Quartic plane curve

The ampersand curve is a quartic plane curve given by the equation:

${\displaystyle \ (y^{2}-x^{2})(x-1)(2x-3)=4(x^{2}+y^{2}-2x)^{2}.}$

It has genus zero, with three ordinary double points, all in the real plane. [1]

The bean curve is a quartic plane curve with the equation:

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

The bean curve has genus zero. It has one singularity at the origin, an ordinary triple point. [2][3]

The bicuspid is a quartic plane curve with the equation

${\displaystyle (x^{2}-a^{2})(x-a)^{2}+(y^{2}-a^{2})^{2}=0\,}$

where a determines the size of the curve. The bicuspid has only the two cusps as singularities, and hence is a curve of genus one. [4]

The bow curve is a quartic plane curve with the equation:

${\displaystyle x^{4}=x^{2}y-y^{3}.\,}$

The bow curve has a single triple point at x=0, y=0, and consequently is a rational curve, with genus zero. [5]

The cruciform curve, or cross curve is a quartic plane curve given by the equation

${\displaystyle x^{2}y^{2}-b^{2}x^{2}-a^{2}y^{2}=0\,}$

where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a²x² + b²y² = 1, and is therefore a rational plane algebraic curve of genus zero. The cruciform curve has three double points in the real projective plane, at x=0 and y=0, x=0 and z=0, and y=0 and z=0. [6]

Because the curve is rational, it can be parametrized by rational functions. For instance, if a=1 and b=2, then

${\displaystyle x=-{\frac {t^{2}-2t+5}{t^{2}-2t-3}},\quad y={\frac {t^{2}-2t+5}{2t-2}}}$

parametrizes the points on the curve outside of the exceptional cases where a denominator is zero.

Illustration of the inverse Pythagorean and regular Pythagorean theorems

The inverse Pythagorean theorem is obtained from the above equation by substituting x with AC, y with BC, and each a and b with CD, where A, B are the endpoints of the hypotenuse of a right triangle ABC, and D is the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse:

${\displaystyle {\begin{aligned}AC^{2}BC^{2}-CD^{2}AC^{2}-CD^{2}BC^{2}&=0\\AC^{2}BC^{2}&=CD^{2}BC^{2}+CD^{2}AC^{2}\\{\frac {1}{CD^{2}}}&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\\therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}}$

Spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y axes. Spiric sections are included in the family of toric sections and include the family of hippopedes and the family of Cassini ovals. The name is from σπειρα meaning torus in ancient Greek.

The Cartesian equation can be written as

${\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f,}$

and the equation in polar coordinates as

${\displaystyle r^{4}=dr^{2}\cos ^{2}\theta +er^{2}\sin ^{2}\theta +f.\,}$

The three-leaved clover or trifolium[7] is the quartic plane curve

${\displaystyle x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}=0.\,}$

By solving for y, the curve can be described by the following function:

${\displaystyle y=\pm {\sqrt {\frac {-2x^{2}-3x\pm {\sqrt {16x^{3}+9x^{2}}}}{2}}},}$

where the two appearances of ± are independent of each other, giving up to four distinct values of y for each x.

The parametric equation of curve is

${\displaystyle x=\cos(3t)\cos t,\quad y=\cos(3t)\sin t.\,}$[8]

In polar coordinates (x = r cos φ, y = r sin φ) the equation is

${\displaystyle r=\cos(3\varphi ).\,}$

It is a special case of rose curve with k = 3. This curve has a triple point at the origin (0, 0) and has three double tangents.