Function Repository Resource:

# CesaroEquation

Compute the Cesàro equation for a planar curve

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["CesaroEquation"][c,t,{κ,s}] computes the Cesàro equation for curvature κ and arc length s of a planar curve c parametrized by t.

## Examples

### Basic Examples (3)

Cesàro equation for the circle:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

Cesàro equation for the logarithmic spiral:

 In[3]:=
 Out[3]=

Equations can be simplified using assumptions:

 In[4]:=
 Out[4]=

Cesàro equations for several curves:

 In[5]:=
 Out[5]=

### Properties and Relations (2)

For some curves, entities can be used to get the Cesàro equation:

 In[6]:=
 Out[6]=

This can instead be computed by CesaroEquation with κ=1/ρ:

 In[7]:=
 In[8]:=
 Out[8]=

Cesàro equation for the Cornu spiral:

 In[9]:=
 Out[9]=
 In[10]:=
 Out[10]=

Curvature is linear:

 In[11]:=
 Out[11]=

Alfred Gray’s generalization of the Cornu spiral:

 In[12]:=

The clothoid for n, a both set to unity gives the Cornu spiral:

 In[13]:=
 Out[13]=

Plot the clothoid:

 In[14]:=
 Out[14]=

Different values give other types of special functions:

 In[15]:=
 Out[15]=

The clothoid for general values n, a:

 In[16]:=
 Out[16]=

Find the Cesàro equation for the clothoid:

 In[17]:=
 Out[17]=

The curvature takes the simpler form:

 In[18]:=
 Out[18]=

A plot of quadratic intrinsic curvature:

 In[19]:=
 Out[19]=

Enrique Zeleny

## Version History

• 1.0.0 – 01 June 2020