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BOSL2/vectors.scad
Adrian Mariano cee36c13f2 add vector_perp
2024-12-03 20:34:17 -05:00

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//////////////////////////////////////////////////////////////////////
// LibFile: vectors.scad
// This file provides some mathematical operations that apply to each
// entry in a vector. It provides normalization and angle computation, and
// it provides functions for searching lists of vectors for matches to
// a given vector.
// Includes:
// include <BOSL2/std.scad>
// FileGroup: Math
// FileSummary: Vector arithmetic, angle, and searching.
// FileFootnotes: STD=Included in std.scad
//////////////////////////////////////////////////////////////////////
// Section: Vector Testing
// Function: is_vector()
// Synopsis: Returns true if the given value is a vector.
// Topics: Vectors, Math
// See Also: is_matrix(), is_path(), is_region()
// Usage:
// bool = is_vector(v, [length], [zero=], [all_nonzero=], [eps=]);
// Description:
// Returns true if v is a list of finite numbers.
// Arguments:
// v = The value to test to see if it is a vector.
// length = If given, make sure the vector is `length` items long.
// ---
// zero = If false, require that the `norm()` of the vector is not approximately zero. If true, require the `norm()` of the vector to be approximately zero. Default: `undef` (don't check vector `norm()`.)
// all_nonzero = If true, requires all elements of the vector to be more than `eps` different from zero. Default: `false`
// eps = The minimum vector length that is considered non-zero. Default: `EPSILON` (`1e-9`)
// Example:
// is_vector(4); // Returns false
// is_vector([4,true,false]); // Returns false
// is_vector([3,4,INF,5]); // Returns false
// is_vector([3,4,5,6]); // Returns true
// is_vector([3,4,undef,5]); // Returns false
// is_vector([3,4,5],3); // Returns true
// is_vector([3,4,5],4); // Returns true
// is_vector([]); // Returns false
// is_vector([0,4,0],3,zero=false); // Returns true
// is_vector([0,0,0],zero=false); // Returns false
// is_vector([0,0,1e-12],zero=false); // Returns false
// is_vector([0,1,0],all_nonzero=false); // Returns false
// is_vector([1,1,1],all_nonzero=false); // Returns true
// is_vector([],zero=false); // Returns false
function is_vector(v, length, zero, all_nonzero=false, eps=EPSILON) =
is_list(v) && len(v)>0 && []==[for(vi=v) if(!is_finite(vi)) 0]
&& (is_undef(length) || (assert(is_num(length))len(v)==length))
&& (is_undef(zero) || ((norm(v) >= eps) == !zero))
&& (!all_nonzero || all_nonzero(v)) ;
// Section: Scalar operations on vectors
// Function: add_scalar()
// Synopsis: Adds a scalar value to every item in a vector.
// Topics: Vectors, Math
// See Also: add_scalar(), v_mul(), v_div()
// Usage:
// v_new = add_scalar(v, s);
// Description:
// Given a vector and a scalar, returns the vector with the scalar added to each item in it.
// Arguments:
// v = The initial array.
// s = A scalar value to add to every item in the array.
// Example:
// a = add_scalar([1,2,3],3); // Returns: [4,5,6]
function add_scalar(v,s) =
assert(is_vector(v), "Input v must be a vector")
assert(is_finite(s), "Input s must be a finite scalar")
[for(entry=v) entry+s];
// Function: v_mul()
// Synopsis: Returns the element-wise multiplication of two equal-length vectors.
// Topics: Vectors, Math
// See Also: add_scalar(), v_mul(), v_div()
// Usage:
// v3 = v_mul(v1, v2);
// Description:
// Element-wise multiplication. Multiplies each element of `v1` by the corresponding element of `v2`.
// Both `v1` and `v2` must be the same length. Returns a vector of the products. Note that
// the items in `v1` and `v2` can be anything that OpenSCAD will multiply.
// Arguments:
// v1 = The first vector.
// v2 = The second vector.
// Example:
// v_mul([3,4,5], [8,7,6]); // Returns [24, 28, 30]
function v_mul(v1, v2) =
assert( is_list(v1) && is_list(v2) && len(v1)==len(v2), "Incompatible input")
[for (i = [0:1:len(v1)-1]) v1[i]*v2[i]];
// Function: v_div()
// Synopsis: Returns the element-wise division of two equal-length vectors.
// Topics: Vectors, Math
// See Also: add_scalar(), v_mul(), v_div()
// Usage:
// v3 = v_div(v1, v2);
// Description:
// Element-wise vector division. Divides each element of vector `v1` by
// the corresponding element of vector `v2`. Returns a vector of the quotients.
// Arguments:
// v1 = The first vector.
// v2 = The second vector.
// Example:
// v_div([24,28,30], [8,7,6]); // Returns [3, 4, 5]
function v_div(v1, v2) =
assert( is_vector(v1) && is_vector(v2,len(v1)), "Incompatible vectors")
[for (i = [0:1:len(v1)-1]) v1[i]/v2[i]];
// Function: v_abs()
// Synopsis: Returns the absolute values of the given vector.
// Topics: Vectors, Math
// See Also: v_abs(), v_floor(), v_ceil()
// Usage:
// v2 = v_abs(v);
// Description: Returns a vector of the absolute value of each element of vector `v`.
// Arguments:
// v = The vector to get the absolute values of.
// Example:
// v_abs([-1,3,-9]); // Returns: [1,3,9]
function v_abs(v) =
assert( is_vector(v), "Invalid vector" )
[for (x=v) abs(x)];
// Function: v_floor()
// Synopsis: Returns the values of the given vector, rounded down.
// Topics: Vectors, Math
// See Also: v_abs(), v_floor(), v_ceil()
// Usage:
// v2 = v_floor(v);
// Description:
// Returns the given vector after performing a `floor()` on all items.
function v_floor(v) =
assert( is_vector(v), "Invalid vector" )
[for (x=v) floor(x)];
// Function: v_ceil()
// Synopsis: Returns the values of the given vector, rounded up.
// Topics: Vectors, Math
// See Also: v_abs(), v_floor(), v_ceil()
// Usage:
// v2 = v_ceil(v);
// Description:
// Returns the given vector after performing a `ceil()` on all items.
function v_ceil(v) =
assert( is_vector(v), "Invalid vector" )
[for (x=v) ceil(x)];
// Function: v_lookup()
// Synopsis: Like `lookup()`, but it can interpolate between vector results.
// Topics: Vectors, Math
// See Also: v_abs(), v_floor(), v_ceil()
// Usage:
// v2 = v_lookup(x, v);
// Description:
// Works just like the built-in function [`lookup()`](https://en.wikibooks.org/wiki/OpenSCAD_User_Manual/Mathematical_Functions#lookup), except that it can also interpolate between vector result values of the same length.
// Arguments:
// x = The scalar value to look up.
// v = A list of [KEY,VAL] pairs. KEYs are scalars. VALs should either all be scalar, or all be vectors of the same length.
// Example:
// x = v_lookup(4.5, [[4, [3,4,5]], [5, [5,6,7]]]); // Returns: [4,5,6]
function v_lookup(x, v) =
is_num(v[0][1])? lookup(x,v) :
let(
i = lookup(x, [for (i=idx(v)) [v[i].x,i]]),
vlo = v[floor(i)],
vhi = v[ceil(i)],
lo = vlo[1],
hi = vhi[1]
)
assert(is_vector(lo) && is_vector(hi),
"Result values must all be numbers, or all be vectors.")
assert(len(lo) == len(hi), "Vector result values must be the same length")
vlo.x == vhi.x? vlo[1] :
let( u = (x - vlo.x) / (vhi.x - vlo.x) )
lerp(lo,hi,u);
// Section: Vector Properties
// Function: unit()
// Synopsis: Returns the unit length of a given vector.
// Topics: Vectors, Math
// See Also: v_abs(), v_floor(), v_ceil()
// Usage:
// v = unit(v, [error]);
// Description:
// Returns the unit length normalized version of vector v. If passed a zero-length vector,
// asserts an error unless `error` is given, in which case the value of `error` is returned.
// Arguments:
// v = The vector to normalize.
// error = If given, and input is a zero-length vector, this value is returned. Default: Assert error on zero-length vector.
// Example:
// v1 = unit([10,0,0]); // Returns: [1,0,0]
// v2 = unit([0,10,0]); // Returns: [0,1,0]
// v3 = unit([0,0,10]); // Returns: [0,0,1]
// v4 = unit([0,-10,0]); // Returns: [0,-1,0]
// v5 = unit([0,0,0],[1,2,3]); // Returns: [1,2,3]
// v6 = unit([0,0,0]); // Asserts an error.
function unit(v, error=[[["ASSERT"]]]) =
assert(is_vector(v), "Invalid vector")
norm(v)<EPSILON? (error==[[["ASSERT"]]]? assert(norm(v)>=EPSILON,"Cannot normalize a zero vector") : error) :
v/norm(v);
// Function: v_theta()
// Synopsis: Returns the angle counter-clockwise from X+ on the XY plane.
// Topics: Vectors, Math
// See Also: unit()
// Usage:
// theta = v_theta([X,Y]);
// Description:
// Given a vector, returns the angle in degrees counter-clockwise from X+ on the XY plane.
function v_theta(v) =
assert( is_vector(v,2) || is_vector(v,3) , "Invalid vector")
atan2(v.y,v.x);
// Function: vector_angle()
// Synopsis: Returns the minor angle between two vectors.
// Topics: Vectors, Math
// See Also: unit(), v_theta()
// Usage:
// ang = vector_angle(v1,v2);
// ang = vector_angle([v1,v2]);
// ang = vector_angle(PT1,PT2,PT3);
// ang = vector_angle([PT1,PT2,PT3]);
// Description:
// If given a single list of two vectors, like `vector_angle([V1,V2])`, returns the angle between the two vectors V1 and V2.
// If given a single list of three points, like `vector_angle([A,B,C])`, returns the angle between the line segments AB and BC.
// If given two vectors, like `vector_angle(V1,V2)`, returns the angle between the two vectors V1 and V2.
// If given three points, like `vector_angle(A,B,C)`, returns the angle between the line segments AB and BC.
// Arguments:
// v1 = First vector or point.
// v2 = Second vector or point.
// v3 = Third point in three point mode.
// Example:
// ang1 = vector_angle(UP,LEFT); // Returns: 90
// ang2 = vector_angle(RIGHT,LEFT); // Returns: 180
// ang3 = vector_angle(UP+RIGHT,RIGHT); // Returns: 45
// ang4 = vector_angle([10,10], [0,0], [10,-10]); // Returns: 90
// ang5 = vector_angle([10,0,10], [0,0,0], [-10,10,0]); // Returns: 120
// ang6 = vector_angle([[10,0,10], [0,0,0], [-10,10,0]]); // Returns: 120
function vector_angle(v1,v2,v3) =
assert( ( is_undef(v3) && ( is_undef(v2) || same_shape(v1,v2) ) )
|| is_consistent([v1,v2,v3]) ,
"Bad arguments.")
assert( is_vector(v1) || is_consistent(v1), "Bad arguments.")
let( vecs = ! is_undef(v3) ? [v1-v2,v3-v2] :
! is_undef(v2) ? [v1,v2] :
len(v1) == 3 ? [v1[0]-v1[1], v1[2]-v1[1]]
: v1
)
assert(is_vector(vecs[0],2) || is_vector(vecs[0],3), "Bad arguments.")
let(
norm0 = norm(vecs[0]),
norm1 = norm(vecs[1])
)
assert(norm0>0 && norm1>0, "Zero length vector.")
// NOTE: constrain() corrects crazy FP rounding errors that exceed acos()'s domain.
acos(constrain((vecs[0]*vecs[1])/(norm0*norm1), -1, 1));
// Function: vector_axis()
// Synopsis: Returns the perpendicular axis between two vectors.
// Topics: Vectors, Math
// See Also: unit(), v_theta(), vector_angle()
// Usage:
// axis = vector_axis(v1,v2);
// axis = vector_axis([v1,v2]);
// axis = vector_axis(PT1,PT2,PT3);
// axis = vector_axis([PT1,PT2,PT3]);
// Description:
// If given a single list of two vectors, like `vector_axis([V1,V2])`, returns the vector perpendicular the two vectors V1 and V2.
// If given a single list of three points, like `vector_axis([A,B,C])`, returns the vector perpendicular to the plane through a, B and C.
// If given two vectors, like `vector_axis(V1,V2)`, returns the vector perpendicular to the two vectors V1 and V2.
// If given three points, like `vector_axis(A,B,C)`, returns the vector perpendicular to the plane through a, B and C.
// Arguments:
// v1 = First vector or point.
// v2 = Second vector or point.
// v3 = Third point in three point mode.
// Example:
// axis1 = vector_axis(UP,LEFT); // Returns: [0,-1,0] (FWD)
// axis2 = vector_axis(RIGHT,LEFT); // Returns: [0,-1,0] (FWD)
// axis3 = vector_axis(UP+RIGHT,RIGHT); // Returns: [0,1,0] (BACK)
// axis4 = vector_axis([10,10], [0,0], [10,-10]); // Returns: [0,0,-1] (DOWN)
// axis5 = vector_axis([10,0,10], [0,0,0], [-10,10,0]); // Returns: [-0.57735, -0.57735, 0.57735]
// axis6 = vector_axis([[10,0,10], [0,0,0], [-10,10,0]]); // Returns: [-0.57735, -0.57735, 0.57735]
function vector_axis(v1,v2=undef,v3=undef) =
is_vector(v3)
? assert(is_consistent([v3,v2,v1]), "Bad arguments.")
vector_axis(v1-v2, v3-v2)
: assert( is_undef(v3), "Bad arguments.")
is_undef(v2)
? assert( is_list(v1), "Bad arguments.")
len(v1) == 2
? vector_axis(v1[0],v1[1])
: vector_axis(v1[0],v1[1],v1[2])
: assert( is_vector(v1,zero=false) && is_vector(v2,zero=false) && is_consistent([v1,v2])
, "Bad arguments.")
let(
eps = 1e-6,
w1 = point3d(v1/norm(v1)),
w2 = point3d(v2/norm(v2)),
w3 = (norm(w1-w2) > eps && norm(w1+w2) > eps) ? w2
: (norm(v_abs(w2)-UP) > eps)? UP
: RIGHT
) unit(cross(w1,w3));
// Function: vector_bisect()
// Synopsis: Returns the vector that bisects two vectors.
// Topics: Vectors, Math
// See Also: unit(), v_theta(), vector_angle(), vector_axis()
// Usage:
// newv = vector_bisect(v1,v2);
// Description:
// Returns a unit vector that exactly bisects the minor angle between two given vectors.
// If given two vectors that are directly opposed, returns `undef`.
function vector_bisect(v1,v2) =
assert(is_vector(v1))
assert(is_vector(v2))
assert(!approx(norm(v1),0), "Zero length vector.")
assert(!approx(norm(v2),0), "Zero length vector.")
assert(len(v1)==len(v2), "Vectors are of different sizes.")
let( v1 = unit(v1), v2 = unit(v2) )
approx(v1,-v2)? undef :
let(
axis = vector_axis(v1,v2),
ang = vector_angle(v1,v2),
v3 = unit(rot(ang/2, v=axis, p=v1))
) v3;
// Function: vector_perp()
// Synopsis: Returns component of a vector perpendicular to a second vector
// Topics: Vectors, Math
// Usage:
// perp = vector_perp(v,w);
// Description:
// Returns the component of vector w that is perpendicular to vector v. Vectors must have the same length.
// Arguments:
// v = reference vector
// w = vector whose perpendicular component is returned
// Example(2D): We extract the component of the red vector that is perpendicular to the yellow vector. That component appears in blue.
// v = [12,6];
// w = [13,22];
// stroke([[0,0],v],endcap2="arrow2");
// stroke([[0,0],w],endcap2="arrow2",color="red");
// stroke([[0,0],vector_perp(v,w)], endcap2="arrow2", color="blue");
function vector_perp(v,w) =
assert(is_vector(v) && is_vector(w) && len(v)==len(w), "Invalid or mismatched inputs")
w - w*v*v/(v*v);
// Section: Vector Searching
// Function: pointlist_bounds()
// Synopsis: Returns the min and max bounding coordinates for the given list of points.
// Topics: Geometry, Bounding Boxes, Bounds
// See Also: closest_point()
// Usage:
// pt_pair = pointlist_bounds(pts);
// Description:
// Finds the bounds containing all the points in `pts` which can be a list of points in any dimension.
// Returns a list of two items: a list of the minimums and a list of the maximums. For example, with
// 3d points `[[MINX, MINY, MINZ], [MAXX, MAXY, MAXZ]]`
// Arguments:
// pts = List of points.
function pointlist_bounds(pts) =
assert(is_path(pts,dim=undef,fast=true) , "Invalid pointlist." )
let(
select = ident(len(pts[0])),
spread = [
for(i=[0:len(pts[0])-1])
let( spreadi = pts*select[i] )
[ min(spreadi), max(spreadi) ]
]
) transpose(spread);
// Function: closest_point()
// Synopsis: Finds the closest point in a list of points.
// Topics: Geometry, Points, Distance
// See Also: pointlist_bounds(), furthest_point(), closest_point()
// Usage:
// index = closest_point(pt, points);
// Description:
// Given a list of `points`, finds the index of the closest point to `pt`.
// Arguments:
// pt = The point to find the closest point to.
// points = The list of points to search.
function closest_point(pt, points) =
assert( is_vector(pt), "Invalid point." )
assert(is_path(points,dim=len(pt)), "Invalid pointlist or incompatible dimensions." )
min_index([for (p=points) norm(p-pt)]);
// Function: furthest_point()
// Synopsis: Finds the furthest point in a list of points.
// Topics: Geometry, Points, Distance
// See Also: pointlist_bounds(), furthest_point(), closest_point()
// Usage:
// index = furthest_point(pt, points);
// Description:
// Given a list of `points`, finds the index of the furthest point from `pt`.
// Arguments:
// pt = The point to find the farthest point from.
// points = The list of points to search.
function furthest_point(pt, points) =
assert( is_vector(pt), "Invalid point." )
assert(is_path(points,dim=len(pt)), "Invalid pointlist or incompatible dimensions." )
max_index([for (p=points) norm(p-pt)]);
// Function: vector_search()
// Synopsis: Finds points in a list that are close to a given point.
// Topics: Search, Points, Closest
// See Also: vector_search_tree(), vector_nearest()
// Usage:
// indices = vector_search(query, r, target);
// Description:
// Given a list of query points `query` and a `target` to search,
// finds the points in `target` that match each query point. A match holds when the
// distance between a point in `target` and a query point is less than or equal to `r`.
// The returned list will have a list for each query point containing, in arbitrary
// order, the indices of all points that match that query point.
// The `target` may be a simple list of points or a search tree.
// When `target` is a large list of points, a search tree is constructed to
// speed up the search with an order around O(log n) per query point.
// For small point lists, a direct search is done dispensing a tree construction.
// Alternatively, `target` may be a search tree built with `vector_search_tree()`.
// In that case, that tree is parsed looking for matches.
// An empty list of query points will return a empty output list.
// An empty list of target points will return a output list with an empty list for each query point.
// Arguments:
// query = list of points to find matches for.
// r = the search radius.
// target = list of the points to search for matches or a search tree.
// Example: A set of four queries to find points within 1 unit of the query. The circles show the search region and all have radius 1.
// $fn=32;
// k = 2000;
// points = list_to_matrix(rands(0,10,k*2,seed=13333),2);
// queries = [for(i=[3,7],j=[3,7]) [i,j]];
// search_ind = vector_search(queries, points, 1);
// move_copies(points) circle(r=.08);
// for(i=idx(queries)){
// color("blue")stroke(move(queries[i],circle(r=1)), closed=true, width=.08);
// color("red") move_copies(select(points, search_ind[i])) circle(r=.08);
// }
// Example: when a series of searches with different radius are needed, its is faster to pre-compute the tree
// $fn=32;
// k = 2000;
// points = list_to_matrix(rands(0,10,k*2),2,seed=13333);
// queries1 = [for(i=[3,7]) [i,i]];
// queries2 = [for(i=[3,7]) [10-i,i]];
// r1 = 1;
// r2 = .7;
// search_tree = vector_search_tree(points);
// search_1 = vector_search(queries1, r1, search_tree);
// search_2 = vector_search(queries2, r2, search_tree);
// move_copies(points) circle(r=.08);
// for(i=idx(queries1)){
// color("blue")stroke(move(queries1[i],circle(r=r1)), closed=true, width=.08);
// color("red") move_copies(select(points, search_1[i])) circle(r=.08);
// }
// for(i=idx(queries2)){
// color("green")stroke(move(queries2[i],circle(r=r2)), closed=true, width=.08);
// color("red") move_copies(select(points, search_2[i])) circle(r=.08);
// }
function vector_search(query, r, target) =
query==[] ? [] :
is_list(query) && target==[] ? is_vector(query) ? [] : [for(q=query) [] ] :
assert( is_finite(r) && r>=0,
"The query radius should be a positive number." )
let(
tgpts = is_matrix(target), // target is a point list
tgtree = is_list(target) // target is a tree
&& (len(target)==2)
&& is_matrix(target[0])
&& is_list(target[1])
&& (len(target[1])==4 || (len(target[1])==1 && is_list(target[1][0])) )
)
assert( tgpts || tgtree,
"The target should be a list of points or a search tree compatible with the query." )
let(
dim = tgpts ? len(target[0]) : len(target[0][0]),
simple = is_vector(query, dim)
)
assert( simple || is_matrix(query,undef,dim),
"The query points should be a list of points compatible with the target point list.")
tgpts
? len(target)<=400
? simple ? [for(i=idx(target)) if(norm(target[i]-query)<=r) i ] :
[for(q=query) [for(i=idx(target)) if(norm(target[i]-q)<=r) i ] ]
: let( tree = _bt_tree(target, count(len(target)), leafsize=25) )
simple ? _bt_search(query, r, target, tree) :
[for(q=query) _bt_search(q, r, target, tree)]
: simple ? _bt_search(query, r, target[0], target[1]) :
[for(q=query) _bt_search(q, r, target[0], target[1])];
//Ball tree search
function _bt_search(query, r, points, tree) =
assert( is_list(tree)
&& ( ( len(tree)==1 && is_list(tree[0]) )
|| ( len(tree)==4 && is_num(tree[0]) && is_num(tree[1]) ) ),
"The tree is invalid.")
len(tree)==1
? assert( tree[0]==[] || is_vector(tree[0]), "The tree is invalid." )
[for(i=tree[0]) if(norm(points[i]-query)<=r) i ]
: norm(query-points[tree[0]]) > r+tree[1] ? [] :
concat(
[ if(norm(query-points[tree[0]])<=r) tree[0] ],
_bt_search(query, r, points, tree[2]),
_bt_search(query, r, points, tree[3]) ) ;
// Function: vector_search_tree()
// Synopsis: Makes a distance search tree for a list of points.
// Topics: Search, Points, Closest
// See Also: vector_nearest(), vector_search()
// Usage:
// tree = vector_search_tree(points,leafsize);
// Description:
// Construct a search tree for the given list of points to be used as input
// to the function `vector_search()`. The use of a tree speeds up the
// search process. The tree construction stops branching when
// a tree node represents a number of points less or equal to `leafsize`.
// Search trees are ball trees. 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. (See https://en.wikipedia.org/wiki/Ball_tree)
// For a small lists of points, the search with a tree may be more expensive
// than direct comparisons. The argument `treemin` sets the minimum length of
// point set for which a tree search will be done by `vector_search`.
// For an empty list of points it returns an empty list.
// Arguments:
// points = list of points to store in the search tree.
// leafsize = the size of the tree leaves. Default: 25
// treemin = the minimum size of the point list for which a tree search is done. Default: 400
// Example: A set of four queries to find points within 1 unit of the query. The circles show the search region and all have radius 1.
// $fn=32;
// k = 2000;
// points = random_points(k, scale=10, dim=2,seed=13333);
// queries = [for(i=[3,7],j=[3,7]) [i,j]];
// search_tree = vector_search_tree(points);
// search_ind = vector_search(queries,1,search_tree);
// move_copies(points) circle(r=.08);
// for(i=idx(queries)){
// color("blue") stroke(move(queries[i],circle(r=1)), closed=true, width=.08);
// color("red") move_copies(select(points, search_ind[i])) circle(r=.08);
// }
function vector_search_tree(points, leafsize=25, treemin=400) =
points==[] ? [] :
assert( is_matrix(points), "The input list entries should be points." )
assert( is_int(leafsize) && leafsize>=1,
"The tree leaf size should be an integer greater than zero.")
len(points)<treemin ? points :
[ points, _bt_tree(points, count(len(points)), leafsize) ];
//Ball tree construction
function _bt_tree(points, ind, leafsize=25) =
len(ind)<=leafsize ? [ind] :
let(
bounds = pointlist_bounds(select(points,ind)),
coord = max_index(bounds[1]-bounds[0]),
projc = [for(i=ind) points[i][coord] ],
meanpr = mean(projc),
pivot = min_index([for(p=projc) abs(p-meanpr)]),
radius = max([for(i=ind) norm(points[ind[pivot]]-points[i]) ]),
Lind = [for(i=idx(ind)) if(projc[i]<=meanpr && i!=pivot) ind[i] ],
Rind = [for(i=idx(ind)) if(projc[i] >meanpr && i!=pivot) ind[i] ]
)
[ ind[pivot], radius, _bt_tree(points, Lind, leafsize), _bt_tree(points, Rind, leafsize) ];
// Function: vector_nearest()
// Synopsis: Finds the `k` nearest points in a list to a given point.
// Topics: Search, Points, Closest
// See Also: vector_search(), vector_search_tree()
// Usage:
// indices = vector_nearest(query, k, target);
// Description:
// Search `target` for the `k` points closest to point `query`.
// The input `target` is either a list of points to search or a search tree
// pre-computed by `vector_search_tree(). A list is returned containing the indices
// of the points found in sorted order, closest point first.
// Arguments:
// query = point to search for
// k = number of neighbors to return
// target = a list of points or a search tree to search in
// Example: Four queries to find the 15 nearest points. The circles show the radius defined by the most distant query result. Note they are different for each query.
// $fn=32;
// k = 1000;
// points = list_to_matrix(rands(0,10,k*2,seed=13333),2);
// tree = vector_search_tree(points);
// queries = [for(i=[3,7],j=[3,7]) [i,j]];
// search_ind = [for(q=queries) vector_nearest(q, 15, tree)];
// move_copies(points) circle(r=.08);
// for(i=idx(queries)){
// circle = circle(r=norm(points[last(search_ind[i])]-queries[i]));
// color("red") move_copies(select(points, search_ind[i])) circle(r=.08);
// color("blue") stroke(move(queries[i], circle), closed=true, width=.08);
// }
function vector_nearest(query, k, target) =
assert(is_int(k) && k>0)
assert(is_vector(query), "Query must be a vector.")
let(
tgpts = is_matrix(target,undef,len(query)), // target is a point list
tgtree = is_list(target) // target is a tree
&& (len(target)==2)
&& is_matrix(target[0],undef,len(query))
&& (len(target[1])==4 || (len(target[1])==1 && is_list(target[1][0])) )
)
assert( tgpts || tgtree,
"The target should be a list of points or a search tree compatible with the query." )
assert((tgpts && (k<=len(target))) || (tgtree && (k<=len(target[0]))),
"More results are requested than the number of points.")
tgpts
? let( tree = _bt_tree(target, count(len(target))) )
column(_bt_nearest( query, k, target, tree),0)
: column(_bt_nearest( query, k, target[0], target[1]),0);
//Ball tree nearest
function _bt_nearest(p, k, points, tree, answers=[]) =
assert( is_list(tree)
&& ( ( len(tree)==1 && is_list(tree[0]) )
|| ( len(tree)==4 && is_num(tree[0]) && is_num(tree[1]) ) ),
"The tree is invalid.")
len(tree)==1
? _insert_many(answers, k, [for(entry=tree[0]) [entry, norm(points[entry]-p)]])
: let( d = norm(p-points[tree[0]]) )
len(answers)==k && ( d > last(answers)[1]+tree[1] ) ? answers :
let(
answers1 = _insert_sorted(answers, k, [tree[0],d]),
answers2 = _bt_nearest(p, k, points, tree[2], answers1),
answers3 = _bt_nearest(p, k, points, tree[3], answers2)
)
answers3;
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
: assert(is_vector(newlist[i],2), "The tree is invalid.")
_insert_many(_insert_sorted(list,k,newlist[i]),k,newlist,i+1);
// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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