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Add vantage point tree for searching vector lists. Add "concave"

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method to vnf_vertex_array.
This commit is contained in:
Adrian Mariano 2021-05-11 20:51:09 -04:00
parent a9cd600782
commit c4ace59ccd
3 changed files with 171 additions and 4 deletions
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23
math.scad
View file

@ -744,6 +744,29 @@ function mean(v) =
sum(v)/len(v);
// Function: ninther()
// Usage:
// med = ninther(v)
// Description:
// Finds a value in the input list of numbers `v` that is the median of a
// sample of 9 entries of `v`.
// It is a much faster approximation of the true median computation.
// Arguments:
// v = an array of numbers
function ninther(v) =
let( l=len(v) )
l<=4 ? l<=2 ? v[0] : _med3(v[0], v[1], v[2]) :
l==5 ? _med3(v[0], _med3(v[1], v[2], v[3]), v[4]) :
_med3(_med3(v[0],v[floor(l/6)],v[floor(l/3)]),
_med3(v[floor(l/3)],v[floor(l/2)],v[floor(2*l/3)]),
_med3(v[floor(2*l/3)],v[floor((5*l/3 -1)/2)],v[l-1]) );
// the median of a triple
function _med3(a,b,c) =
a < c ? a < b ? min(b,c) : min(a,c) :
b < c ? min(a,c) : min(a,b);
// Function: convolve()
// Usage:
// x = convolve(p,q);

136
vectors.scad
View file

@ -218,5 +218,141 @@ function vector_axis(v1,v2=undef,v3=undef) =
) unit(cross(w1,w3));
// Section: Vector Searching
// Function: vp_tree()
// Usage:
// tree = vp_tree(points, <leafsize>)
// Description:
// Organizes n-dimensional data into a Vantage Point Tree, which can be
// efficiently searched for for nearest matches. The Vantage Point Tree
// is an effort to generalize binary search to n dimensions. Constructing the
// tree should be O(n log n) and searches should be O(log n), though real life
// performance depends on how the data is distributed, and it will deteriorate
// for high data dimensions. This data structure is useful when you will be
// performing many searches of the same data, so that the cost of constructing
// the tree is justified.
// .
// The vantage point tree at a given level chooses vp, the
// "vantage point", and a radius, R, and divides the data based
// on distance to vp. Points closer than R go in on branch
// of the tree and points farther than R go in the other branch.
// .
// The tree has the form [vp, R, inside, outside], where vp is
// the vantage point index, R is the radius, inside is a
// recursively computed tree for the inside points (distance less than
// or equal to R from the vantage point), and outside
// is a tree for the outside points (distance greater than R from the
// vantage point).
// .
// If the number of points is less than or equal to leafsize then
// vp_tree instead returns the list [ind] where ind is a list of
// the indices of the points. This means the list has the form
// [[i0, i1, i2,...]], so tree[0] is a list of indices. You can
// tell that a node is a leaf node by checking if tree[0] is a list.
// The leafsize parameter determines how many points can be
// store in the leaf nodes. The default value of 25 was found
// emperically to be a reasonable option for 3d data searched with vp_search().
// .
// Vantage point tree is described here: http://web.cs.iastate.edu/~honavar/nndatastructures.pdf
// Arguments:
// points = list of points to store in the tree
// leafsize = maximum number of points to store in the tree's leaf nodes. Default: 25
function vp_tree(points, leafsize=25) =
assert(is_matrix(points),"points must be a consistent list of data points")
_vp_tree(points, count(len(points)), leafsize);
function _vp_tree(ptlist, ind, leafsize) =
len(ind)<=leafsize ? [ind] :
let(
center = mean(select(ptlist,ind)),
cdistances = [for(i=ind) norm(ptlist[i]-center)],
vpind = ind[max_index(cdistances)],
vp = ptlist[vpind],
vp_dist = [for(i=ind) norm(vp-ptlist[i])],
r = ninther(vp_dist),
inside = [for(i=idx(ind)) if (vp_dist[i]<=r && ind[i]!=vpind) ind[i]],
outside = [for(i=idx(ind)) if (vp_dist[i]>r) ind[i]]
)
[vpind, r, _vp_tree(ptlist,inside,leafsize),_vp_tree(ptlist,outside,leafsize)];
// Function: vp_search()
// Usage:
// indices = vp_search(points, tree, p, r);
// Description:
// Search a vantage point tree for all points whose distance from p
// is less than or equal to r. Returns a list of indices of the points it finds
// in arbitrary order. The input points is a list of points to search and tree is the
// vantage point tree computed from that point list. The search should be
// around O(log n).
// Arguments:
// points = points indexed by the vantage point tree
// tree = vantage point tree from vp_tree
// p = point to search for
// r = search radius
function _vp_search(points, tree, p, r) =
is_list(tree[0]) ? [for(i=tree[0]) if (norm(points[i]-p)<=r) i]
:
let(
d = norm(p-points[tree[0]]) // dist to vantage point
)
[
if (d <= r) tree[0],
if (d-r <= tree[1]) each _vp_search(points, tree[2], p, r),
if (d+r > tree[1]) each _vp_search(points, tree[3], p, r)
];
function vp_search(points, tree, p, r) =
assert(is_list(tree) && (len(tree)==4 || (len(tree)==1 && is_list(tree[0]))), "Vantage point tree not valid")
assert(is_matrix(points), "Parameter points is not a consistent point list")
assert(is_vector(p,len(points[0])), "Query must be a vector whose length matches the point list")
assert(all_positive(r),"Radius r must be a positive number")
_vp_search(points, tree, p, r);
// Function: vp_nearest()
// Usage:
// indices = vp_nearest(points, tree, p, k)
// Description:
// Search the vantage point tree for the k points closest to point p.
// The input points is the list of points to search and tree is
// the vantage point tree computed from that point list. The list is
// returned in sorted order, closest point first.
// Arguments:
// points = points indexed by the vantage point tree
// tree = vantage point tree from vp_tree
// p = point to search for
// k = number of neighbors to return
function _insert_sorted(list, k, new) =
len(list)==k && new[1]>= last(list)[1] ? list
: [
for(entry=list) if (entry[1]<=new[1]) entry,
new,
for(i=[0:1:min(k-1,len(list))-1]) if (list[i][1]>new[1]) list[i]
];
function _insert_many(list, k, newlist,i=0) =
i==len(newlist) ? list :
_insert_many(_insert_sorted(list,k,newlist[i]),k,newlist,i+1);
function _vp_nearest(points, tree, p, k, answers=[]) =
is_list(tree[0]) ? _insert_many(answers, k, [for(entry=tree[0]) [entry, norm(points[entry]-p)]]) :
let(
d = norm(p-points[tree[0]]),
answers1 = _insert_sorted(answers, k, [tree[0],d]),
answers2 = d-last(answers1)[1] <= tree[1] ? _vp_nearest(points, tree[2], p, k, answers1) : answers1,
answers3 = d+last(answers2)[1] > tree[1] ? _vp_nearest(points, tree[3], p, k, answers2) : answers2
)
answers3;
function vp_nearest(points, tree, p, k) =
assert(is_int(k) && k>0)
assert(k<=len(points), "You requested more results that contained in the set")
assert(is_matrix(points), "Parameter points is not a consistent point list")
assert(is_vector(p,len(points[0])), "Query must be a vector whose length matches the point list")
assert(is_list(tree) && (len(tree)==4 || (len(tree)==1 && is_list(tree[0]))), "Vantage point tree not valid")
subindex(_vp_nearest(points, tree, p, k),0);
// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap

16
vnf.scad
View file

@ -221,8 +221,8 @@ function vnf_triangulate(vnf) =
// triangles. The default style is an arbitrary, systematic subdivision in the same direction. The "alt" style
// is the uniform subdivision in the other (alternate) direction. The "min_edge" style picks the shorter edge to
// subdivide for each quadrilateral, so the division may not be uniform across the shape. The "quincunx" style
// adds a vertex in the center of each quadrilateral and creates four triangles, and the "convex" style
// chooses the locally convex subdivision.
// adds a vertex in the center of each quadrilateral and creates four triangles, and the "convex" and "concave" styles
// chooses the locally convex/concave subdivision.
// Arguments:
// points = A list of vertices to divide into columns and rows.
// caps = If true, add endcap faces to the first AND last rows.
@ -231,7 +231,7 @@ function vnf_triangulate(vnf) =
// col_wrap = If true, add faces to connect the last column to the first.
// row_wrap = If true, add faces to connect the last row to the first.
// reverse = If true, reverse all face normals.
// style = The style of subdividing the quads into faces. Valid options are "default", "alt", "min_edge", "quincunx", and "convex".
// style = The style of subdividing the quads into faces. Valid options are "default", "alt", "min_edge", "quincunx","convex" and "concave".
// vnf = If given, add all the vertices and faces to this existing VNF structure.
// Example(3D):
// vnf = vnf_vertex_array(
@ -297,7 +297,7 @@ function vnf_vertex_array(
) =
assert(!(any([caps,cap1,cap2]) && !col_wrap), "col_wrap must be true if caps are requested")
assert(!(any([caps,cap1,cap2]) && row_wrap), "Cannot combine caps with row_wrap")
assert(in_list(style,["default","alt","quincunx", "convex","min_edge"]))
assert(in_list(style,["default","alt","quincunx", "convex","concave", "min_edge"]))
assert(is_consistent(points), "Non-rectangular or invalid point array")
let(
pts = flatten(points),
@ -358,6 +358,14 @@ function vnf_vertex_array(
: [[i1,i3,i2],[i1,i4,i3]]
)
convexfaces
: style=="concave"?
let( // Find normal for 3 of the points. Is the other point above or below?
n = (reverse?-1:1)*cross(pts[i2]-pts[i1],pts[i3]-pts[i1]),
concavefaces = n==0 ? [[i1,i4,i3]]
: n*pts[i4] <= n*pts[i1] ? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
concavefaces
: [[i1,i3,i2],[i1,i4,i3]],
// remove degenerate faces
culled_faces= [for(face=faces)