radius of curvature
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In
differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
, the radius of curvature, , is the reciprocal of the
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

curvature
. For a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
, it equals the
radius In classical geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties ...

radius
of the
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
which best approximates the curve at that point. For
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
s, the radius of curvature is the radius of a circle that best fits a normal section or
combinations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
thereof.


Definition

In the case of a
space curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

space curve
, the radius of curvature is the length of the
curvature vector Differential geometry of curves is the branch of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
. In the case of a
plane curve In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought o ...
, then is the
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

absolute value
of : R \equiv \left, \frac \ = \frac, where is the
arc length Arc length is the distance between two points along a section of a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. In ...

arc length
from a fixed point on the curve, is the tangential angle and is the
curvature In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

curvature
.


Formula


In 2D

If the curve is given in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
as , i.e., as the
graph of a function In mathematics, the graph of a Function (mathematics), function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space ...

graph of a function
, then the radius of curvature is (assuming the curve is differentiable up to order 2): : R =\left, \frac \, \qquad\mbox\quad y' = \frac,\quad y'' = \frac, and denotes the absolute value of . If the curve is given parametrically by functions and , then the radius of curvature is :R = \left, \frac\ = \left, \frac \, \qquad\mbox\quad \dot = \frac,\quad\ddot = \frac,\quad \dot = \frac,\quad\ddot = \frac. Heuristically, this result can be interpreted as : R = \frac, \qquad\mbox\quad \left, \mathbf \ = \big, (\dot x, \dot y) \big, = R \frac.


In n dimensions

If is a parametrized curve in then the radius of curvature at each point of the curve, , is given by :\rho = \frac. As a special case, if is a function from to , then the radius of curvature of its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
, , is :\rho(t)=\frac.


Derivation

Let be as above, and fix . We want to find the radius of a parametrized circle which matches in its zeroth, first, and second derivatives at . Clearly the radius will not depend on the position , only on the velocity and acceleration . There are only three independent scalars that can be obtained from two vectors and , namely , , and . Thus the radius of curvature must be a function of the three scalars , and . The general equation for a parametrized circle in is :\mathbf(u) = \mathbf a \cos h(u) + \mathbf b \sin h(u) + \mathbf c where is the center of the circle (irrelevant since it disappears in the derivatives), are perpendicular vectors of length (that is, and ), and is an arbitrary function which is twice differentiable at . The relevant derivatives of work out to be :\begin , \mathbf g', ^2 &= \rho^2 (h')^2 \\ \mathbf g' \cdot \mathbf g'' &= \rho^2 h' h'' \\ , \mathbf g'', ^2 &= \rho^2 \left((h')^4 + (h'')^2 \right) \end If we now equate these derivatives of to the corresponding derivatives of at we obtain :\begin , \boldsymbol\gamma'(t), ^ &= \rho^2 h'^(t) \\ \boldsymbol\gamma'(t) \cdot \boldsymbol\gamma''(t) &= \rho^2 h'(t) h''(t) \\ , \boldsymbol\gamma''(t), ^ &= \rho^2 \left(h'^(t) + h''^(t)\right) \end These three equations in three unknowns (, and ) can be solved for , giving the formula for the radius of curvature: :\rho(t) = \frac or, omitting the parameter for readability, :\rho = \frac.


Examples


Semicircles and circles

For a
semi-circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of radius in the upper half-plane :y=\sqrt, \quad y'=\frac, \quad y''=\frac,\quad R=, -a, =a. For a semi-circle of radius in the lower half-plane : y=-\sqrt, \quad R=, a, =a. The
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
of radius has a radius of curvature equal to .


Ellipses

In an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
with major axis and minor axis , the vertices on the major axis have the smallest radius of curvature of any points, ; and the vertices on the minor axis have the largest radius of curvature of any points, . The ellipse's radius of curvature, as a function of θR(\theta)=\sqrt


Applications

*For the use in
differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ...
, see Cesàro equation. *For the radius of curvature of the earth (approximated by an oblate ellipsoid); see also: arc measurement *Radius of curvature is also used in a three part equation for bending of beams. *
Radius of curvature (optics) Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis 300px, Optical axis (co ...
*Thin films technologies *
Printed electronics Printed electronics is a set of printing Printing is a process for mass reproducing text and images using a master form or template. The earliest non-paper products involving printing include cylinder seals and objects such as the Cyrus Cy ...
*
Minimum railway curve radius The minimum railway curve radius is the shortest allowable design radius for the centerline of railway tracks under a particular set of conditions. It has an important bearing on construction costs and operating costs and, in combination with sup ...


Stress in semiconductor structures

Stress in the
semiconductor A semiconductor material has an value falling between that of a , such as metallic copper, and an , such as glass. Its falls as its temperature rises; metals behave in the opposite way. Its conducting properties may be altered in useful ways ...
structure involving evaporated
thin films A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometre, micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step i ...
usually results from the
thermal expansion Thermal expansion is the tendency of matter to change its shape A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface ...
(thermal stress) during the manufacturing process. Thermal stress occurs because film depositions are usually made above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress. Intrinsic stress results from the microstructure created in the film as atoms are deposited on the substrate. Tensile stress results from microvoids (small holes, considered to be defects) in the thin film, because of the attractive interaction of atoms across the voids. The stress in thin film semiconductor structures results in the buckling of the wafers. The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula. The topography of the stressed structure including radii of curvature can be measured using optical scanner methods. The modern scanner tools have capability to measure full topography of the substrate and to measure both principal radii of curvature, while providing the accuracy of the order of 0.1% for radii of curvature of 90 meters and more.


See also

*AFM probe *Base curve radius *Bend radius *Curve *Curvature *Degree of curvature (civil engineering) *Diameter *Osculating circle *Reverse curve *Track transition curve *Transition curve


References


Further reading

*


External links


The Geometry Center: Principal Curvatures


* * {{curvature Differential geometry Curvature (mathematics) Curves Integral calculus Multivariable calculus Theoretical physics