Quartic plane curve

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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, it can be multiplied by any non-zero factor 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 in general position.

A quartic curve can have a maximum of:

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

[edit] Examples

• Deltoid curve
• Lemniscate of Gerono
• Klein quartic
• Toric section

Ampersand curve	Bean curve	Bicuspid curve	Bow curve
Crooked egg curve	Cruciform curves	Spiric sections	Three-leaved clover
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