Function Repository Resource:

# FociPointEllipse

Find the equation for an ellipse given two foci and a point

Contributed by: Ed Pegg Jr and Jan Mangaldan
 ResourceFunction["FociPointEllipse"][{f1,f2,p},{x,y}] returns the ellipse A x2+B x y+C y2+D x+E y+F in the variables x and y, given the foci f1,f2 and a point p through which the ellipse passes. ResourceFunction["FociPointEllipse"][{f1,f2,p},t] returns a parametric equation in the variable t. ResourceFunction["FociPointEllipse"][{f1,f2,p}] returns an Ellipsoid object representing the ellipse.

## Examples

### Basic Examples (2)

Find the Cartesian equation of an ellipse with foci (3,3) and ) that goes through point ):

 In[1]:=
 Out[2]=

Show the ellipse:

 In[3]:=
 Out[3]=

### Scope (4)

Generate the parametric equations of an ellipse with foci (3,3) and ) that goes through point ):

 In[4]:=
 Out[5]=

Show the ellipse:

 In[6]:=
 Out[6]=

Generate the corresponding Ellipsoid object:

 In[7]:=
 Out[7]=

Show the ellipse:

 In[8]:=
 Out[8]=

### Properties and Relations (4)

Use a different set of variables:

 In[9]:=
 Out[9]=

Use formal variables:

 In[10]:=
 Out[10]=

Use FociPointEllipse to generate the implicit Cartesian equation of an ellipse:

 In[11]:=
 Out[12]=

Use GroebnerBasis to get an equivalent result:

 In[13]:=
 Out[13]=

Generate an equivalent parametric equation:

 In[14]:=
 Out[14]=

Use GroebnerBasis to derive the implicit Cartesian equation from the parametric equation:

 In[15]:=
 Out[15]=

Use the resource function EllipseProperties to generate properties of the ellipse:

 In[16]:=
 Out[16]=

Get an ellipse equation:

 In[17]:=
 Out[17]=

Show positions for coefficients in A x2+B x y+C y2+D x+E y+F=0:

 In[18]:=
 Out[18]=

Get the coefficients:

 In[19]:=
 Out[19]=

See the coefficients in the standard order:

 In[20]:=
 Out[20]=

### Neat Examples (4)

Construct an ellipse from given foci and a point:

 In[21]:=
 Out[22]=

Use three mysterious points to create a complex cubic, take the derivative and solve the quadratic to find foci that happen to be F and G:

 In[23]:=
 Out[24]=

In Marden's theorem, the above step finds the foci of an inellipse that is tangent to the midpoints of the sides of the triangle generated by vertices PQR:

 In[25]:=
 Out[25]=

The resulting inellipse is a scaled version of the Steiner circumellipse, with a scaling factor of 1/2. Use the resource function SteinerCircumellipse to produce an equivalent figure:

 In[26]:=
 Out[27]=

## Version History

• 1.0.0 – 18 July 2022