mirror of
https://github.com/BelfrySCAD/BOSL2.git
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409 lines
18 KiB
OpenSCAD
409 lines
18 KiB
OpenSCAD
//////////////////////////////////////////////////////////////////////
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// LibFile: vectors.scad
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// Vector math functions.
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// Includes:
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// include <BOSL2/std.scad>
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//////////////////////////////////////////////////////////////////////
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// Section: Vector Manipulation
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// Function: is_vector()
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// Usage:
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// is_vector(v, <length>, ...);
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// Description:
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// Returns true if v is a list of finite numbers.
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// Arguments:
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// v = The value to test to see if it is a vector.
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// length = If given, make sure the vector is `length` items long.
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// zero = If false, require that the length/`norm()` of the vector is not approximately zero. If true, require the length/`norm()` of the vector to be approximately zero-length. Default: `undef` (don't check vector length/`norm()`.)
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// all_nonzero = If true, requires all elements of the vector to be more than `eps` different from zero. Default: `false`
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// eps = The minimum vector length that is considered non-zero. Default: `EPSILON` (`1e-9`)
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// Example:
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// is_vector(4); // Returns false
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// is_vector([4,true,false]); // Returns false
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// is_vector([3,4,INF,5]); // Returns false
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// is_vector([3,4,5,6]); // Returns true
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// is_vector([3,4,undef,5]); // Returns false
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// is_vector([3,4,5],3); // Returns true
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// is_vector([3,4,5],4); // Returns true
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// is_vector([]); // Returns false
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// is_vector([0,4,0],3,zero=false); // Returns true
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// is_vector([0,0,0],zero=false); // Returns false
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// is_vector([0,0,1e-12],zero=false); // Returns false
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// is_vector([0,1,0],all_nonzero=false); // Returns false
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// is_vector([1,1,1],all_nonzero=false); // Returns true
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// is_vector([],zero=false); // Returns false
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function is_vector(v, length, zero, all_nonzero=false, eps=EPSILON) =
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is_list(v) && len(v)>0 && []==[for(vi=v) if(!is_num(vi)) 0]
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&& (is_undef(length) || len(v)==length)
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&& (is_undef(zero) || ((norm(v) >= eps) == !zero))
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&& (!all_nonzero || all_nonzero(v)) ;
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// Function: v_theta()
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// Usage:
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// theta = v_theta([X,Y]);
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// Description:
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// Given a vector, returns the angle in degrees counter-clockwise from X+ on the XY plane.
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function v_theta(v) =
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assert( is_vector(v,2) || is_vector(v,3) , "Invalid vector")
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atan2(v.y,v.x);
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// Function: v_mul()
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// Description:
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// Element-wise multiplication. Multiplies each element of `v1` by the corresponding element of `v2`.
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// Both `v1` and `v2` must be the same length. Returns a vector of the products.
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// Arguments:
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// v1 = The first vector.
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// v2 = The second vector.
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// Example:
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// v_mul([3,4,5], [8,7,6]); // Returns [24, 28, 30]
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function v_mul(v1, v2) =
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assert( is_list(v1) && is_list(v2) && len(v1)==len(v2), "Incompatible input")
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[for (i = [0:1:len(v1)-1]) v1[i]*v2[i]];
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// Function: v_div()
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// Description:
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// Element-wise vector division. Divides each element of vector `v1` by
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// the corresponding element of vector `v2`. Returns a vector of the quotients.
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// Arguments:
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// v1 = The first vector.
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// v2 = The second vector.
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// Example:
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// v_div([24,28,30], [8,7,6]); // Returns [3, 4, 5]
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function v_div(v1, v2) =
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assert( is_vector(v1) && is_vector(v2,len(v1)), "Incompatible vectors")
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[for (i = [0:1:len(v1)-1]) v1[i]/v2[i]];
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// Function: v_abs()
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// Description: Returns a vector of the absolute value of each element of vector `v`.
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// Arguments:
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// v = The vector to get the absolute values of.
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// Example:
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// v_abs([-1,3,-9]); // Returns: [1,3,9]
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function v_abs(v) =
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assert( is_vector(v), "Invalid vector" )
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[for (x=v) abs(x)];
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// Function: v_floor()
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// Description:
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// Returns the given vector after performing a `floor()` on all items.
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function v_floor(v) =
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assert( is_vector(v), "Invalid vector" )
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[for (x=v) floor(x)];
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// Function: v_ceil()
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// Description:
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// Returns the given vector after performing a `ceil()` on all items.
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function v_ceil(v) =
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assert( is_vector(v), "Invalid vector" )
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[for (x=v) ceil(x)];
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// Function: unit()
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// Usage:
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// unit(v, [error]);
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// Description:
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// Returns the unit length normalized version of vector v. If passed a zero-length vector,
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// asserts an error unless `error` is given, in which case the value of `error` is returned.
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// Arguments:
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// v = The vector to normalize.
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// error = If given, and input is a zero-length vector, this value is returned. Default: Assert error on zero-length vector.
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// Examples:
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// unit([10,0,0]); // Returns: [1,0,0]
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// unit([0,10,0]); // Returns: [0,1,0]
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// unit([0,0,10]); // Returns: [0,0,1]
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// unit([0,-10,0]); // Returns: [0,-1,0]
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// unit([0,0,0],[1,2,3]); // Returns: [1,2,3]
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// unit([0,0,0]); // Asserts an error.
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function unit(v, error=[[["ASSERT"]]]) =
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assert(is_vector(v), str("Expected a vector. Got: ",v))
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norm(v)<EPSILON? (error==[[["ASSERT"]]]? assert(norm(v)>=EPSILON,"Tried to normalize a zero vector") : error) :
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v/norm(v);
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// Function: vector_angle()
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// Usage:
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// vector_angle(v1,v2);
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// vector_angle([v1,v2]);
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// vector_angle(PT1,PT2,PT3);
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// vector_angle([PT1,PT2,PT3]);
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// Description:
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// If given a single list of two vectors, like `vector_angle([V1,V2])`, returns the angle between the two vectors V1 and V2.
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// If given a single list of three points, like `vector_angle([A,B,C])`, returns the angle between the line segments AB and BC.
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// If given two vectors, like `vector_angle(V1,V2)`, returns the angle between the two vectors V1 and V2.
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// If given three points, like `vector_angle(A,B,C)`, returns the angle between the line segments AB and BC.
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// Arguments:
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// v1 = First vector or point.
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// v2 = Second vector or point.
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// v3 = Third point in three point mode.
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// Examples:
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// vector_angle(UP,LEFT); // Returns: 90
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// vector_angle(RIGHT,LEFT); // Returns: 180
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// vector_angle(UP+RIGHT,RIGHT); // Returns: 45
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// vector_angle([10,10], [0,0], [10,-10]); // Returns: 90
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// vector_angle([10,0,10], [0,0,0], [-10,10,0]); // Returns: 120
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// vector_angle([[10,0,10], [0,0,0], [-10,10,0]]); // Returns: 120
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function vector_angle(v1,v2,v3) =
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assert( ( is_undef(v3) && ( is_undef(v2) || same_shape(v1,v2) ) )
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|| is_consistent([v1,v2,v3]) ,
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"Bad arguments.")
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assert( is_vector(v1) || is_consistent(v1), "Bad arguments.")
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let( vecs = ! is_undef(v3) ? [v1-v2,v3-v2] :
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! is_undef(v2) ? [v1,v2] :
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len(v1) == 3 ? [v1[0]-v1[1], v1[2]-v1[1]]
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: v1
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)
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assert(is_vector(vecs[0],2) || is_vector(vecs[0],3), "Bad arguments.")
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let(
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norm0 = norm(vecs[0]),
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norm1 = norm(vecs[1])
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)
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assert(norm0>0 && norm1>0, "Zero length vector.")
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// NOTE: constrain() corrects crazy FP rounding errors that exceed acos()'s domain.
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acos(constrain((vecs[0]*vecs[1])/(norm0*norm1), -1, 1));
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// Function: vector_axis()
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// Usage:
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// vector_axis(v1,v2);
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// vector_axis([v1,v2]);
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// vector_axis(PT1,PT2,PT3);
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// vector_axis([PT1,PT2,PT3]);
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// Description:
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// If given a single list of two vectors, like `vector_axis([V1,V2])`, returns the vector perpendicular the two vectors V1 and V2.
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// 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.
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// If given two vectors, like `vector_axis(V1,V2)`, returns the vector perpendicular to the two vectors V1 and V2.
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// If given three points, like `vector_axis(A,B,C)`, returns the vector perpendicular to the plane through a, B and C.
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// Arguments:
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// v1 = First vector or point.
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// v2 = Second vector or point.
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// v3 = Third point in three point mode.
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// Examples:
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// vector_axis(UP,LEFT); // Returns: [0,-1,0] (FWD)
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// vector_axis(RIGHT,LEFT); // Returns: [0,-1,0] (FWD)
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// vector_axis(UP+RIGHT,RIGHT); // Returns: [0,1,0] (BACK)
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// vector_axis([10,10], [0,0], [10,-10]); // Returns: [0,0,-1] (DOWN)
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// vector_axis([10,0,10], [0,0,0], [-10,10,0]); // Returns: [-0.57735, -0.57735, 0.57735]
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// vector_axis([[10,0,10], [0,0,0], [-10,10,0]]); // Returns: [-0.57735, -0.57735, 0.57735]
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function vector_axis(v1,v2=undef,v3=undef) =
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is_vector(v3)
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? assert(is_consistent([v3,v2,v1]), "Bad arguments.")
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vector_axis(v1-v2, v3-v2)
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: assert( is_undef(v3), "Bad arguments.")
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is_undef(v2)
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? assert( is_list(v1), "Bad arguments.")
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len(v1) == 2
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? vector_axis(v1[0],v1[1])
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: vector_axis(v1[0],v1[1],v1[2])
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: assert( is_vector(v1,zero=false) && is_vector(v2,zero=false) && is_consistent([v1,v2])
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, "Bad arguments.")
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let(
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eps = 1e-6,
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w1 = point3d(v1/norm(v1)),
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w2 = point3d(v2/norm(v2)),
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w3 = (norm(w1-w2) > eps && norm(w1+w2) > eps) ? w2
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: (norm(v_abs(w2)-UP) > eps)? UP
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: RIGHT
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) unit(cross(w1,w3));
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// Section: Vector Searching
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// Function: vp_tree()
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// Usage:
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// tree = vp_tree(points, <leafsize>)
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// Description:
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// Organizes n-dimensional data into a Vantage Point Tree, which can be
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// efficiently searched for for nearest matches. The Vantage Point Tree
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// is an effort to generalize binary search to n dimensions. Constructing the
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// tree should be O(n log n) and searches should be O(log n), though real life
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// performance depends on how the data is distributed, and it will deteriorate
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// for high data dimensions. This data structure is useful when you will be
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// performing many searches of the same data, so that the cost of constructing
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// the tree is justified.
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// .
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// The vantage point tree at a given level chooses vp, the
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// "vantage point", and a radius, R, and divides the data based
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// on distance to vp. Points closer than R go in on branch
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// of the tree and points farther than R go in the other branch.
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// .
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// The tree has the form [vp, R, inside, outside], where vp is
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// the vantage point index, R is the radius, inside is a
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// recursively computed tree for the inside points (distance less than
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// or equal to R from the vantage point), and outside
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// is a tree for the outside points (distance greater than R from the
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// vantage point).
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// .
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// If the number of points is less than or equal to leafsize then
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// vp_tree instead returns the list [ind] where ind is a list of
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// the indices of the points. This means the list has the form
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// [[i0, i1, i2,...]], so tree[0] is a list of indices. You can
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// tell that a node is a leaf node by checking if tree[0] is a list.
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// The leafsize parameter determines how many points can be
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// store in the leaf nodes. The default value of 25 was found
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// emperically to be a reasonable option for 3d data searched with vp_search().
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// .
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// Vantage point tree is described here: http://web.cs.iastate.edu/~honavar/nndatastructures.pdf
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// Arguments:
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// points = list of points to store in the tree
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// leafsize = maximum number of points to store in the tree's leaf nodes. Default: 25
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function vp_tree(points, leafsize=25) =
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assert(is_matrix(points),"points must be a consistent list of data points")
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_vp_tree(points, count(len(points)), leafsize);
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function _vp_tree(ptlist, ind, leafsize) =
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len(ind)<=leafsize ? [ind] :
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let(
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center = mean(select(ptlist,ind)),
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cdistances = [for(i=ind) norm(ptlist[i]-center)],
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vpind = ind[max_index(cdistances)],
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vp = ptlist[vpind],
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vp_dist = [for(i=ind) norm(vp-ptlist[i])],
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r = ninther(vp_dist),
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inside = [for(i=idx(ind)) if (vp_dist[i]<=r && ind[i]!=vpind) ind[i]],
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outside = [for(i=idx(ind)) if (vp_dist[i]>r) ind[i]]
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)
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[vpind, r, _vp_tree(ptlist,inside,leafsize),_vp_tree(ptlist,outside,leafsize)];
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// Function: vp_search()
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// Usage:
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// indices = vp_search(points, tree, p, r);
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// Description:
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// Search a vantage point tree for all points whose distance from p
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// is less than or equal to r. Returns a list of indices of the points it finds
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// in arbitrary order. The input points is a list of points to search and tree is the
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// vantage point tree computed from that point list. The search should be
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// around O(log n).
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// Arguments:
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// points = points indexed by the vantage point tree
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// tree = vantage point tree from vp_tree
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// p = point to search for
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// r = search radius
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// 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.
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// $fn=32;
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// k = 2000;
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// points = array_group(rands(0,10,k*2,seed=13333),2);
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// vp = vp_tree(points);
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// queries = [for(i=[3,7],j=[3,7]) [i,j]];
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// search_ind = [for(q=queries) vp_search(points, vp, q, 1)];
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// move_copies(points) circle(r=.08);
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// for(i=idx(queries)){
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// color("blue")stroke(move(queries[i],circle(r=1)), closed=true, width=.08);
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// color("red")move_copies(select(points, search_ind[i])) circle(r=.08);
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// }
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function _vp_search(points, tree, p, r) =
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is_list(tree[0]) ? [for(i=tree[0]) if (norm(points[i]-p)<=r) i]
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:
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let(
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d = norm(p-points[tree[0]]) // dist to vantage point
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)
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[
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if (d <= r) tree[0],
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if (d-r <= tree[1]) each _vp_search(points, tree[2], p, r),
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if (d+r > tree[1]) each _vp_search(points, tree[3], p, r)
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];
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function vp_search(points, tree, p, r) =
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assert(is_list(tree) && (len(tree)==4 || (len(tree)==1 && is_list(tree[0]))), "Vantage point tree not valid")
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assert(is_matrix(points), "Parameter points is not a consistent point list")
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assert(is_vector(p,len(points[0])), "Query must be a vector whose length matches the point list")
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assert(all_positive(r),"Radius r must be a positive number")
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_vp_search(points, tree, p, r);
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// Function: vp_nearest()
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// Usage:
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// indices = vp_nearest(points, tree, p, k)
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// Description:
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// Search the vantage point tree for the k points closest to point p.
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// The input points is the list of points to search and tree is
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// the vantage point tree computed from that point list. The list is
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// returned in sorted order, closest point first.
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// Arguments:
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// points = points indexed by the vantage point tree
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// tree = vantage point tree from vp_tree
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// p = point to search for
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// k = number of neighbors to return
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// 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.
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// $fn=32;
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// k = 2000;
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// points = array_group(rands(0,10,k*2,seed=13333),2);
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// vp = vp_tree(points);
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// queries = [for(i=[3,7],j=[3,7]) [i,j]];
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// search_ind = [for(q=queries) vp_nearest(points, vp, q, 15)];
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// move_copies(points) circle(r=.08);
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// for(i=idx(queries)){
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// color("red")move_copies(select(points, search_ind[i])) circle(r=.08);
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// color("blue")stroke(move(queries[i],
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// circle(r=norm(points[last(search_ind[i])]-queries[i]))),
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// closed=true, width=.08);
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// }
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function _insert_sorted(list, k, new) =
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len(list)==k && new[1]>= last(list)[1] ? list
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: [
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for(entry=list) if (entry[1]<=new[1]) entry,
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new,
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for(i=[0:1:min(k-1,len(list))-1]) if (list[i][1]>new[1]) list[i]
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];
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function _insert_many(list, k, newlist,i=0) =
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i==len(newlist) ? list :
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_insert_many(_insert_sorted(list,k,newlist[i]),k,newlist,i+1);
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function _vp_nearest(points, tree, p, k, answers=[]) =
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is_list(tree[0]) ? _insert_many(answers, k, [for(entry=tree[0]) [entry, norm(points[entry]-p)]]) :
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let(
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d = norm(p-points[tree[0]]),
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answers1 = _insert_sorted(answers, k, [tree[0],d]),
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answers2 = d-last(answers1)[1] <= tree[1] ? _vp_nearest(points, tree[2], p, k, answers1) : answers1,
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answers3 = d+last(answers2)[1] > tree[1] ? _vp_nearest(points, tree[3], p, k, answers2) : answers2
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)
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answers3;
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function vp_nearest(points, tree, p, k) =
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assert(is_int(k) && k>0)
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assert(k<=len(points), "You requested more results that contained in the set")
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assert(is_matrix(points), "Parameter points is not a consistent point list")
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assert(is_vector(p,len(points[0])), "Query must be a vector whose length matches the point list")
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assert(is_list(tree) && (len(tree)==4 || (len(tree)==1 && is_list(tree[0]))), "Vantage point tree not valid")
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subindex(_vp_nearest(points, tree, p, k),0);
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// Function: search_radius()
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// Usage:
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// index_list = search_radius(points, queries, r, <leafsize>);
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// Description:
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// Given a list of points and a compatible list of queries, for each query
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// search the points list for all points whose distance from the query
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// is less than or equal to r. The return value index_list[i] lists the indices
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// in points of all matches to query q[i]. This list can be in arbitrary order.
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// .
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// This function is advantageous to use especially when both `points` and `queries`
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// are large sets. The method contructs a vantage point tree and then uses it
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// to check all the queries. If you use queries=points and set r to epsilon then
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// you can find all of the approximate duplicates in a large list of vectors.
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// Example: Finding duplicates in a list of vectors. With exact equality the order of the output is consistent, but with small variations [2,4] could occur in one position and [4,2] in the other one.
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// v = array_group(rands(0,10,5*3,seed=9),3);
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// points = [v[0],v[1],v[2],v[3],v[2],v[3],v[3],v[4]];
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// echo(search_radius(points,points,1e-9)); // Prints [[0],[1],[2,4],[3,5,6],[2,4],[3,5,6],[3,5,6],[7]]
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//
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function search_radius(points, queries, r, leafsize=25) =
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assert(is_matrix(points),"Invalid points list")
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assert(is_matrix(queries),"Invalid query list")
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assert(len(points[0])==len(queries[0]), "Query vectors don't match length of points")
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let(
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vptree = vp_tree(points, leafsize)
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)
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[for(q=queries) vp_search(points, vptree, q, r)];
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// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
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