algebraic curve

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A curve is algebraic when its defining Cartesian equation is algebraic, that is a polynomial in x and y. The order or degree of the curve is the maximum degree of each of its terms xⁱ y^j.
An algebraic curve with degree greater than 2 is called a higher plane curve.

An algebraic curve is called a circular algebraic curve, when the points (1, i) and (1, -i) are on the curve. In that case the highest degree of the Cartesian equation is divisible by (x² + y²).
The curve got its name from the fact that it contains the two imaginary circular points: replace x by x/w and y by y/w, and let the variable w go to zero, we obtain the circular points.
Examples of circular algebraic curves:

the circle is the only circular conic section.
the cubic conchoid of de Sluze
the sextic atriphtaloid

A bicircular algebraic curve passes twice through the points (1, i) and (1, -i). In this case the highest degree of the Cartesian equation is divisible by (x² + y²)².
There are several examples of bicircular quartics.

And so on, a tricircular algebraic curve passes three times through the points (1, i) and (1, -i). In this case the highest degree of the Cartesian equation is divisible by (x² + y²)³.
An example of a tricircular algebraic curve is the sextic Watt's curve.

When a curve is not algebraic, we call the curve (and its function) transcendental.

In the case a function is sufficiently sophisticated it is said to be a special function.