Function Repository Resource:

OsculatingCircleProperties

Source Notebook

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]:=

$ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^3, {x, 2}, {y, 8}]$

Out[1]=

Return the osculating circle for a parametrically defined curve:

In[2]:=

$ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][{t, t^2}, {t, s}, x, y]$

Out[2]=

Return the osculating circle for an implicitly defined curve:

In[3]:=

$ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][y^2 == x, {x, 1}, {y, 1}, All]$

Out[3]=

Applications (4)

Return the osculating circle:

In[4]:=

$ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2, {x, 0}, y, All]$

Out[4]=

Return only the circle as a Graphics primitive:

In[5]:=

$res = ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2, {x, 0}, y, "Circle"]$

Out[5]=

Plot the osculating circle and expression:

In[6]:=

$Show[{Graphics[res], Plot[x^2, {x, -1, 1}]}]$

Out[6]=

Or view the plot directly:

In[7]:=

$ResourceFunction[ "OsculatingCircleProperties", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^2, {x, 0}, y, "Plot"]$