I want to specify a subspace of a high dimension Euclidean space. The subspace consists of those points whose closest proximity to a specified hyperplane is a fixed value r. Here are some examples.
Example 1. Consider a 3D space, and a line (the hyperplane) running though it. The subspace is the surface of cylinder of radius r whose axis is the line.
Example 2. Consider a 3D space, and a plane (the hyperplane) running though it. The subspace consists of the two planes parallel to the original one and a distance r from it.
I want this subspace specified in the form of a locus of points. In other words, a set of parameters uniquely specifies any point in the subspace. For instance in example 1, any point in the subspace could be specified by 3 parameters, e.g.
- one parameter would specify a point on the line
- r would specify a circle radius r, centred on the point and orthogonal to the line
- theta (an angle) would specify a point on that circle. (You would have to define where theta=zero was and the direction of rotation of theta but I don't think you would need more parameters for that, just the adoption of some convention.)
I've been able to specify a b dimensional hyperplane in a b + c dimensional euclidean space (b < b+c), but am now stuck.
I appreciate any help you can give me.
What about this. One could start with the polar coordinates for a point on a b dimensional sphere. Covert from polar to Cartesian, to express these b parameters as a b-vector in b dimensional Euclidean space. Place the b dimensional sphere in a b + c dimensional Euclidean space. Place it at position zero in each of the c new dimensions. Now add c parameters which are the positions of the b-sphere in the c new dimensions. You have now created a parameterisation for one example of the desired subspace. The c dimensions are the hyperpane. You can then rotate and translate your c+b parameterisation in c+b dimensional space to make it coincide with any specific c dimensional hyperplane. You set the radius of the b-sphere to a specific value. Your final parameterisation has b+c-1 parameters.
You could think about stereographic projection: If you have a linear subspace $U$ of dimension $m$ of a Euclidean vector space $V$ of dimension $n$, you could pick a vector in $U^\bot$, call it $w$, and consider (orthonormal) co-ordinates $(x_1,\ldots,x_{n-1})$ on the space $w^\bot$ orthogonal to $w$, chosen such that $(x_1,\ldots,x_m)$ give co-ordinates on $U$. Now apply the inverse stereographic projection map (you can look up the formula I guess) to the co-ordinates $(x_{m+1},\ldots, x_{n-1})$, and leave the variables $x_1,\ldots,x_m$ fixed. That gives you a map from $\mathbb{R}^{n-1}$ to your locus. Unfortunately this is not quite surjective, and misses out a translate of $U$ in the direction $w$. Still, the formulas are nicer than finding spherical co-ordinates I think.
One small point: 'hyperplane' generally refers to a subspace of dimension one less than the ambient vector space.