Quartic plane curve

A quartic plane curve is a plane curve of the fourth degree. It can be defined by a quartic equation:

$Ax^{4}+By^{4}+Cx^{3}y+Dx^{2}y^{2}+Exy^{3}+Fx^{3}+Gy^{3}+Hx^{2}y+Ixy^{2}+Jx^{2}+Ky^{2}+Lxy+Mx+Ny+P=0.$

This equation has fifteen constants. However, it can be multiplied by any non-zero constant without changing the curve. Therefore, the space of quartic curves can be identified with the real projective space $\mathbb {RP} ^{14}$. It also follows that there is exactly one quartic curve that passes through a set of fourteen distinct points in general position, since a quartic has 14 degrees of freedom.

A quartic curve can have a maximum of:

• Four connected components
• Twenty-eight bi-tangents
• Three ordinary double points.

Examples

• Bicorn curve
• Bullet-nose curve
• Cartesian oval
• Cassini oval
• Deltoid curve
• Hippopede
• Lemniscate of Bernoulli
• Lemniscate of Gerono
• Klein quartic
• Spiric section
• Toric section
• Kampyle of Eudoxus

• 
  Ampersand curve
• 
  Bean curve
• 
  Bicuspid curve
• 
  Bow curve
• 
  Cruciform curves
• 
  Spiric sections
• 
  Three-leaved clover
• Trott curve
  Trott curve