We want to describe some "surface" or "submanifold" of R

^{n}, which is a set of points/vectors {x = x

_{1}e

_{1}+ x

_{2}e

_{2}+ ... + x

_{n}e

_{n}}.

An implicit description of the manifold is a set of equations

f

_{1}(x) = 0 , f

_{2}(x) = 0 , ... , f

_{k}(x) = 0,

where each f

_{i}is a function from R

^{n}to R. These can be combined into a single vector equation f(x) = 0 (or f(x) = constant), where f is a function from R

^{n}to R

^{k}. The manifold is (n-k)-dimensional (in general).

A parametric description of the manifold is an equation x = f(t), where f is a function from R

^{k}to R

^{n}. We call (t

_{1}, t

_{2}, ... , t

_{k}) the "parameters" or "coordinates". The manifold is k-dimensional (in general).

I'll allow the function f to have multivector inputs and outputs. This isn't really a generalization, because multivectors are equivalent to vectors in an abstract sense; they just have different rules for multiplication, and different interpretations.

Let's focus on implicit equations first.

2 dimensions

Circle : x

^{2}= a

^{2}

Line : x^a = B

By taking the dual (multiplying by e

_{1}e

_{2}), this equation is equivalent to

x.c = d

The line's direction (tangent vector) is a, and its normal vector is c. If ||c|| = 1, then d is the line's distance from the origin.

Pair of parallel lines : ||x^a||

^{2}= d

^{2}

or equivalently (x.c)

^{2}= d

^{2}

Point : x^a = b

or equivalently x.C = d

where C is a bivector. The wedge product with a scalar (a) is the same as scalar multiplication, which is invertible: x = b/a = constant.

Entire plane : x^A = 0

where A is a bivector.

3 dimensions

Sphere : x

^{2}= a

^{2}

Plane : x^A = B

or equivalently x.c = d

The plane's direction (tangent bivector) is A, and its normal vector is c.

Pair of parallel planes : ||x^A||

^{2}= d

^{2}

or equivalently (x.c)

^{2}= d

^{2}

Line : x^a = B

or equivalently x.C = d

The line's direction is a, and its normal bivector is C.

Cylinder: ||x^a||

^{2}= d

^{2}

or equivalently ||x.C||

^{2}= d

^{2}

Point : x^a = b

or equivalently x.C = D

where C is a trivector.

Entire space : x^A = 0

where A is a trivector.

4 dimensions

Point : x^a = b

or equivalently x.C = D

where C is a quadvector.

Glome : ||x^a||

^{2}= d

^{2}

This is getting tedious. Here's the general pattern.

A k-dimensional linear subspace has a direction (tangent k-blade) A, and a normal (n-k)-blade C which is the dual of A. The equation is x^A = constant, or x.C = constant.

A "boundary of a normal tube", or I guess a "spheration", of the subspace x^A = 0, is the set of all points at a certain distance from it. This is isomorphic to the Cartesian product of R

^{k}with the sphere S

^{n-k-1}. The equation is ||x^A||

^{2}= constant, or ||x.C||

^{2}= constant.