PARAMETRIC | EQUATION(S)
| CODIMEN | COMMENT | |

Point
| (x, y) = (p, q) | x=p, y=q | 2 | no tangent vector |

Line
| (x, y) = (a, b) + t (v_{1}, v_{2}) | Ax + By + C = 0 | 1 | (v_{1}, v_{2}) is tangent; (A, B) is normal |

Curve
| (x, y) = (g(t), h(t)) | F(x, y) = 0 | 1 | (g'(t), h'(t)) is tangent; (F_{x}, F_{y}) is normal |

1. In the ``line'' row above the vectors
(v_{1}, v_{2}) and (A, B) are perpendicular; therefore,
their *dot product* satisfies:
(A, B) **.** (v_{1}, v_{2}) =
Av_{1} + Bv_{2} = 0 .

2. A ``line'' is a special kind of ``curve''. If one takes F(x, y) = Ax + By + C , then the gradient vector of the function F, i.e., its vector of partial derivatives, is the vector (A, B).

PARAMETRIC | EQUATION(S)
| CODIMEN | COMMENT | |

Point
| (x, y, z) = (p, q, r) | x=p, y=q, z = r | 3 | no tangent vector |

Line
| (x, y, z) = (a, b, c) + | two linear equations | 2 | (v_{1}, v_{2}, v_{3}) |

t (v_{1}, v_{2}, v_{3}) | is a tangent vector | |||

Curve
| (x, y, z) = (g(t), h(t), k(t)) | two equations | 2 | (g'(t), h'(t), k'(t)) is tangent |

Plane
| (x, y, z) = (a, b, c) + | Ax + By + Cz = D | 1 | (A, B, C) |

u (v_{1}, v_{2}, v_{3}) + v (w_{1}, w_{2}, w_{3}) | (A, B, C) || (V \times W) | is normal | ||

Surface
| (x, y, z) = R(u, v) = | F(x, y, z) = 0 | 1 | the gradient \nabla F = |

(g(u, v), h(u, v), k(u, v)) | \nabla f || (\partial R/\partial u \times\partial R /\partial v) | (F_{x}, F_{y}, F_{z}) is normal |

1. A parametric representation consists of either one vector equation or a collection of scalar equations, one scalar equation for each coordinate. The number of parameters is the dimension of the object represented unless the representation is ``degenerate''.

2. A representation of a geometric object as the set of points
satisfying a collection of equations in the coordinates involves at
least as many equations (in the coordinates) as the **codimension**
of the object being represented. It is not always true, but is
frequently true and always true for lines and planes, that the
smallest number of equations is the codimension.

3. In the parametric form of a plane in space the vectors (v_{1}, v_{2}, v_{3}) and (w_{1}, w_{2}, w_{3}) are vectors that are parallel to line segments in the represented plane. Likewise in the case of a parameterized surface for the vectors \partial R /\partial u and \partial R /\partial v, where R(u, v) = (g(u, v), h(u, v), k(u, v)).

AUTHOR | COMMENT