Quartic plane curve

From Wikipedia, the free encyclopedia

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

Ax⁴ + By⁴ + Cx³y + Dx²y² + Exy³ + Fx³ + Gy³ + Hx²y + Ixy² + Jx² + Ky² + Lxy + Mx + Ny + P = 0

This equation has fifteen constants. However, any one of them can be set equal to one without changing the shape of the curve. Therefore, quartic curves form a space of dimension fourteen. It also follows that there is exactly one quartic curve that passes through a set of fourteen distinct points.

A quartic curve can have a maximum of:

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

[edit] Examples

• Toric section
• Klein quartic

Ampersand curve	Bean curve	Bicuspid curve	Bow curve
Crooked egg curve	Cruciform curves	Spiric sections	Trott curve

• The bean curve is a special case of the crooked egg curve
• A spiric section is a special case of a toric section