Function Repository Resource:

# OsculatingCircleProperties

Compute the defining properties of the osculating circle for a curve at a point

Contributed by: Wolfram|Alpha Math Team
 ResourceFunction["OsculatingCircleProperties"][f,{x,a}, y] gives the osculating circle of the curve f at the point x=a with respect to the variables x and y. ResourceFunction["OsculatingCircleProperties"][f,{x,a},{y,b}] gives the osculating circle of f at the point (a,b) with respect to the variables x and y. ResourceFunction["OsculatingCircleProperties"][{f[t],g[t]},{t,t0},x,y] gives the osculating circle to the parameterized curve at the point t0. ResourceFunction["OsculatingCircleProperties"][…,type] gives the property type of the osculating circle.

## Details

The property type can be All, "Radius", "Curvature", "Center", "Equation", "Function", "Circle", "Normal" and "Plot".
If f is an expression in ResourceFunction["OsculatingCircleProperties"][f,{x,a},y], the implicit curve is taken to be y==f.

## Examples

### Basic Examples (3)

Return the osculating circle for a given expression:

 In[1]:=
 Out[1]=

Return the osculating circle for a parametrically defined curve:

 In[2]:=
 Out[2]=

Return the osculating circle for an implicitly defined curve:

 In[3]:=
 Out[3]=

### Applications (4)

Return the osculating circle:

 In[4]:=
 Out[4]=

Return only the circle as a Graphics primitive:

 In[5]:=
 Out[5]=

Plot the osculating circle and expression:

 In[6]:=
 Out[6]=

Or view the plot directly:

 In[7]:=
 Out[7]=

### Possible Issues (2)

OsculatingCircleProperties returns unevaluated if the curvature does not exist at a given point:

 Out[8]=

OsculatingCircleProperties returns unevaluated if the gradient is zero:

 Out[9]=

## Publisher

Wolfram|Alpha Math Team

## Version History

• 2.0.0 – 23 March 2023
• 1.0.1 – 04 March 2021
• 1.0.0 – 31 March 2020

## Author Notes

To view the full source code for OsculatingCircleProperties, evaluate the following:

 In[1]:=