OjaNP (version 0.9-2)

hyperplane: Hyperplane Passing Through k Points in the k-dimensional Space.

Description

The function computes a $(k\!-\!1)$-dimensional hyperplane passing through $k$ given points in the $k$-dimensional space.

Usage

hyperplane(X)

Arguments

X

a numeric $k \times k$ matrix containing $k$ data points as rows.

Value

• a vector of length $(k\!+\!1)$ describing the hyperplane, see details above.

Details

A $(k\!-\!1)$-dimensional hyperplane in $R^k$ consists of all points $x$ that satisfy $$d^T x + c = 0,$$ where $d$ is a $k$-vector and $c$ is a scalar. The function returns the $(k\!+\!1)$-vector $(d,c)$. It is normalized such that the length of $d$ equals $(k\!-\!1)!$ times the $(k\!-\!1)$-dimensional volume of the simplex formed by the points on the plane. (If $k = 3$, this is a triangle.) Hence this function can also easily be used to compute volumes of simplices. The direction of $d$, that is, whether it points towards the origin or not, is not fixed. It depends on the order of the data points within the matrix X. If the $k$ points do not uniquely define a $(k\!-\!1)$-dimensional hyperplane (i.e. they lie on a $(k\!-\!2)$-dimensional hyperplane), a vector containing zeros is returned.

Examples

### ----<< Example 1 >>---- : line in R^2
X <- rbind(c(4,5),c(8,2))
hyperplane(X)
# The line through the the points c(4,5) and c(8,2) is given by 
#    3*x + 4*y - 32 = 0.
# The norm of the first two components of the return value 
# of hyperplane() (i.e. the vector d above) equals the 
# distance of both points.

X <- rbind(c(8,2),c(4,5))
hyperplane(X)
# If the order of the points is changed, the direction of d 
# (see details) may also change.



### ----<< Example 2 >>---- : unit vectors in R^3
X <- diag(rep(1,3))
hyperplane(X)
# The plane passing through all three unit vectors is given by 
#   -x - y - z + 1 = 0.
# These three points form a equilateral triangle on the plane 
# with side length sqrt(2) and hence area sqrt(3)/2.
# The norm of d (see details) equals twice this number.