About the Representation of Geometric Objects

1 In the Plane

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

Observations

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).

2 In Space

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)
s(v_{1}, v_{2}, v_{3}) + t(w_{1}, w_{2}, w_{3}) (A, B, C) ∥ (V × W)  is normal
Surface  (x, y, z) = R(u, v) =  F (x, y, z) = 0  1  the gradient ∇F =
(g(s, t), h(s, t), k(s, t))  ∇f ∥ (∂R/∂u × ∂R/∂v)  (F_{x}, F_{y}, F_{z}) is normal

Observations

1. A parametric representation consists of either a vector-valued function 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. The same thing is true in the case of a parameterized surface for the vectors ∂R/∂s and ∂R/∂t when R(s, t) = (g(s, t), h(s, t), k(s, t)).