Quartic plane curve

⁠ It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom.

Various combinations of coefficients in the above equation give rise to various important families of curves as listed below.

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.

For instance, if a=1 and b=2, then 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: Spiric sections can be defined as bicircular quartic curves that are symmetric with respect to the x and y axes.

The Cartesian equation can be written as and the equation in polar coordinates as The three-leaved clover or trifolium[7] is the quartic plane curve By solving for y, the curve can be described by the following function: 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 In polar coordinates (x = r cos φ, y = r sin φ) the equation is It is a special case of rose curve with k = 3.