with at least one of A, B, C, D, E not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space$\mathbb {RP} ^{14}.$ 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.

One may also consider quartic curves over other fields (or even rings), for instance the complex numbers. In this way, one gets Riemann surfaces, which are one-dimensional objects over $\mathbb {C} ,$ but are two-dimensional over $\mathbb {R} .$ An example is the Klein quartic. Additionally, one can look at curves in the projective plane, given by homogeneous polynomials.

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

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

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 a^{2}x^{2} + b^{2}y^{2} = 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

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. 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.

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Gibson, C. G., Elementary Geometry of Algebraic Curves, an Undergraduate Introduction, Cambridge University Press, Cambridge, 2001, ISBN978-0-521-64641-3. Pages 12 and 78.