In

The Geometry Center: Principal Curvatures

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

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). ...

. 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 ...

, 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 ...

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 aspace 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 ...

, 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 − ...

of
: $R\; \backslash equiv\; \backslash left,\; \backslash frac\; \backslash \; =\; \backslash frac,$
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 ...

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). ...

.
Formula

In 2D

If the curve is given inCartesian 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'') ...

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 ...

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

If is a parametrized curve in then the radius of curvature at each point of the curve, , is given by :$\backslash rho\; =\; \backslash frac$. As a special case, if is a function from to , then the radius of curvature of itsgraph
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 ...

, , is
:$\backslash rho(t)=\backslash frac.$
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 :$\backslash mathbf(u)\; =\; \backslash mathbf\; a\; \backslash cos\; h(u)\; +\; \backslash mathbf\; b\; \backslash sin\; h(u)\; +\; \backslash mathbf\; c$ 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 :$\backslash begin\; ,\; \backslash mathbf\; g\text{'},\; ^2\; \&=\; \backslash rho^2\; (h\text{'})^2\; \backslash \backslash \; \backslash mathbf\; g\text{'}\; \backslash cdot\; \backslash mathbf\; g\text{'}\text{'}\; \&=\; \backslash rho^2\; h\text{'}\; h\text{'}\text{'}\; \backslash \backslash \; ,\; \backslash mathbf\; g\text{'}\text{'},\; ^2\; \&=\; \backslash rho^2\; \backslash left((h\text{'})^4\; +\; (h\text{'}\text{'})^2\; \backslash right)\; \backslash end$ If we now equate these derivatives of to the corresponding derivatives of at we obtain :$\backslash begin\; ,\; \backslash boldsymbol\backslash gamma\text{'}(t),\; ^\; \&=\; \backslash rho^2\; h\text{'}^(t)\; \backslash \backslash \; \backslash boldsymbol\backslash gamma\text{'}(t)\; \backslash cdot\; \backslash boldsymbol\backslash gamma\text{'}\text{'}(t)\; \&=\; \backslash rho^2\; h\text{'}(t)\; h\text{'}\text{'}(t)\; \backslash \backslash \; ,\; \backslash boldsymbol\backslash gamma\text{'}\text{'}(t),\; ^\; \&=\; \backslash rho^2\; \backslash left(h\text{'}^(t)\; +\; h\text{'}\text{'}^(t)\backslash right)\; \backslash end$ These three equations in three unknowns (, and ) can be solved for , giving the formula for the radius of curvature: :$\backslash rho(t)\; =\; \backslash frac$ or, omitting the parameter for readability, :$\backslash rho\; =\; \backslash frac.$Examples

Semicircles and circles

For asemi-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
:$y=\backslash sqrt,\; \backslash quad\; y\text{'}=\backslash frac,\; \backslash quad\; y\text{'}\text{'}=\backslash frac,\backslash quad\; R=,\; -a,\; =a.$
For a semi-circle of radius in the lower half-plane
:$y=-\backslash sqrt,\; \backslash quad\; R=,\; a,\; =a.$
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 ...

of radius has a radius of curvature equal to .
Ellipses

In anellipse
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 ...

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 θ$$R(\backslash theta)=\backslash sqrt$$
Applications

*For the use indifferential 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 thesemiconductor
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 curveReferences

Further reading

*External links

The Geometry Center: Principal Curvatures

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