In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (
κ
) at a point of the curve to the arc length (
s
) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (
R
) to arc length. (These are equivalent because
R
=
1
/
κ
.) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro.
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
Line:
κ
=
0
.
Circle:
κ
=
1
/
α
, where
α
is the radius.
Logarithmic spiral:
κ
=
C
/
s
, where
C
is a constant.
Circle involute:
κ
=
C
/
s
, where
C
is a constant.
Cornu spiral:
κ
=
C
s
, where
C
is a constant.
Catenary:
κ
=
a
s
2
+
a
2
.
The Cesàro equation of a curve is related to its Whewell equation in the following way. If the Whewell equation is
φ
=
f
(
s
)
then the Cesàro equation is
κ
=
f
′
(
s
)
.
