Algebraic Surface

Synonyms

Implicit polynomial surface

Related Concepts

Algebraic Curve

Definition

Similar to an algebraic curve, an algebraic surface is determined by a 3-D implicit polynomial (IP) of degree n:

$$ \begin{array}{lll} f_n ({\rm x}) = \,\mathop \Sigma \limits_{0 \le i,j,k;i+j+k \le n} \,a_{ij} x^i y^j z^k \\ \quad = \,\underbrace {(1\,x\,y\,z \ldots \,z^n )}_{{\mathbf m}^{\rm T} }\underbrace {(a_{000} \,a_{100} \, a_{010} \, a_{001} \ldots \,a_{00n} )}_{\mathbf a}^{\rm T} \\ \quad = 0, \end{array} $$

(1)

where $\mathbf{x} = (x,y,z)$ is a 3-D point on a surface, that is, the surface is always represented by f _n ’s zero level set: $\{\mathbf{x}\vert f_{n}(\mathbf{x}) = 0\}$. The polynomial function can be denoted by an inner product of two vectors: monomial vector m and coefficient vector a. For the entries in these vectors, indices {i, j, k} can be arranged in different orders, such as lexicographical order or inverse lexicographical order. In addition, the homogeneous binary polynomial...