Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. •
Bicorn •
Bullet-nose curve •
Cartesian oval •
Cassini oval •
Deltoid curve •
Devil's curve •
Hippopede •
Kampyle of Eudoxus •
Klein quartic •
Lemniscate •
Lemniscate of Bernoulli •
Lemniscate of Gerono •
Limaçon •
Lüroth quartic •
Spiric section •
Squircle •
Lamé's special quartic •
Toric section •
Trott curve Ampersand curve The
ampersand curve is a quartic plane curve given by the equation: :\ (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.
Bean curve The
bean curve is a quartic plane curve with the equation: :x^4+x^2y^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.
Bicuspid curve The
bicuspid is a quartic plane curve with the equation :(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.
Bow curve The
bow curve is a quartic plane curve with the equation: :x^4=x^2y-y^3. \, The bow curve has a single triple point at
x=0,
y=0, and consequently is a rational curve, with genus zero.
Cruciform curve The
cruciform curve, or
cross curve is a quartic plane curve given by the equation :x^2y^2-b^2x^2-a^2y^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
a2
x2 +
b2
y2 = 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. Because the curve is rational, it can be parametrized by rational functions. For instance, if
a=1 and
b=2, then :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. 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: :\begin{align} 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{align}
Spiric section 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 :(x^2+y^2)^2=dx^2+ey^2+f , and the equation in polar coordinates as :r^4=dr^2\cos^2\theta+er^2\sin^2\theta+f. \,
Three-leaved clover (trifolium) The
three-leaved clover or
trifolium is the quartic plane curve : x^4+2x^2y^2+y^4-x^3+3xy^2=0. \, By solving for
y, the curve can be described by the following function: : 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 : x = \cos(3t) \cos t,\quad y = \cos(3t) \sin t. \, In polar coordinates (
x =
r cos φ,
y =
r sin φ) the equation is :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. ==See also==