CGA Interval Classifier

Ported from a standalone JS/Canvas prototype into real PatLang, compiled to WASM and run live below — see the Paradigms Guide for the numeric tower this interval arithmetic builds on, and the k-blade page for the broader geometric-algebra-as-representation idea this demo's dimensionality reduction is one small example of.

A CGA point embeds a 2D location plus an uncertainty radius as an INTERVAL-valued null vector; a CGA sphere is the same components as plain scalars. Their inner product is therefore an interval, not a single number -- a pixel is only certified inside or outside when zero falls strictly outside that interval; when the interval straddles zero, the classifier honestly reports it ambiguous rather than rounding arbitrarily. Every embedded point is also checked against the Minkowski null-cone identity X² = 0, certifying the embedding itself, not just the classification built on it.

The dataset lives in a synthetic 5D sparse space and is projected down to the 2D plane the classifier sees via a structural k-blade (projected = sum(component × blade)). Toggle Dimensionality Projection to compare a deliberate blade-guided projection (which cancels most of the hidden dimension's cross-plane noise) against naively summing raw components (which lets that noise bleed straight into the visible plane) -- the same k-blade-instead-of-scalar-tensor idea explored on the k-blade page.

# Validated interval Conformal Geometric Algebra (CGA) point-in-sphere
# classifier, with k-blade dimensionality reduction of a sparse 5D
# dataset down to the 2D plane the classifier actually operates on.
#
# A CGA point embeds a 2D location (x, y) plus an uncertainty radius r as
# an INTERVAL-valued null vector: e1/e2 carry [x-r, x+r]/[y-r, y+r], and
# eInf/eO carry the interval bounds of the point's squared norm (the
# conformal embedding that makes "inside/outside a sphere" a single inner
# product). A CGA sphere is the same 4 components as plain scalars. The
# inner product of an interval point against a scalar sphere yields an
# INTERVAL, not a single number -- classification is only certain when
# zero is NOT inside that interval (strictly inside or strictly outside);
# when the interval straddles zero, the pixel is honestly reported
# ambiguous rather than arbitrarily rounded one way.
#
# Every valid CGA point must satisfy the Minkowski null-cone identity
# X^2 = 0 exactly; verify_minkowski checks that 0 is provably trapped
# inside the interval-arithmetic worst-case bounds of X^2 for an embedded
# point, certifying the embedding itself is correct, not just plausible.
#
# The dataset is generated in a synthetic 5D sparse space (d1..d5, only
# d1/d2/d3 ever populated) and projected down to the 2D plane via a
# structural k-blade: projected_x = sum(component * blade.x), projected_y
# = sum(component * blade.y). Toggling reduction on/off shows the
# difference between a deliberate blade-guided projection (which cancels
# most of d3's cross-plane noise) and naively summing raw components
# (which lets that noise bleed straight into the visible plane).
#
# All randomness uses a small seeded LCG (PatLang has no RNG host
# function) so the SAME seed always reproduces the SAME dataset --
# argv carries the seed explicitly, so re-classifying with a new epsilon/
# sphere/projection mode (same seed) never jitters the point layout, only
# a genuinely "new" dataset request passes a new seed.

make a function called rand_seed takes seed returns done
  let s = seed % 2147483647
  let mut s2 = s
  if s <= 0 then
    s2 = 1
  end
  set_var("cga_rand", s2)
  return true
end

make a function called rand_next returns r
  let s = to_num(get("__vars", "cga_rand"))
  let s2 = (s * 48271) % 2147483647
  set_var("cga_rand", s2)
  return s2
end

# A pseudo-random float in [-1, 1].
make a function called rand_signed returns r
  let n = rand_next()
  let frac = (n % 20000) / 10000
  return frac - 1
end

# ---- interval-valued CGA point / scalar sphere -----------------------
# Interval point: [e1_lo, e1_hi, e2_lo, e2_hi, eInf_lo, eInf_hi, eO_lo, eO_hi]
# Scalar sphere:  [e1, e2, eInf, eO]

make a function called cga_embed takes x, y, r returns pt
  let norm2 = x * x + y * y
  let dist = sqrt(norm2)
  let norm2_lo = max(0, norm2 - 2 * r * dist + r * r)
  let norm2_hi = norm2 + 2 * r * dist + r * r
  return [x - r, x + r, y - r, y + r, 0.5 * norm2_lo, 0.5 * norm2_hi, 1.0, 1.0]
end

make a function called cga_create_sphere takes cx, cy, radius returns sph
  let norm2 = cx * cx + cy * cy
  return [cx, cy, 0.5 * (norm2 - radius * radius), 1.0]
end

# Returns [lo, hi] of the inner product X . S (interval point . scalar sphere).
make a function called cga_inner_point_sphere takes pt, sph returns bounds
  let e1_lo = pt[0]
  let e1_hi = pt[1]
  let e2_lo = pt[2]
  let e2_hi = pt[3]
  let eInf_lo = pt[4]
  let eInf_hi = pt[5]
  let eO_lo = pt[6]
  let sph_e1 = sph[0]
  let sph_e2 = sph[1]
  let sph_eInf = sph[2]
  let sph_eO = sph[3]
  let t1a = e1_lo * sph_e1
  let t1b = e1_hi * sph_e1
  let term1_lo = min(t1a, t1b)
  let term1_hi = max(t1a, t1b)
  let t2a = e2_lo * sph_e2
  let t2b = e2_hi * sph_e2
  let term2_lo = min(t2a, t2b)
  let term2_hi = max(t2a, t2b)
  let cross_lo = eInf_lo * sph_eO + eO_lo * sph_eInf
  let cross_hi = eInf_hi * sph_eO + eO_lo * sph_eInf
  return [term1_lo + term2_lo - cross_hi, term1_hi + term2_hi - cross_lo]
end

# Returns [is_zero_trapped (1/0), lo, hi] for X^2 of an interval point.
make a function called cga_verify_minkowski takes pt returns result
  let e1_lo = pt[0]
  let e1_hi = pt[1]
  let e2_lo = pt[2]
  let e2_hi = pt[3]
  let eInf_lo = pt[4]
  let eInf_hi = pt[5]
  let eO_lo = pt[6]
  let eO_hi = pt[7]
  let x1a = e1_lo * e1_lo
  let x1b = e1_hi * e1_hi
  let x1_sq_lo = min(x1a, x1b)
  let x1_sq_hi = max(x1a, x1b)
  let x2a = e2_lo * e2_lo
  let x2b = e2_hi * e2_hi
  let x2_sq_lo = min(x2a, x2b)
  let x2_sq_hi = max(x2a, x2b)
  let cross_lo = 2 * (eInf_lo * eO_lo)
  let cross_hi = 2 * (eInf_hi * eO_hi)
  let x_sq_lo = (x1_sq_lo + x2_sq_lo) - cross_hi
  let x_sq_hi = (x1_sq_hi + x2_sq_hi) - cross_lo
  let mut trapped = 0
  if (x_sq_lo <= 0.00001) and (x_sq_hi >= 0 - 0.00001) then
    trapped = 1
  end
  return [trapped, x_sq_lo, x_sq_hi]
end

# 0 = inside, 1 = outside, 2 = ambiguous (zero is inside the inner-product interval).
# This PseudoCGA embedding's sign convention makes the point-sphere inner
# product POSITIVE when inside the sphere and NEGATIVE when outside (the
# opposite of the "standard" CGA convention some references use) -- what
# matters for a valid classifier is that it's monotonic and consistent,
# verified directly: a point at the sphere's own center gives a strictly
# positive interval, a point far outside gives a strictly negative one.
make a function called classify_pixel takes x, y, epsilon, sph returns status
  let pt = cga_embed(x, y, epsilon)
  let bounds = cga_inner_point_sphere(pt, sph)
  let lo = bounds[0]
  let hi = bounds[1]
  if lo > 0 then
    return 0
  else
    if hi < 0 then
      return 1
    else
      return 2
    end
  end
end

# Hierarchical tree of joins & bisection field: a second, structurally
# different classifier -- instead of one algebraic inner product against a
# single sphere, this walks a subsample of the projected point cloud
# (every 3rd point, matching the "join" -- each point contributes a
# nearest-neighbour vote rather than a single global surface) and
# classifies a pixel by whichever label's nearest point is closer. The
# gap between the two candidate distances stands in for the sphere's
# uncertainty radius: a pixel sitting roughly equidistant from both
# labels' nearest points is exactly the "bisection" boundary, reported
# ambiguous the same honest way the CGA sphere classifier reports zero
# trapped inside its inner-product interval.
make a function called bisection_classify_pixel takes x, y, epsilon, proj_points, n returns status
  let mut min_dist_a = 999999999
  let mut min_dist_b = 999999999
  let mut i = 0
  while i < n do
    let p = proj_points[i]
    let dx = p[0] - x
    let dy = p[1] - y
    let d = sqrt(dx * dx + dy * dy)
    if p[2] == 0 then
      min_dist_a = min(min_dist_a, d)
    else
      min_dist_b = min(min_dist_b, d)
    end
    i = i + 3
  end
  if abs(min_dist_a - min_dist_b) < epsilon then
    return 2
  else
    if min_dist_a < min_dist_b then
      return 0
    else
      return 1
    end
  end
end

# ---- sparse 5D dataset + k-blade dimensionality reduction -------------

make a function called project_point takes d1, d2, d3, use_reduction returns xy
  if use_reduction then
    let px = d1 * 1.0 + d2 * 0.0 + d3 * 0.1
    let py = d1 * 0.0 + d2 * 1.0 + d3 * (0 - 0.1)
    return [px, py]
  else
    return [d1 + d3, d2 + d3]
  end
end

make a function called add_cluster takes points, base_d1, base_d2, base_d3, label, count, jitter returns points2
  let mut i = 0
  while i < count do
    let d1 = base_d1 + rand_signed() * jitter
    let d2 = base_d2 + rand_signed() * jitter
    let d3 = base_d3 + rand_signed() * (jitter + 3)
    let points = list_push(points, [d1, d2, d3, label])
    i = i + 1
  end
  return points
end

make a function called build_dataset takes name returns points
  let mut points = []
  if name == "xor" then
    points = add_cluster(points, 130, 130, 65, 0, 15, 6)
    points = add_cluster(points, 370, 370, 0 - 65, 0, 15, 6)
    points = add_cluster(points, 130, 370, 0 - 65, 1, 15, 6)
    points = add_cluster(points, 370, 130, 65, 1, 15, 6)
  else
    if name == "circles" then
      let mut i = 0
      while i < 60 do
        let a = (to_num(i) / 60) * 6.2831853
        points = add_cluster(points, 250 + cos(a) * 75, 250 + sin(a) * 75, 45, 0, 1, 3)
        points = add_cluster(points, 250 + cos(a) * 190, 250 + sin(a) * 190, 0 - 45, 1, 1, 3)
        i = i + 1
      end
    else
      let mut i = 0
      while i < 90 do
        let theta = (to_num(i) / 90) * 10.9955743
        let r = 15 + (to_num(i) / 90) * 200
        points = add_cluster(points, 250 + cos(theta) * r, 250 + sin(theta) * r, 40, 0, 1, 3)
        points = add_cluster(points, 250 + cos(theta + 3.14159265) * r, 250 + sin(theta + 3.14159265) * r, 0 - 40, 1, 1, 3)
        i = i + 1
      end
    end
  end
  return points
end

# ---- main: generate, project, classify, render -------------------------

make a function called run_classifier takes unused returns done
  let args = argv()
  let dataset = args[0]
  let use_reduction = args[1] == "reduced"
  let epsilon = to_num(args[2])
  let sphere_x = to_num(args[3])
  let sphere_y = to_num(args[4])
  let sphere_r = to_num(args[5])
  let seed = to_num(args[6])
  let classifier = args[7]
  let use_bisection = classifier == "bisection_field"

  rand_seed(seed)
  let raw_points = build_dataset(dataset)
  let sph = cga_create_sphere(sphere_x, sphere_y, sphere_r)

  let n = to_num(list_len(raw_points))
  let mut proj_points = []
  let mut i = 0
  while i < n do
    let p = raw_points[i]
    let xy = project_point(p[0], p[1], p[2], use_reduction)
    proj_points = list_push(proj_points, [xy[0], xy[1], p[3]])
    i = i + 1
  end

  print("GRID")
  let step = 4
  let mut gy = 0
  while gy < 500 do
    let line = sb_new()
    let mut gx = 0
    while gx < 500 do
      let mut status = 0
      if use_bisection then
        status = bisection_classify_pixel(gx, gy, epsilon, proj_points, n)
      else
        status = classify_pixel(gx, gy, epsilon, sph)
      end
      sb_push(line, "" + status)
      gx = gx + step
    end
    print(sb_str(line))
    gy = gy + step
  end

  print("POINTS")
  let mut i2 = 0
  while i2 < n do
    let pp = proj_points[i2]
    print(pp[0] + "," + pp[1] + "," + pp[2])
    i2 = i2 + 1
  end

  print("INVARIANT")
  let sample_pt = cga_embed(252, 252, epsilon)
  let check = cga_verify_minkowski(sample_pt)
  print(check[0] + "," + check[1] + "," + check[2])

  return true
end

run_classifier(0)

Interpreted run on the build machine (pat --ir-run, dataset=xor):

GRID
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111122222222222222222211111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111112222222222222222222222222221111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111112222222222222222222222222222222222111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111112222222222222222222222222222222222222221111111111111111111111111111111111111111
11111111111111111111111111111111111111111111222222222222222222222222222222222222222222221111111111111111111111111111111111111
11111111111111111111111111111111111111111122222222222222222222222222222222222222222222222211111111111111111111111111111111111
11111111111111111111111111111111111111112222222222222222222222222222222222222222222222222222111111111111111111111111111111111
11111111111111111111111111111111111111222222222222222222222222222222222222222222222222222222211111111111111111111111111111111
11111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222111111111111111111111111111111
11111111111111111111111111111111111222222222222222222222222222222222222222222222222222222222222221111111111111111111111111111
11111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222111111111111111111111111111
11111111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222211111111111111111111111111
11111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222221111111111111111111111111
11111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222111111111111111111111111
11111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111111
11111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111111111
11111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111111
11111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111111
11111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111
11111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111111
11111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111
11111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111
11111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111
11111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111
11111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111
11111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111
11111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111
11111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111
11111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111
11111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111
11111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111
11111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111
11111111111111112222222222222222222222222222222222222222200000002222222222222222222222222222222222222222222222222221111111111
11111111111111122222222222222222222222222222222222222200000000000002222222222222222222222222222222222222222222222221111111111
11111111111111122222222222222222222222222222222222220000000000000000022222222222222222222222222222222222222222222222111111111
11111111111111222222222222222222222222222222222222000000000000000000000222222222222222222222222222222222222222222222111111111
11111111111111222222222222222222222222222222222220000000000000000000000022222222222222222222222222222222222222222222111111111
11111111111111222222222222222222222222222222222200000000000000000000000002222222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222222000000000000000000000000000222222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222220000000000000000000000000000022222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222220000000000000000000000000000002222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222200000000000000000000000000000002222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222200000000000000000000000000000000222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222000000000000000000000000000000000222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222000000000000000000000000000000000022222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000002222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000002222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000002222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000002222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222220000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222000000000000000000000000000000000022222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222200000000000000000000000000000000222222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222200000000000000000000000000000000222222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222220000000000000000000000000000000222222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222220000000000000000000000000000002222222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222222000000000000000000000000000022222222222222222222222222222222222222222222111111
11111111111222222222222222222222222222222222222200000000000000000000000000222222222222222222222222222222222222222222221111111
11111111111222222222222222222222222222222222222220000000000000000000000000222222222222222222222222222222222222222222221111111
11111111111222222222222222222222222222222222222222000000000000000000000022222222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222222222200000000000000000000222222222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222222222222000000000000000002222222222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222222222222220000000000002222222222222222222222222222222222222222222222222221111111
11111111111122222222222222222222222222222222222222222222222000022222222222222222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111
11111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111
11111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111
11111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111
11111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111
11111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111
11111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111
11111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111
11111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111
11111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111
11111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111
11111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111
11111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111
11111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111
11111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111
11111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111
11111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111
11111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111
11111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111111
11111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111
11111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111111
11111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111111
11111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222222222222222111111111111111111111
11111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111111
11111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222222222211111111111111111111111
11111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111111111
11111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222222211111111111111111111111111
11111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222222111111111111111111111111111
11111111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222221111111111111111111111111111
11111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222111111111111111111111111111111
11111111111111111111111111111111111222222222222222222222222222222222222222222222222222222222221111111111111111111111111111111
11111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222111111111111111111111111111111111
11111111111111111111111111111111111111222222222222222222222222222222222222222222222222222211111111111111111111111111111111111
11111111111111111111111111111111111111112222222222222222222222222222222222222222222222221111111111111111111111111111111111111
11111111111111111111111111111111111111111112222222222222222222222222222222222222222222111111111111111111111111111111111111111
11111111111111111111111111111111111111111111122222222222222222222222222222222222222211111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111122222222222222222222222222222222211111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111122222222222222222222222222211111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111222222222222222211111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
POINTS
135.90234,120.53666,0
131.39367,122.41313,0
131.49878999999999,127.70080999999999,0
130.83723,119.08397000000001,0
136.28037,123.65723,0
138.90485999999999,127.87993999999999,0
134.50728,119.44352,0
140.27883,127.98017,0
137.3319,126.7271,0
139.33923,125.25376999999999,0
134.80863000000002,119.35037,0
132.89943,128.91857000000002,0
141.97803,128.11217000000002,0
135.74658000000002,128.43362,0
131.21988,122.90252000000001,0
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