Algebraic Curve

Synonyms

Implicit polynomial curve

Related Concepts

Algebraic Surface

Definition

An algebraic curve is a curve determined by a 2-D implicit polynomial (IP) of degree n:

$$\displaystyle\begin{array}{rcl} f_{n}(\mathbf{x})& =& \displaystyle\sum _{0\leq i,j;i+j\leq n}a_{ij}{x}^{i}{y}^{j} \\ & =& (\mathop{\underbrace{1\ x\ \ldots \ {y}^{n}}}\limits _{{\mathbf{m}}^{\mathrm{T}}}){(\mathop{\underbrace{a_{00}\ a_{10}\ \ldots \ a_{0n}}}\limits _{\mathbf{a}})}^{\mathrm{T}} = 0,\end{array}$$

(1)

where x = (x, y)^T is the coordinate of a point on a curve. That is, the curve is always represented by f _n ’s zero level set: {x | f _n (x) = 0}. The polynomial function is usually denoted by an inner product between two vectors: monomial vector m and coefficient vector a. For the entries in these vectors, indices {i, j} can be arranged in different orders, such as lexicographical order or inverse lexicographical order. In addition, the homogeneous binary polynomial of degree r in x and y, \(\sum...