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Encyclopedia > Radius of curvature

The distance from the center of a sphere or ellipsoid to its surface is its radius. The equivalent "surface radius" that is described by radial distances at points along the body's surface is its radius of curvature (more formally, the radius of curvature of a curve at a point is the radius of the osculating circle at that point). With a sphere, the radius of curvature equals the radius. With an oblate ellipsoid (or, more properly, an oblate spheroid), however, not only does it differ from the radius, but it varies, depending on the direction being faced. The extremes are known as the principal radii of curvature. A sphere is a perfectly symmetrical geometrical object. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ... An osculating circle A circle with 4-point contact at a vertex of a curve In differential geometry, the osculating circle of a curve at a point, is a circle which: Touches the curve at that point Has its unit tangent vector , equal to the unit tangent of the curve... An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ...

Contents

Explanation in lay terms

Imagine driving a car on a curved road on a completely flat plain (so that the geographic plain is a geometric plane). At one point along the way, lock the steering wheel in its position, so that the car thereafter follows a perfect circle, possibly deviating from the road, which may be a more complicated curve than a circle. That circle is the osculating circle to the curve at the point at which the steering wheel was locked. The radius of that circle is the radius of curvature of the curved road at the point at which the steering wheel was locked: The more sharply curved, the smaller the radius of curvature. In geography, a plain is a large area of land with relatively low relief. ... An osculating circle A circle with 4-point contact at a vertex of a curve In differential geometry, the osculating circle of a curve at a point, is a circle which: Touches the curve at that point Has its unit tangent vector , equal to the unit tangent of the curve...


Formula

If we are dealing with a function of t f(t), then radius of curvature is:

rho(t)=frac{|1+f '(t)^2|^{3/2}}{|f ' '(t)|}

Elliptic, latitudinal components

The radius extremes of an oblate spheroid are the equatorial radius, or semi-major axis, a, and the polar radius, or semi-minor axis, b. The "ellipticalness" of any ellipsoid, like any ellipse, is measured in different ways (e.g., eccentricity and flattening), any and all of which are trigonometric functions of its angular eccentricity, o!varepsilon,!: The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... (This page refers to eccentricity in mathematics. ... The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ... In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ...

begin{matrix}{}_{color{white}.}o!varepsilon=arccos!left(frac{b}{a}right)=2arctan!left(sqrt{frac{a-b}{a+b}},right).{}^{color{white}.}end{matrix},!

The primary parameter utilized in identifying a point's vertical position is its latitude. A latitude can be expressed either directly or from the arcsine of a trigonometric product, the arguments (i.e., a function's "input") of the factors being the arc path (which defines, and is the azimuth at the equator of, a given great circle, or its elliptical counterpart) and the transverse colatitude, which is a corresponding, vertical latitude ring that defines a point along an arc path/great circle. The relationship can be remembered by the terms' initial letter, L-A-T: Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... In mathematical analysis, Clairauts theorem states that if has continuous second partial derivatives at then for In words, the partial derivatives of this function commute. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

sin(boldsymbol{L})=cos(boldsymbol{A})sin(boldsymbol{T}).,!

Therefore, along a north-south arc path (which equals 0°), the primary quadrant form of latitude equals the transverse colatitude's at a given point. As most introductory discussions of curvature and their radius identify position in terms of latitude, this article will too, with only the added inclusion of a "0" placeholder for more advanced discussions where the arc path is actively utilized: F(L)rightarrow F(0,L)=F(A,T).,! There are two types of latitude commonly employed in these discussions, the planetographic (or planetodetic; for Earth, the customized terms are "geographic" and "geodetic") and reduced latitudes, phi,! and beta! (respectively): Latitude,usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ...

begin{matrix}{}_{color{white}.}beta&=&arctan(cos(o!varepsilon)tan(phi));phi&=&arctan(sec(o!varepsilon)tan(beta)).{}^{color{white}.}end{matrix},!

The calculation of elliptic quantities usually involves different elliptic integrals, the most basic integrands being E'(0,L),! and its complement, C'(0,L),!: In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

begin{matrix}{}_{color{white}.}E'(0,phi)=C'(0,frac{pi}{2}-phi)!!&=&!!!!!!asqrt{1-(cos(0)sin(phi)sin(o!varepsilon))^2},qquad=quadfrac{ab}{C'(0,beta)}!!!&=&!!asqrt{cos(o!varepsilon)^2+(cos(phi)sin(o!varepsilon))^2},&=&!!!!!!!!sqrt{(acos(phi))^2+(bsin(phi))^2},qquad&=&!!!!!!!!absqrt{left(frac{sin(phi)}{a}right)^2+left(frac{cos(phi)}{b}right)^2};qquad{}^{color{white}.}end{matrix},!
begin{matrix}{}_{color{white}.}C'(0,beta)=E'(0,frac{pi}{2}-beta)!!&=&!!asqrt{cos(o!varepsilon)^2+(cos(0)sin(beta)sin(o!varepsilon))^2},qquad=quadfrac{ab}{E'(0,phi)}!!!&=&!!!!!!!asqrt{1-(cos(beta)sin(o!varepsilon))^2},qquadqquadquad&=&!!!!!!sqrt{(asin(beta))^2+(bcos(beta))^2},qquadqquadquad&=&!!!!!!!!absqrt{left(frac{cos(beta)}{a}right)^2+left(frac{sin(beta)}{b}right)^2};qquadqquadquad{}^{color{white}.}end{matrix},!

Thus E'(0,phi)C'(0,beta)=ab,!.


Curvature

A simple, if crude, definition of a circle is "a curved line bent in equal proportions, where its endpoints meet". Curvature, then, is the state and degree of deviation from a straight line—i.e., an "arced line". There are different interpretations of curvature, depending on such things as the planular angle the given arc is dividing and the direction being faced at the surface's point. What is concerned with here is normal curvature, where "normal" refers to orthogonality, or perpendicularity. There are two principal curvatures identified, a maximum, κ1, and a minimum, κ2. In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...


Meridional maximum

kappa_1=frac{E'(0,phi)^3}{(ab)^2}=frac{ab}{C'(0,beta)^3};,!
The arc in the meridional, north-south vertical direction at the planetographic equator possesses the maximum curvature, where it "pinches", thereby being the least straight.

Perpendicular minimum

kappa_2=frac{E'(0,phi)}{a^2}=frac{cos(o!varepsilon)}{C'(0,beta)};,!
The perpendicular, horizontally directed arc contains the least curvature at the equator, as the equatorial circumference is——at least in mathematical definition——perfectly circular.

The spot of least curvature on an oblate spheroid is at the poles, where the principal curvatures converge (as there is only one facing direction——towards the planetographic equator!) and the surface is most flattened.


Merged curvature

There are two universally recognized blendings of the principal curvatures: The arithmetic mean is known as the mean curvature, H, while the squared geometric mean——or simply the product——is known as the Gaussian curvature, K:
H=frac{kappa_1+kappa_2}{2};qquadKappa=kappa_1kappa_2;,!

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, mean curvature of a surface is a notion from differential geometry. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... Curvature is the amount by which a geometric object deviates from being flat. ...

Principal radii of curvature

A curvature's radius, RoC, is simply its reciprocal:

mathrm{RoC} = frac{1}mathrm{curvature};qquad mathrm{curvature} = frac{1}mathrm{RoC};,!

Therefore, there are two principal radii of curvature: A vertical, corresponding to κ1, and a horizontal, corresponding to κ2. Most introductions to the principal radii of curvature provide explanations independent to their curvature counterparts, focusing more on positioning and angle, rather than shape and contortion.


Meridional radius of curvature

The vertical radius of curvature is parallel to the "principal vertical", which is the facing, central meridian and is known as the meridional radius of curvature, M (alternatively, R1 or p):
M=M_v(0,phi)=;frac{(ab)^2}{E'(0,phi)^3};=frac{1}{kappa_1}=;frac{C'(0,beta)^3}{ab};=M_p(0,beta);,!
(Crossing the planetographic equator, {}_{M=bcos(o!varepsilon)=frac{b^2}{a}},!.}

Normal radius of curvature

The horizontal radius of curvature is perpendicular (again, meaning "normal" or "orthogonal") to the central meridian, but parallel to a great arc (be it spherical or elliptical) as it crosses the "prime vertical", or transverse equator (i.e., the meridian 90° away from the facing principal meridian——the "horizontal meridian"), and is known as the transverse (equatorial), or normal, radius of curvature, N (alternatively, R2 or v):
N=N_v(0,phi)=;frac{a^2}{E'(0,phi)};=frac{1}{kappa_2}=;frac{a}{b}C'(0,beta);=N_p(0,beta);,!
(Along the planetographic equator, which is an ellipsoid's
 only true great circle, {}_{N=bsec(o!varepsilon)=a},!.)

In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

Polar convergence

Just as with the curvature, at the poles M and N converge, resulting in an equal radius of curvature:
M=N=asec(o!varepsilon)=frac{a^2}{b}.,!

Merged radius of curvature

There are two possible, basic "means":
frac{M+N}{2}=frac{frac{1}{kappa_1}+frac{1}{kappa_2}}{2}=frac{M}{2}!cdot!left(1+frac{a^4}{(bN)^2}right)=frac{N}{2}!cdot!left(frac{(bN)^2}{a^4}+1right);,!
frac{2}{frac{1}{M}+frac{1}{N}}=frac{2}{kappa_1+kappa_2}=frac{1}{H}=frac{2M}{1+frac{(bN)^2}{a^4}}=frac{2N}{frac{a^4}{(bN)^2}+1}.,!
If these means are then arithmetically and harmonically averaged together, with the results reaveraged until the two averages converge, the result will be the arithmetic-harmonic mean, which equals the geometric mean and, in turn, equals the square root of the inverse of Gaussian curvature!
sqrt{M!N}=sqrt{frac{1}{Kappa}}=sqrt{frac{1}{kappa_1kappa_2}}=frac{b}{a^2}N^2;,!
While, at first glance, the squared form may be regarded as either the "radius of Gaussian curvature", "radius of Gaussian curvature2" or "radius2 of Gaussian Curvature", none of these terms quite fit, as Gaussian Curvature is the product of two curvatures, rather than a singular curvature.

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

Applications and examples

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Important points in intrinsic coordinates Intrinsic coordinates is a coordinate system which defines points upon a curve partly by the nature of the tangents to the curve at that point. ...

See also

The term radius of curvature has specific meaning and sign convention in optical design. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ... Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, or hose without kinking, damaging it, or shortening its life. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... DIAMETER is an AAA protocol (Authentication, Authorization and Accounting) succeeding its predecessor RADIUS. // The name is a pun on the RADIUS protocol, which is the predecessor (a diameter is twice the radius). ...

External links


  Results from FactBites:
 
Curvature - Wikipedia, the free encyclopedia (1206 words)
There is a key distinction between extrinsic curvature which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature which is defined at each point in a differential manifold.
For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
Radius - Wikipedia, the free encyclopedia (159 words)
In classical geometry, a radius of a circle or sphere is any line segment from its center to its boundary.
More generally—in geometry, engineering, graph theory, and many other contexts—the radius of something (e.g., a cylinder, a polygon, a graph, or a mechanical part) is the distance from its center or axis of symmetry to its outermost points.
The relationship between the radius and the circumference of a circle is
  More results at FactBites »

 
 

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