Radius of curvature (applications)

Radius of curvature (applications)
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Radius_of_curvature_(applications)".

Radius of curvature is sometimes used as a synonym for radius.

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1 Explanation
2 Formula
- 2.1 Derivation
3 Elliptic, latitudinal components
4 Curvature
5 Principal radii of curvature
6 Applications and examples
7 See also
8 External links

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.

Explanation

Imagine driving a car on a curvy road on a completely flat plain (so that the geographic plain is a geometric plane). At any one point along the way, lock the steering wheel in its position, so that the car thereafter follows a perfect circle. The car will, or course, deviate from the road, unless the road is also a perfect circle. The radius of that circle the car makes is the radius of curvature of the curvy road at the point at which the steering wheel was locked. The more sharply curved the road is at the point you locked the steering wheel, the smaller the radius of curvature.

Formula

If $\gamma : \mathbb{R} \rightarrow \mathbb{R}^n$ is a parameterized curve in $\mathbb{R}^n$ then the radius of curvature at each point of the curve, $\rho : \mathbb{R} \rightarrow \mathbb{R}$ , is given by

$\rho = \frac{|\gamma'|^3}{\sqrt{|\gamma'|^2 \; |\gamma''|^2 - (\gamma' \cdot \gamma'')^2}}$ .

As a special case, if f(t) is a function from $\mathbb{R}$ to $\mathbb{R}$ , then the curvature of its graph, $γ(t) = (t, f (t))$ , is

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

Derivation

Let $γ$ be as above, and fix $t$ . We want to find the radius $ρ$ of a parameterized circle which matches $γ$ in its zeroth, first, and second derivatives at $t$ . Clearly the radius will not depend on the position ( $γ(t)$ ), only on the velocity ( $γ'(t)$ ) and acceleration ( $γ''(t)$ ). There are only three independent scalars that can be obtained from two vectors v and w, namely v·v, v·w, and w·w. Thus the radius of curvature must be a function of the three scalars $|\gamma'^2(t)|\,\!$ , $|\gamma''^2(t)|\,\!$ and $\gamma'(t) \cdot \gamma''(t)$ .

The general equation for a parameterized circle in $\mathbb{R}^n$ is

$g(u) = A \cos(h(u)) + B \sin(h(u)) + C \,\!$

where $C \in \mathbb{R}^n$ is the center of the circle (irrelevant since it disappears in the derivatives), $A,B \in \mathbb{R}^n$ are perpendicular vectors of length $ρ$ (that is, $A\cdot A = B\cdot B = \rho^2$ and $A\cdot B = 0$ ), and $h : \mathbb{R} \rightarrow \mathbb{R}$ is an arbitrary function which is twice differentiable at t.

The relevant derivatives of g work out to be

$\begin{array}{lll} |g'|^2 &=& \rho^2 (h')^2 \\ g' \cdot g'' &=& \rho^2 h' h'' \\ |g''|^2 &=& \rho^2 \left( (h')^4 + (h'')^2 \right) \end{array}$

If we now equate these derivatives of g to the corresponding derivatives of $γ$ at t we obtain

$\begin{array}{lll} |\gamma'^2(t)| &=& \rho^2 h'^2(t) \\ \gamma'(t) \cdot \gamma''(t) &=& \rho^2 h'(t) h''(t) \\ |\gamma''^2(t)| &=& \rho^2 (h'^4(t) + h''^2(t)) \end{array}$

These three equations in three unknowns ( $ρ$ , $h'(t)$ and $h''(t)$ ) can be solved for $ρ$ , giving the formula for the radius of curvature:

$\rho(t) = \frac{|\gamma'^3(t)|}{\sqrt{|\gamma'^2(t)| \; |\gamma''^2(t)| - (\gamma'(t) \cdot \gamma''(t))^2}}$

or, omitting the parameter (t) for readability,

$\rho = \frac{|\gamma'|^3}{\sqrt{|\gamma'|^2 \; |\gamma''|^2 - (\gamma' \cdot \gamma'')^2}}$ .

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, $o\!\varepsilon\,\!$ :

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

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:

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

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):

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

The calculation of elliptic quantities usually involves different elliptic integrals, the most basic integrands being $E'(0,L)\,\!$ and its complement, $C'(0,L)\,\!$ :

$\begin{matrix}{}_{\color{white}.}\\E'(0,\phi)=C'(0,\frac{\pi}{2}-\phi)\!\!&=&\!\!\!\!\!\!a\sqrt{1-(\cos(0)\sin(\phi)\sin(o\!\varepsilon))^2},\\\\\qquad=\quad\frac{ab}{C'(0,\beta)}\!\!\!&=&\!\!a\sqrt{\cos^2(o\!\varepsilon)+(\cos(\phi)\sin(o\!\varepsilon))^2},\\\\&=&\!\!\!\!\!\!\!\!\sqrt{(a\cos(\phi))^2+(b\sin(\phi))^2},\qquad\\\\&=&\!\!\!\!\!\!\!\!ab\sqrt{\left(\frac{\sin(\phi)}{a}\right)^2+\left(\frac{\cos(\phi)}{b}\right)^2};\qquad\\{}^{\color{white}.}\end{matrix}\,\!$

$\begin{matrix}{}_{\color{white}.}\\C'(0,\beta)=E'(0,\frac{\pi}{2}-\beta)\!\!&=&\!\!a\sqrt{\cos^2(o\!\varepsilon)+(\cos(0)\sin(\beta)\sin(o\!\varepsilon))^2},\\\\\qquad=\quad\frac{ab}{E'(0,\phi)}\!\!\!&=&\!\!\!\!\!\!\!a\sqrt{1-(\cos(\beta)\sin(o\!\varepsilon))^2},\qquad\qquad\quad\\\\&=&\!\!\!\!\!\!\sqrt{(a\sin(\beta))^2+(b\cos(\beta))^2},\qquad\qquad\quad\\\\&=&\!\!\!\!\!\!\!\!ab\sqrt{\left(\frac{\cos(\beta)}{a}\right)^2+\left(\frac{\sin(\beta)}{b}\right)^2};\qquad\qquad\quad\\{}^{\color{white}.}\end{matrix}\,\!$

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, κ_{_₁}, and a minimum, κ_{_₂}.

Meridional maximum

$\kappa_1=\frac{E'^3(0,\phi)}{(ab)^2}=\frac{ab}{C'^3(0,\beta)};\,\!$

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:

$H=\frac{\kappa_1+\kappa_2}{2};\qquad\Kappa=\kappa_1\kappa_2;\,\!$

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 κ_{_₁}, and a horizontal, corresponding to κ_{_₂}. 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):

$M=\widehat{M}(0,\phi)=\;\frac{(ab)^2}{E'^3(0,\phi)}\;=\frac{1}{\kappa_1}=\;\frac{C'^3(0,\beta)}{ab}\;=\tilde{M}(0,\beta);\,\!$

(Crossing the planetographic equator, ${}_{M=b\cos(o\!\varepsilon)=\frac{b^2}{a}}\,\!$ .}

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):

$N=\widehat{N}(0,\phi)=\;\frac{a^2}{E'(0,\phi)}\;=\frac{1}{\kappa_2}=\;\frac{a}{b}C'(0,\beta)\;=\tilde{N}(0,\beta);\,\!$

(Along the planetographic equator, which is an ellipsoid's

only true great circle, ${}_{N=b\sec(o\!\varepsilon)=a}\,\!$ .)

Polar convergence

Just as with the curvature, at the poles M and N converge, resulting in an equal radius of curvature:

$M=N=a\sec(o\!\varepsilon)=\frac{a^2}{b}.\,\!$

Merged radius of curvature

There are two possible, basic "means":

Mean radius of curvature, which is the arithmetic mean:

$\frac{M+N}{2}=\frac{\frac{1}{\kappa_1}+\frac{1}{\kappa_2}}{2}=\frac{M}{2}\!\cdot\!\left(1+\frac{a^4}{(bN)^2}\right)=\frac{N}{2}\!\cdot\!\left(\frac{(bN)^2}{a^4}+1\right);\,\!$

Radius of mean curvature, which is the harmonic mean:

$\frac{2}{\frac{1}{M}+\frac{1}{N}}=\frac{2}{\kappa_1+\kappa_2}=\frac{1}{H}=\frac{2M}{1+\frac{(bN)^2}{a^4}}=\frac{2N}{\frac{a^4}{(bN)^2}+1}.\,\!$

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!

$\sqrt{M\!N}=\sqrt{\frac{1}{\Kappa}}=\sqrt{\frac{1}{\kappa_1\kappa_2}}=\frac{b}{a^2}N^2;\,\!$

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.

Applications and examples

For the use in differential geometry, see Cesàro equation.

Radius of curvature is also used in a three part equation for bending of beams.

External links

USIGS Glossary (Definitions of "transverse" terms)
The Geometry Center: Principal Curvatures
15.3 Curvature and Radius of Curvature
MathWorld: Principal Curvatures
MathWorld: Principal Radius of Curvature
The History of Curvature

v • d • e Different notions of curvature defined in differential geometry

Differential geometry of curves	Curvature · Torsion of curve · Frenet-Serret formulas · Radius of curvature (applications) · Affine curvature · Total curvature

Differential geometry of surfaces	Principal curvatures · Gaussian curvature · Mean curvature · Darboux frame · Gauss–Codazzi equations · First fundamental form · Second fundamental form

Riemannian geometry	Curvature of Riemannian manifolds · Riemann curvature tensor · Ricci curvature · Scalar curvature · Sectional curvature

Curvature of connections	Curvature form · Torsion tensor · Cocurvature · Holonomy

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