In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Cesàro equation of a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
is an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
relating the
curvature () at a point of the curve to the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
() from the start of the curve to the given point. It may also be given as an equation relating the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
() to
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
. (These are equivalent because .) Two
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
curves will have the same Cesàro equation. Cesàro equations are named after
Ernesto Cesàro
__NOTOC__
Ernesto Cesàro (12 March 1859 – 12 September 1906) was an Italian mathematician who worked in the field of differential geometry. He wrote a book, ''Lezioni di geometria intrinseca'' (Naples, 1890), on this topic, in which he also ...
.
Examples
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
*
Line:
.
*
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
:
, where is the radius.
*
Logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
:
, where is a constant.
*
Circle involute:
, where is a constant.
*
Cornu spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.
Eu ...
:
, where is a constant.
*
Catenary
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superfici ...
:
.
Related parameterizations
The Cesàro equation of a curve is related to its
Whewell equation
The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tangential angle is the angle between the tangent to the curve and the -axis, and the arc length is the distance along the cur ...
in the following way: if the Whewell equation is then the Cesàro equation is .
References
*
*
*
External links
*
*
Curvature Curvesat 2dcurves.com.
{{DEFAULTSORT:Cesaro equation
Curves