flat subspace : minimal characterization In the euclidean space ${\mathbb R}^3$, I can define a plane by three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$, using six reals.
Of course I can give an equation $ax+by+cz=d$, using only 4 reals.
But I can define a plane using a normalize equation ($a^2+b^2+c^2=1$), so I can rewrite it as $\cos\theta.x+\sin\theta\cos\phi.y+\sin\theta\sin\phi.z=d$, and I only need 3 reals.
I was wondering how many reals I need to define "canonically" a flat subspace of dimension $d$ into ${\mathbb R}^n$. If $d=0$, the flat is a point and the obvious answer is $n$. But what happens for other value ? How can I define canonical equations or an orthogonal base from such a definition ?
Of course, as ${\mathbb R}^n$ has the same cardinal as ${\mathbb R}$, you can answer there is a bijection between $\mathbb R$ and the set of flat subspaces but this is not my question as I want something constructive (and in some way continuous and naive).
So I want some function $f$ from $\mathbb R^k$ into the set of flat subspace of dimension $d$ (of $\mathbb R^n$). And I want $f$ to be surjective, continuous and $k$ minimum. So there must be some function $\phi(d,n)=k$, that gives the smallest such $k$ for given $d,n$.
 A: I believe the minimal characterization may be the degrees-of-freedom necessary to specify the Grassmann coordinates of the directions parallel to the flat (along with a point on the flat) or orthogonal to the flat (along with the offset along the orthogonal directions). Points on the Grassmann manifold represent unique linear subspaces.
The parallel representation is akin to the parametric representation for the flat $p + Bz$, where $p$ is the point on the flat and $B$ is a matrix whose columns are orthonormal. The minimal number of degrees-of-freedom for $B$ are those necessary to specify a linear subspace aka a point on the Grassmann manifold. $z$ are the parameters, one for each column of $B$. For a plane in $\mathbb{R}^3$, the columns of $B$ would be two vectors spanning the plane.
The orthogonal representation is akin to the implicit representation for the flat $Ax = a$, where $A$ is a matrix whose rows are orthonormal, $x$ is a point in the ambient space, and $a$ is the flat's signed distance from the origin along each direction. For a plane in $\mathbb{R}^3$, the row of $A$ would be the plane's unit normal and $a$ would be the plane's (scalar) distance from the origin along the normal.
According to "The Representation and Parametrization of Orthogonal Matrices" [Ron Shepard, Scott R. Brozell, Gergely Gidofalvi 2015] in Journal of Physical Chemistry A], there are $(m-n)n$ degrees of freedom to uniquely represent a linear subspace, where $m$ is the dimension of the ambient space and $n$ is the dimension of the subspace.
