---
title: "Ampersand curve"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Ampersand_curve"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:04:06.292461+00:00"
instance: "kb-cron"
---

In geometry, the ampersand curve is a type of quartic plane curve. It was named after its resemblance to the ampersand symbol by Henry Cundy and Arthur Rollett.

The ampersand curve is the graph of the equation

  
    
      
        6
        
          x
          
            4
          
        
        +
        4
        
          y
          
            4
          
        
        −
        21
        
          x
          
            3
          
        
        +
        6
        
          x
          
            2
          
        
        
          y
          
            2
          
        
        +
        19
        
          x
          
            2
          
        
        −
        11
        x
        
          y
          
            2
          
        
        −
        3
        
          y
          
            2
          
        
        =
        0.
      
    
    {\displaystyle 6x^{4}+4y^{4}-21x^{3}+6x^{2}y^{2}+19x^{2}-11xy^{2}-3y^{2}=0.}
  

The graph of the ampersand curve has three crunode points where it intersects itself at (0,0), (1,1), and (1,-1). The curve has a genus of 0.
The curve was originally constructed by Julius Plücker as a quartic plane curve that has 28 real bitangents, the maximum possible for bitangents of a quartic.
It is the special case of the Plücker quartic

  
    
      
        (
        x
        +
        y
        )
        (
        y
        −
        x
        )
        (
        x
        −
        1
        )
        (
        x
        −
        
          
            
              3
              2
            
          
        
        )
        −
        2
        (
        
          y
          
            2
          
        
        +
        x
        (
        x
        −
        2
        )
        
          )
          
            2
          
        
        −
        k
        =
        0
        ,
      
    
    {\displaystyle (x+y)(y-x)(x-1)(x-{\tfrac {3}{2}})-2(y^{2}+x(x-2))^{2}-k=0,}
  

with 
  
    
      
        k
        =
        0.
      
    
    {\displaystyle k=0.}
  

The curve has 6 real horizontal tangents at

  
    
      
        
          (
          
            
              
                1
                2
              
            
            ,
            ±
            
              
                
                  5
                
                2
              
            
          
          )
        
        ,
      
    
    {\displaystyle \left({\frac {1}{2}},\pm {\frac {\sqrt {5}}{2}}\right),}
  

  
    
      
        
          (
          
            
              
                
                  159
                  −
                  
                    
                      201
                    
                  
                
                120
              
            
            ,
            ±
            
              
                
                  1389
                  +
                  67
                  
                    
                      67
                      
                        /
                      
                      3
                    
                  
                
                40
              
            
          
          )
        
        ,
      
    
    {\displaystyle \left({\frac {159-{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389+67{\sqrt {67/3}}}}{40}}\right),}
  
 and

  
    
      
        
          (
          
            
              
                
                  159
                  +
                  
                    
                      201
                    
                  
                
                120
              
            
            ,
            ±
            
              
                
                  1389
                  −
                  67
                  
                    
                      67
                      
                        /
                      
                      3
                    
                  
                
                40
              
            
          
          )
        
        .
      
    
    {\displaystyle \left({\frac {159+{\sqrt {201}}}{120}},\pm {\frac {\sqrt {1389-67{\sqrt {67/3}}}}{40}}\right).}
  

And 4 real vertical tangents at 
  
    
      
        
          (
          
            −
            
              
                
                  1
                  10
                
              
            
            ,
            ±
            
              
                
                  
                    23
                  
                  10
                
              
            
          
          )
        
      
    
    {\displaystyle \left(-{\tfrac {1}{10}},\pm {\tfrac {\sqrt {23}}{10}}\right)}
  
 and 
  
    
      
        
          (
          
            
              
                
                  3
                  2
                
              
            
            ,
            
              
                
                  
                    3
                  
                  2
                
              
            
          
          )
        
        .
      
    
    {\displaystyle \left({\tfrac {3}{2}},{\tfrac {\sqrt {3}}{2}}\right).}
  

It is an example of a curve that has no value of x in its domain with only one y value.


== Notes ==


== References ==
Piene, Ragni, Cordian Riener, and Boris Shapiro. "Return of the plane evolute." Annales de l'Institut Fourier. 2023
Figure 2 in Kohn, Kathlén, et al. "Adjoints and canonical forms of polypols." Documenta Mathematica 30.2 (2025): 275-346.
Julius Plücker, Theorie der algebraischen Curven, 1839, [1]
Frost, Percival, Elementary treatise on curve tracing, 1960, [2]


== Further reading ==
"Plücker's Quartic". mathworld.wolfram.com.
"Ampersand Curve Points". mathworld.wolfram.com.