Snell's law.

External links

- [http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml Equations of the Straight Line] at cut-the-knot
- [http://mathworld.wolfram.com/Line.html Rigorous definition of a line] Category:Elementary geometry ja:線分 ja:直線 simple:Line ja:半直線

Ellipse

A x^2 + B xy + C y^2 + D x + E y + F = 0

such that

B^2 < 4 AC

A = PDP^T

Parametrisation

\frac + \frac = 1

x = a\,\cos t

y = b\,\sin t

0 \leq t < 2\pi

\frac + \frac = 1

where (h,k) is the center. A Gauss-mapped form: :

\left(\frac,\frac\right)

has normal

(\cos\phi,\sin\phi)

Eccentricity

e = \sqrt

or where

c

(the linear eccentricity of the ellipse) equals the distance from the center to either focus :

e = \frac

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted

l\,\!

(lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to

a\,\!

and

b\,\!

(the ellipse's semi-axes) by the formula

al=b^2\,\!

or, if using the eccentricity,

l=a(1-e^2)\,\!

. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation :

r (1 + e \cos \theta) = l \,\!

Area

The area enclosed by an ellipse is

\pi ab\,\!

, where

\pi

is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :

c = 2\pi a \left[\right]\!\,

A good approximation is Ramanujan's: :

c \approx \pi \left[3(a+b) - \sqrt\right]\!\,

which can also be written as: :

c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\,

Stretching and Projection

Reflection property

Ellipses in physics

Ellipses in computer graphics

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.

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

2\pi \times r.

- Area of a circle =

\pi \times r^2.

Properties

limit limit

Chord properties

Tangent properties

Inscribed angle theorem

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

External links

Focus (geometry)

In geometry, the focus (pl. foci) is a special point used in describing conic sections. A conic section can be defined as the set of points whose distance to its focus is equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section. Note that (non-circular) ellipses and hyperbolas each have a pair of foci. An ellipse can be described as the set of points for which the sum of the distances to the foci is constant, while a hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is constant. In the gravitational two-body problem, the orbits of the two bodies are described by conic sections with foci at the center of mass. Category:Conic sections ja:焦点

Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an ellipse is greater than zero and less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 and less than infinity.
- The eccentricity of a straight line is infinity. It is given by: :

e = \sqrt

Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola. It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as: :

e' = \sqrt

And is related to the first eccentricity by the equation: :

1 = (1 - e^2)(1 + e'^2)\,\!

Ellipse

semiminor axis For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by: :

e = \sqrt

The eccentricity is the ratio of the distance between the foci (

F_1

and

F_2

) to the major axis; i.e.

\left ( \frac \right )

. The term linear eccentricity is used for

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by: :

e = \sqrt

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links

- [http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Category:Conic sections als:Exzentrizität (Mathematik)

Semi-minor axis

In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas.

Ellipse

The semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci, and to the edge of the ellipse. The minor axis is the longest line that runs perpendicular to the major axis. It is related to the semi-major axis

a

through the eccentricity

e

and the semi-latus rectum

l

, as follows: :

b = a \sqrt

al=b^2

. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping l fixed. Thus a and b tend to infinity, a faster than b.

Hyperbola

The semi-minor axis of a hyperbola is the distance from a top, along the tangent line, to each asymptote; if this is in the y-direction it is b in this equation of the hyperbola:

\frac - \frac = 1

It is related to the semi-major axis through the eccentricity, as follows: :

b = a \sqrt

Note that in a hyperbola b can be larger than a! Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Coordinates (elementary mathematics)

This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space. A coordinate system is a plane or space where the origin and axes are defined so that coordinates can be measured.

Cartesian coordinates

Image:cartesiancoordinates2D.JPG In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components

(x, y)

.
-

x

is the signed distance from the y-axis to the point P, and
-

y

is the signed distance from the x-axis to the point P. In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represent by a tuple of three components

(x, y, z)

.
-

x

is the sign

Conic section

Types of conics

Eccentricity

Semi-latus rectum and polar coordinates

Properties

Applications

Dandelin spheres

Derivation

Derivation of the parabola

Derivation of the ellipse

Derivation of the hyperbola

See also

External links

Curve

Definitions

Conventions and terminology

Length of curves

Differential geometry

Algebraic curve

History

See also

External links

Point (geometry)

Points in Euclidean geometry

Points in Cartesian geometry

Points in differential geometry

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

Circle

Mathematical definitions

Properties

Chord properties

Tangent properties

Inscribed angle theorem

Secant, tangent, and chord properties

See also

External links

Ellipse

Parametrisation

Eccentricity

Semi-latus rectum and polar coordinates

Area

Circumference

Stretching and Projection

Reflection property

Ellipses in physics

Ellipses in computer graphics

See also

Closed curve

Definitions

Conventions and terminology

Length of curves

Differential geometry

Algebraic curve

History

See also

External links

Hyperbola

Definitions

Equations

Cartesian

Polar

Parametric

See also

External links

Straight line

Line segment

Ray

See also

External links

Ellipse

Planes embedded in R³