Function Repository Resource:

# Orthotomic

Compute the orthotomic of a curve

Contributed by: Eric W. Weisstein (with contributions by Wolfram Staff)
 ResourceFunction["Orthotomic"][c,t] computes the orthotomic in parameter t of a curve c with respect to the point {0,0}. ResourceFunction["Orthotomic"][c,p,t] computes the orthotomic with respect to the point p. ResourceFunction["Orthotomic"][c,l,t] computes the orthotomic with respect to the infinite line l.

## Details

The orthotomic curve (also known as the secondary caustic) is the reflection of the rays of a source in the tangents of points of a curve.
The orthotomic of a curve is the homothetic image (magnified by a factor of 2 with respect to the center of similarity p) of the pedal.
The orthotomic of a curve with respect to a given line is the locus of points generated by the orthogonal projection of a point on the original curve to the given line, reflected about the tangent to the curve at that point.
In ResourceFunction["Orthotomic"][c,l,t], l can be specified as an InfiniteLine object.

## Examples

### Basic Examples (1)

Orthotomic of a circle with respect to the origin:

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

### Scope (4)

Orthotomic of a circle with respect to the point {1,1}:

 In[3]:=
 Out[3]=
 In[4]:=
 Out[4]=

Orthotomic of an eight curve with respect to a varying point:

 In[5]:=
 In[6]:=
 Out[6]=

Parametric equations for a deltoid:

 In[7]:=
 Out[7]=

Orthotomic of the deltoid with respect to a given line:

 In[8]:=
 Out[8]=
 In[9]:=
 Out[9]=

Orthotomic of a bifolium with respect to a varying line:

 In[10]:=
 In[11]:=
 Out[11]=

### Properties and Relations (2)

The orthotomic is equivalent to the pedal curve, scaled by a factor of 2:

 In[12]:=
 In[13]:=
 Out[13]=
 In[14]:=
 Out[14]=
 In[15]:=
 Out[15]=
 In[16]:=
 Out[16]=

The catacaustic curve is the evolute of the orthotomic:

 In[17]:=
 In[18]:=
 Out[18]=
 In[19]:=
 Out[19]=
 In[20]:=
 Out[20]=
 In[21]:=
 Out[21]=

### Neat Examples (1)

Generate the orthotomic as an envelope of circles:

 In[22]:=
 In[23]:=
 Out[23]=
 In[24]:=
 Out[24]=

Enrique Zeleny

## Version History

• 1.0.1 – 29 March 2022
• 1.0.0 – 15 April 2020