author Dmitriy Morozov <>
Tue, 24 Mar 2009 15:17:39 -0700
changeset 118 c4e25fb4082c
parent 19 efa14432761a
permissions -rw-r--r--
Added consistency zigzag

== Predicates/Constructions
  * circumsphere, side of circumsphere
  * barycentric coordinates
  * degenerate simplex
  * projection on a k-flat, squared distance to the k-flat
  * signed square distance to an oriented (d-1)-flat (?)

== Computing circumcenter and circumradius of a k-simplex in d dimensions.
  * Ken Clarkson's Gram-Schmidt-based algorithm works perfectly over any field. 
    The idea is as follows. 
    Given points p_i, the circumcenter c satisfies (p_i-c)^2 = r^2.  By
    translation, we can assume that some p_i is at the origin, so that c^c=r^2,
    and that equation becomes p_i \cdot c - p_i^2/2 = 0.  That is, [c -1/2] is
    normal to [p_i p_i^2].  Also, c must be an affine combination of the p_i,
    so if we start with a point [c' 0] with c' in the affine hull of the p_i,
    and remove all multiples of all [p_i p_i^2] from it, via Gram-Schmidt for
    example, and then scale the result so that the last entry is -1/2, we get
    c (alternatively we can start with [0 1]). Note that if the simplex is 
    degenerate (its affine hull is less than k-dimensional), the last entry of 
    c is 0.

      P = {p_0, ..., p_k}
      p'_i = p_i - p_0                  # make p_0 the origin
      p''_i = [p'_i p'_i^2]             # lift the points
      c' = 0                            # c is in the affine hull of p'_i
      c'' = [c' 1]                      # c'' is not in the affine hull of p'_i^2

      for i = 1 to k:                   # create an orthogonal basis for aff(p''_i)
        a''_i = p''_i
        for j = k-1 downto 1:
          a''_i -= a''_j * (a''_i \cdot c''_j)/c''_j^2
      for i = 1 to k:
        c'' -= a''_i * (c'' \cdot a''_i) / (a''_i^2)
      c' = c''[1..d]/(-2 * c''[d+1])    # scale c'' down

  * Ulfar Erlingsson's, Erich Kaltofen's and David Musser's "Generic 
    Gram-Schmidt Orthogonalization by Exact Division" (ISSAC'96) gives an 
    algorithm to compute Gram-Schmidt for vector space with integral domain 
    coefficients. I need to understand the paper better to see the advantages 
    over normalizing rationals at every step of the computation above 
    (the paper's introduction says that it avoids "costly GCD computations").

  * Using QR decomposition (or any other orthogonal decomposition), and 
    then using the formula of the circumsphere on the matrix R (which is 
    just the simplex rotated into a k-dimensional space):

    * Only (d-2) determinants of a k by k matrix are required which is nice.
    * All (implemented) computation of the QR decomposition that I have found 
      so far requires square roots (for norms), divisions, and often
      comparisons.  While divisions are acceptable, square roots and
      comparisons aren't.

    * There is a string of papers in late 80s, early 90s on square root free 
      and division free algorithms for QR decomposition out of VLSI community.
      There is a paper that claims to put all such techniques into one
      framework. There is also a paper that says that all such techniques don't
      work, and square root implementation needs to be reconsidered.  (authors
      to look up if searching for these papers are K.J.R. Liu and E.

  * Using the formulas from Herbert's Weighted Alpha Shapes paper which are 
    derived from formulas of the planes that are lifted Voronoi cells. This 
    approach requires (d-k)*(d+2) determinants of k by k matrices, which is not