Quadrifolium

The quadrifolium (also known as four-leaved clover) is a type of <a href="/facts/Rose_(mathematics)/zqzC97vy">rose curve</a> with an <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> of 2. It has the <a href="/facts/Polar_equation/HBko4YoN">polar equation</a>:

r
        =
        a
        cos
        ⁡
        (
        2
        θ
        )
        ,
        
      
    
    {\displaystyle r=a\cos(2\theta ),\,}

with corresponding algebraic equation

(
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        
          )
          
            3
          
        
        =
        
          a
          
            2
          
        
        (
        
          x
          
            2
          
        
        −
        
          y
          
            2
          
        
        
          )
          
            2
          
        
        .
        
      
    
    {\displaystyle (x^{2}+y^{2})^{3}=a^{2}(x^{2}-y^{2})^{2}.\,}

Rotated counter-clockwise by 45°, this becomes

r
        =
        a
        sin
        ⁡
        (
        2
        θ
        )
        
      
    
    {\displaystyle r=a\sin(2\theta )\,}

with corresponding algebraic equation

(
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        
          )
          
            3
          
        
        =
        4
        
          a
          
            2
          
        
        
          x
          
            2
          
        
        
          y
          
            2
          
        
        .
        
      
    
    {\displaystyle (x^{2}+y^{2})^{3}=4a^{2}x^{2}y^{2}.\,}

In either form, it is a <a href="/facts/Algebraic_curve/LnWsPmDP">plane algebraic curve</a> of <a href="/facts/Geometric_genus/S2oQVNCb">genus</a> zero.
The <a href="/facts/Dual_curve/tLXmm50P">dual curve</a> to the quadrifolium is

(
        
          x
          
            2
          
        
        −
        
          y
          
            2
          
        
        
          )
          
            4
          
        
        +
        837
        (
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        
          )
          
            2
          
        
        +
        108
        
          x
          
            2
          
        
        
          y
          
            2
          
        
        =
        16
        (
        
          x
          
            2
          
        
        +
        7
        
          y
          
            2
          
        
        )
        (
        
          y
          
            2
          
        
        +
        7
        
          x
          
            2
          
        
        )
        (
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        )
        +
        729
        (
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        )
        .
        
      
    
    {\displaystyle (x^{2}-y^{2})^{4}+837(x^{2}+y^{2})^{2}+108x^{2}y^{2}=16(x^{2}+7y^{2})(y^{2}+7x^{2})(x^{2}+y^{2})+729(x^{2}+y^{2}).\,}

The area inside the quadrifolium is 
 
 
 
 
 
 
 1
 2
 
 
 
 π
 
 a
 
 2
 
 
 
 
 {\displaystyle {\tfrac {1}{2}}\pi a^{2}}
 
, which is exactly half of the area of the circumcircle of the quadrifolium. The <a href="/facts/Perimeter/yQv1AajA">perimeter</a> of the quadrifolium is

8
        a
        E
        ⁡
        
          (
          
            
              
                3
              
              2
            
          
          )
        
        =
        4
        π
        a
        
          (
          
            
              
                
                  (
                  52
                  
                    
                      3
                    
                  
                  −
                  90
                  )
                  
                    M
                    ′
                  
                  ⁡
                  (
                  1
                  ,
                  7
                  −
                  4
                  
                    
                      3
                    
                  
                  )
                
                
                  
                    M
                    
                      2
                    
                  
                  ⁡
                  (
                  1
                  ,
                  7
                  −
                  4
                  
                    
                      3
                    
                  
                  )
                
              
            
            +
            
              
                
                  7
                  −
                  4
                  
                    
                      3
                    
                  
                
                
                  M
                  ⁡
                  (
                  1
                  ,
                  7
                  −
                  4
                  
                    
                      3
                    
                  
                  )
                
              
            
          
          )
        
      
    
    {\displaystyle 8a\operatorname {E} \left({\frac {\sqrt {3}}{2}}\right)=4\pi a\left({\frac {(52{\sqrt {3}}-90)\operatorname {M} '(1,7-4{\sqrt {3}})}{\operatorname {M} ^{2}(1,7-4{\sqrt {3}})}}+{\frac {7-4{\sqrt {3}}}{\operatorname {M} (1,7-4{\sqrt {3}})}}\right)}

where 
 
 
 
 E
 ⁡
 (
 k
 )
 
 
 {\displaystyle \operatorname {E} (k)}
 
 is the <a href="/facts/Elliptic_integral/q8EGYzQ3">complete elliptic integral of the second kind</a> with modulus 
 
 
 
 k
 
 
 {\displaystyle k}
 
, 
 
 
 
 M
 
 
 {\displaystyle \operatorname {M} }
 
 is the <a href="/facts/Arithmetic%25E2%2580%2593geometric_mean/8PRN9UB4">arithmetic–geometric mean</a> and 
 
 
 
 
 
 ′
 
 
 
 {\displaystyle '}
 
 denotes the <a href="/facts/Derivative/7Ito70Dh">derivative</a> with respect to the second variable.

Quadrifolium open-in-new

Quadrifolium