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Review of geometry.scad for speed
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RonaldoCMP
parent
cdb68ad977
commit
2dcbfeee11
2 changed files with 99 additions and 57 deletions
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@ -888,9 +888,47 @@ function plane3pt_indexed(points, i1, i2, i3) =
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// Example:
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// plane_from_normal([0,0,1], [2,2,2]); // Returns the xy plane passing through the point (2,2,2)
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function plane_from_normal(normal, pt=[0,0,0]) =
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assert( is_matrix([normal,pt],2,3) && !approx(norm(normal),0),
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"Inputs `normal` and `pt` should 3d vectors/points and `normal` cannot be zero." )
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concat(normal, normal*pt) / norm(normal);
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assert( is_matrix([normal,pt],2,3) && !approx(norm(normal),0),
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"Inputs `normal` and `pt` should be 3d vectors/points and `normal` cannot be zero." )
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concat(normal, normal*pt) / norm(normal);
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// Eigenvalues for a 3x3 symmetrical matrix in decreasing order
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// Based on: https://en.wikipedia.org/wiki/Eigenvalue_algorithm
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function _eigenvals_symm_3(M) =
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let( p1 = pow(M[0][1],2) + pow(M[0][2],2) + pow(M[1][2],2) )
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(p1<EPSILON)
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? -sort(-[ M[0][0], M[1][1], M[2][2] ]) // diagonal matrix: eigenvals in decreasing order
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: let( q = (M[0][0]+M[1][1]+M[2][2])/3,
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B = (M - q*ident(3)),
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dB = [B[0][0], B[1][1], B[2][2]],
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p2 = dB*dB + 2*p1,
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p = sqrt(p2/6),
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r = det3(B/p)/2,
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ph = acos(constrain(r,-1,1))/3,
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e1 = q + 2*p*cos(ph),
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e3 = q + 2*p*cos(ph+120),
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e2 = 3*q - e1 - e3 )
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[ e1, e2, e3 ];
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// i-th normalized eigenvector of 3x3 symmetrical matrix M from its eigenvalues
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// using Cayley–Hamilton theorem according to:
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// https://en.wikipedia.org/wiki/Eigenvalue_algorithm
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function _eigenvec_symm_3(M,evals,i=0) =
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let( A = (M - evals[(i+1)%3]*ident(3)) * (M - evals[(i+2)%3]*ident(3)) ,
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k = max_index( [for(i=[0:2]) norm(A[i]) ])
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)
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norm(A[k])<EPSILON ? ident(3)[k] : A[k]/norm(A[k]);
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// eigenvalues of the covariance matrix of points
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function _covariance_evals(points) =
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let( pm = sum(points)/len(points), // mean point
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Y = [ for(i=[0:len(points)-1]) points[i] - pm ],
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M = transpose(Y)*Y , // covariance matrix
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evals = _eigenvals_symm_3(M) )
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[pm, evals, M ];
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// Function: plane_from_points()
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@ -900,7 +938,7 @@ function plane_from_normal(normal, pt=[0,0,0]) =
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// Given a list of 3 or more coplanar 3D points, returns the coefficients of the normalized cartesian equation of a plane,
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// that is [A,B,C,D] where Ax+By+Cz=D is the equation of the plane where norm([A,B,C])=1.
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// If `fast` is false and the points in the list are collinear or not coplanar, then `undef` is returned.
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// if `fast` is true, then the coplanarity test is skipped and a plane passing through 3 non-collinear arbitrary points is returned.
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// If `fast` is true, the polygon coplanarity check is skipped and a best fitted plane is returned.
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// Arguments:
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// points = The list of points to find the plane of.
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// fast = If true, don't verify that all points in the list are coplanar. Default: false
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@ -914,17 +952,18 @@ function plane_from_normal(normal, pt=[0,0,0]) =
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function plane_from_points(points, fast=false, eps=EPSILON) =
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assert( is_path(points,dim=3), "Improper 3d point list." )
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assert( is_finite(eps) && (eps>=0), "The tolerance should be a non-negative value." )
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let(
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indices = noncollinear_triple(points,error=false)
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)
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indices==[] ? undef :
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let(
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p1 = points[indices[0]],
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p2 = points[indices[1]],
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p3 = points[indices[2]],
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plane = plane3pt(p1,p2,p3)
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)
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fast || points_on_plane(points,plane,eps=eps) ? plane : undef;
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len(points) == 3
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? let( plane = plane3pt(points[0],points[1],points[2]) )
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plane==[] ? undef : plane
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: let(
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cov_evals = _covariance_evals(points),
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pm = cov_evals[0],
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evals = cov_evals[1],
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M = cov_evals[2],
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evec = _eigenvec_symm_3(M,evals,i=2) )
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// echo(error_points_plane= abs(max(points*evec)-pm*evec), limit=eps)
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!fast && abs(max(points*evec)-pm*evec)>eps*evals[0] ? undef :
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[ each evec, pm*evec] ;
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// Function: plane_from_polygon()
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@ -934,7 +973,8 @@ function plane_from_points(points, fast=false, eps=EPSILON) =
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// Given a 3D planar polygon, returns the normalized cartesian equation of its plane.
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// Returns [A,B,C,D] where Ax+By+Cz=D is the equation of the plane where norm([A,B,C])=1.
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// If not all the points in the polygon are coplanar, then [] is returned.
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// If `fast` is true, the polygon coplanarity check is skipped and the plane may not contain all polygon points.
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// If `fast` is false and the points in the list are collinear or not coplanar, then `undef` is returned.
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// if `fast` is true, then the coplanarity test is skipped and a plane passing through 3 non-collinear arbitrary points is returned.
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// Arguments:
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// poly = The planar 3D polygon to find the plane of.
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// fast = If true, doesn't verify that all points in the polygon are coplanar. Default: false
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@ -948,12 +988,11 @@ function plane_from_points(points, fast=false, eps=EPSILON) =
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function plane_from_polygon(poly, fast=false, eps=EPSILON) =
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assert( is_path(poly,dim=3), "Invalid polygon." )
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assert( is_finite(eps) && (eps>=0), "The tolerance should be a non-negative value." )
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let(
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poly = deduplicate(poly),
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n = polygon_normal(poly),
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plane = [n.x, n.y, n.z, n*poly[0]]
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)
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fast? plane: coplanar(poly,eps=eps)? plane: [];
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len(poly)==3 ? plane3pt(poly[0],poly[1],poly[2]) :
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let( triple = sort(noncollinear_triple(poly,error=false)) )
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triple==[] ? [] :
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let( plane = plane3pt(poly[triple[0]],poly[triple[1]],poly[triple[2]]))
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fast? plane: points_on_plane(poly, plane, eps=eps)? plane: [];
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// Function: plane_normal()
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@ -1252,9 +1291,11 @@ function coplanar(points, eps=EPSILON) =
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len(points)<=2 ? false
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: let( ip = noncollinear_triple(points,error=false,eps=eps) )
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ip == [] ? false :
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let( plane = plane3pt(points[ip[0]],points[ip[1]],points[ip[2]]),
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normal = point3d(plane) )
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max( points*normal ) - plane[3]< eps*norm(normal);
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let(
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plane = plane3pt(points[ip[0]],points[ip[1]],points[ip[2]]),
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normal = point3d(plane),
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pt_nrm = points*normal )
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abs(max(max(pt_nrm)-plane[3], -min(pt_nrm)+plane[3])) < eps;
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// Function: points_on_plane()
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@ -1665,7 +1706,7 @@ function noncollinear_triple(points,error=true,eps=EPSILON) =
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n = (pb-pa)/nrm,
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distlist = [for(i=[0:len(points)-1]) _dist2line(points[i]-pa, n)]
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)
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max(distlist)<eps
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max(distlist)<eps*nrm
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? assert(!error, "Cannot find three noncollinear points in pointlist.")
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[]
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: [0,b,max_index(distlist)];
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@ -1746,8 +1787,8 @@ function polygon_area(poly, signed=false) =
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v1 = poly[i] - poly[0],
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v2 = poly[i+1] - poly[0]
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)
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cross(v1,v2) * n
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])/2
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cross(v1,v2)
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])* n/2
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)
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signed ? total : abs(total);
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@ -2008,7 +2049,7 @@ function point_in_polygon(point, poly, nonzero=true, eps=EPSILON) =
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// poly = The list of 2D path points for the perimeter of the polygon.
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function polygon_is_clockwise(poly) =
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assert(is_path(poly,dim=2), "Input should be a 2d path")
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polygon_area(poly, signed=true)<0;
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polygon_area(poly, signed=true)<-EPSILON;
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// Function: clockwise_polygon()
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@ -197,8 +197,8 @@ module test_plane_from_polygon(){
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poly1 = [ rands(-1,1,3), rands(-1,1,3)+[2,0,0], rands(-1,1,3)+[0,2,2] ];
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poly2 = concat(poly1, [sum(poly1)/3] );
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info = info_str([["poly1 = ",poly1],["poly2 = ",poly2]]);
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assert_std(plane_from_polygon(poly1),plane3pt(poly1[0],poly1[1],poly1[2]),info);
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assert_std(plane_from_polygon(poly2),plane3pt(poly1[0],poly1[1],poly1[2]),info);
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assert_approx(plane_from_polygon(poly1),plane3pt(poly1[0],poly1[1],poly1[2]),info);
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assert_approx(plane_from_polygon(poly2),plane3pt(poly1[0],poly1[1],poly1[2]),info);
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}
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*test_plane_from_polygon();
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@ -208,8 +208,7 @@ module test_plane_from_normal(){
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displ = normal*point;
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info = info_str([["normal = ",normal],["point = ",point],["displ = ",displ]]);
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assert_approx(plane_from_normal(normal,point)*[each point,-1],0,info);
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assert_std(plane_from_normal(normal,point),normalize_plane([each normal,displ]),info);
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assert_std(plane_from_normal([1,1,1],[1,2,3]),[0.57735026919,0.57735026919,0.57735026919,3.46410161514]);
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assert_approx(plane_from_normal([1,1,1],[1,2,3]),[0.57735026919,0.57735026919,0.57735026919,3.46410161514]);
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}
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*test_plane_from_normal();
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@ -680,23 +679,23 @@ module test_triangle_area() {
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module test_plane3pt() {
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assert_std(plane3pt([0,0,20], [0,10,10], [0,0,0]), [1,0,0,0]);
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assert_std(plane3pt([2,0,20], [2,10,10], [2,0,0]), [1,0,0,2]);
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assert_std(plane3pt([0,0,0], [10,0,10], [0,0,20]), [0,1,0,0]);
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assert_std(plane3pt([0,2,0], [10,2,10], [0,2,20]), [0,1,0,2]);
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assert_std(plane3pt([0,0,0], [10,10,0], [20,0,0]), [0,0,1,0]);
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assert_std(plane3pt([0,0,2], [10,10,2], [20,0,2]), [0,0,1,2]);
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assert_approx(plane3pt([0,0,20], [0,10,10], [0,0,0]), [1,0,0,0]);
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assert_approx(plane3pt([2,0,20], [2,10,10], [2,0,0]), [1,0,0,2]);
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assert_approx(plane3pt([0,0,0], [10,0,10], [0,0,20]), [0,1,0,0]);
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assert_approx(plane3pt([0,2,0], [10,2,10], [0,2,20]), [0,1,0,2]);
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assert_approx(plane3pt([0,0,0], [10,10,0], [20,0,0]), [0,0,1,0]);
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assert_approx(plane3pt([0,0,2], [10,10,2], [20,0,2]), [0,0,1,2]);
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}
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*test_plane3pt();
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module test_plane3pt_indexed() {
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pts = [ [0,0,0], [10,0,0], [0,10,0], [0,0,10] ];
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s13 = sqrt(1/3);
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assert_std(plane3pt_indexed(pts, 0,3,2), [1,0,0,0]);
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assert_std(plane3pt_indexed(pts, 0,2,3), [-1,0,0,0]);
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assert_std(plane3pt_indexed(pts, 0,1,3), [0,1,0,0]);
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assert_std(plane3pt_indexed(pts, 0,3,1), [0,-1,0,0]);
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assert_std(plane3pt_indexed(pts, 0,2,1), [0,0,1,0]);
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assert_approx(plane3pt_indexed(pts, 0,3,2), [1,0,0,0]);
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assert_approx(plane3pt_indexed(pts, 0,2,3), [-1,0,0,0]);
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assert_approx(plane3pt_indexed(pts, 0,1,3), [0,1,0,0]);
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assert_approx(plane3pt_indexed(pts, 0,3,1), [0,-1,0,0]);
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assert_approx(plane3pt_indexed(pts, 0,2,1), [0,0,1,0]);
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assert_approx(plane3pt_indexed(pts, 0,1,2), [0,0,-1,0]);
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assert_approx(plane3pt_indexed(pts, 3,2,1), [s13,s13,s13,10*s13]);
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assert_approx(plane3pt_indexed(pts, 1,2,3), [-s13,-s13,-s13,-10*s13]);
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@ -715,12 +714,12 @@ module test_plane_from_points() {
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module test_plane_normal() {
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assert_std(plane_normal(plane3pt([0,0,20], [0,10,10], [0,0,0])), [1,0,0]);
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assert_std(plane_normal(plane3pt([2,0,20], [2,10,10], [2,0,0])), [1,0,0]);
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assert_std(plane_normal(plane3pt([0,0,0], [10,0,10], [0,0,20])), [0,1,0]);
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assert_std(plane_normal(plane3pt([0,2,0], [10,2,10], [0,2,20])), [0,1,0]);
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assert_std(plane_normal(plane3pt([0,0,0], [10,10,0], [20,0,0])), [0,0,1]);
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assert_std(plane_normal(plane3pt([0,0,2], [10,10,2], [20,0,2])), [0,0,1]);
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assert_approx(plane_normal(plane3pt([0,0,20], [0,10,10], [0,0,0])), [1,0,0]);
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assert_approx(plane_normal(plane3pt([2,0,20], [2,10,10], [2,0,0])), [1,0,0]);
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assert_approx(plane_normal(plane3pt([0,0,0], [10,0,10], [0,0,20])), [0,1,0]);
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assert_approx(plane_normal(plane3pt([0,2,0], [10,2,10], [0,2,20])), [0,1,0]);
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assert_approx(plane_normal(plane3pt([0,0,0], [10,10,0], [20,0,0])), [0,0,1]);
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assert_approx(plane_normal(plane3pt([0,0,2], [10,10,2], [20,0,2])), [0,0,1]);
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}
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*test_plane_normal();
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@ -780,7 +779,7 @@ module test_coplanar() {
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assert(coplanar([ [5,5,1],[0,0,0],[-1,-1,1] ]) == true);
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assert(coplanar([ [0,0,0],[1,0,1],[1,1,1], [0,1,2] ]) == false);
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assert(coplanar([ [0,0,0],[1,0,1],[1,1,2], [0,1,1] ]) == true);
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}
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}
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*test_coplanar();
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@ -836,7 +835,9 @@ module test_cleanup_path() {
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module test_polygon_area() {
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assert(approx(polygon_area([[1,1],[-1,1],[-1,-1],[1,-1]]), 4));
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assert(approx(polygon_area(circle(r=50,$fn=1000),signed=true), -PI*50*50, eps=0.1));
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assert(approx(polygon_area(rot([13,27,75],p=path3d(circle(r=50,$fn=1000),fill=23)),signed=true), PI*50*50, eps=0.1));
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assert(approx(polygon_area(rot([13,27,75],
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p=path3d(circle(r=50,$fn=1000),fill=23)),
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signed=true), -PI*50*50, eps=0.1));
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}
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*test_polygon_area();
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