Function Repository Resource:

# RollingCurve

Compute a rolling curve

Contributed by: Wolfram Staff (original content by Alfred Gray)
 ResourceFunction["RollingCurve"][c,r,h,t0,t] gives the parametrized curve traced out by a point P attached to a circle of radius r rolling along a plane curve c parametrized by variable t. The distance from P to the center of the rolling circle is h, and t0 is the point of the curve at which the circle starts rolling. ResourceFunction["RollingCurve"][c,r,t0,t] takes the distance from the tracing point to the center of the rolling circle to be Abs[r].

## Details

A rolling curve is also referred to as a roulette.
Rolling curves have been extensively studied for many applications in mechanical engineering.

## Examples

### Basic Examples (3)

Define the parametric equations for an ellipse:

 In[1]:=
 Out[1]=

Compute the rolling curve of the ellipse:

 In[2]:=
 Out[2]=

Plot the rolling curve:

 In[3]:=
 Out[3]=

### Scope (4)

Parametric equations for a circle:

 In[4]:=

The epitrochoid is the rolling curve of a circle if the rolling circle is outside the original circle:

 In[5]:=
 Out[5]=

The epicycloid is obtained if :

 In[6]:=
 Out[6]=

An equivalent way to get the epicycloid:

 In[7]:=
 Out[7]=

### Properties and Relations (5)

Define the parametric equations of a parabola:

 In[8]:=
 Out[8]=

Define the rolling curve of the parabola:

 In[9]:=
 Out[9]=

When (the curtate case), the rolling curve is smooth:

 In[10]:=
 Out[10]=

When (the cycloidal case), the rolling curve has cusps:

 In[11]:=
 Out[11]=

When (the prolate case), the rolling curve has self-intersection points:

 In[12]:=
 Out[12]=

### Neat Examples (4)

Define the parametric equations of a trochoid:

 In[13]:=
 Out[13]=

Compute the parallel curve of the trochoid:

 In[14]:=
 Out[14]=

Compute the rolling curve of the trochoid:

 In[15]:=
 Out[15]=

Create a Manipulate that rolls a circle along the curve:

 In[16]:=
 Out[16]=

Enrique Zeleny

## Version History

• 1.0.1 – 29 March 2021
• 1.0.0 – 28 February 2020