/**
* Compute the symmetric Hausdorf distance.
*/
double hausdorf(D2<SBasis>& A, D2<SBasis> const& B,
double m_precision,
double *a_t, double* b_t) {
double h_dist = hausdorfl(A, B, m_precision, a_t, b_t);
double dist = 0;
Point Bx = B.at0();
double t = Geom::nearest_point(Bx, A);
dist = Geom::distance(Bx, A(t));
if (dist > h_dist) {
if(a_t) *a_t = t;
if(b_t) *b_t = 0;
h_dist = dist;
}
Bx = B.at1();
t = Geom::nearest_point(Bx, A);
dist = Geom::distance(Bx, A(t));
if (dist > h_dist) {
if(a_t) *a_t = t;
if(b_t) *b_t = 1;
h_dist = dist;
}
return h_dist;
}
/**
* Compute the Hausdorf distance from A to B only.
*/
double hausdorfl(D2<SBasis>& A, D2<SBasis> const& B,
double m_precision,
double *a_t, double* b_t) {
std::vector< std::pair<double, double> > xs;
std::vector<Point> Az, Bz;
sbasis_to_bezier (Az, A);
sbasis_to_bezier (Bz, B);
find_collinear_normal(xs, Az, Bz, m_precision);
double h_dist = 0, h_a_t = 0, h_b_t = 0;
double dist = 0;
Point Ax = A.at0();
double t = Geom::nearest_point(Ax, B);
dist = Geom::distance(Ax, B(t));
if (dist > h_dist) {
h_a_t = 0;
h_b_t = t;
h_dist = dist;
}
Ax = A.at1();
t = Geom::nearest_point(Ax, B);
dist = Geom::distance(Ax, B(t));
if (dist > h_dist) {
h_a_t = 1;
h_b_t = t;
h_dist = dist;
}
for (size_t i = 0; i < xs.size(); ++i)
{
Point At = A(xs[i].first);
Point Bu = B(xs[i].second);
double distAtBu = Geom::distance(At, Bu);
t = Geom::nearest_point(At, B);
dist = Geom::distance(At, B(t));
//FIXME: we might miss it due to floating point precision...
if (dist >= distAtBu-.1 && distAtBu > h_dist) {
h_a_t = xs[i].first;
h_b_t = xs[i].second;
h_dist = distAtBu;
}
}
if(a_t) *a_t = h_a_t;
if(b_t) *b_t = h_b_t;
return h_dist;
}
/** Make a path from a d2 sbasis.
\param p the d2 Symmetric basis polynomial
\returns a Path
If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
*/
void build_from_sbasis(Geom::PathBuilder &pb, D2<SBasis> const &B, double tol, bool only_cubicbeziers) {
if (!B.isFinite()) {
THROW_EXCEPTION("assertion failed: B.isFinite()");
}
if(tail_error(B, 2) < tol || sbasis_size(B) == 2) { // nearly cubic enough
if( !only_cubicbeziers && (sbasis_size(B) <= 1) ) {
pb.lineTo(B.at1());
} else {
std::vector<Geom::Point> bez;
sbasis_to_bezier(bez, B, 4);
pb.curveTo(bez[1], bez[2], bez[3]);
}
} else {
build_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol, only_cubicbeziers);
build_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol, only_cubicbeziers);
}
}
/**
* Centroid using sbasis integration.
\param p the Element.
\param centroid on return contains the centroid of the shape
\param area on return contains the signed area of the shape.
\relates Piecewise
This approach uses green's theorem to compute the area and centroid using integrals. For curved shapes this is much faster than converting to polyline. Note that without an uncross operation the output is not the absolute area.
* Returned values:
0 for normal execution;
2 if area is zero, meaning centroid is meaningless.
*/
unsigned Geom::centroid(Piecewise<D2<SBasis> > const &p, Point& centroid, double &area) {
Point centroid_tmp(0,0);
double atmp = 0;
for(unsigned i = 0; i < p.size(); i++) {
SBasis curl = dot(p[i], rot90(derivative(p[i])));
SBasis A = integral(curl);
D2<SBasis> C = integral(multiply(curl, p[i]));
atmp += A.at1() - A.at0();
centroid_tmp += C.at1()- C.at0(); // first moment.
}
// join ends
centroid_tmp *= 2;
Point final = p[p.size()-1].at1(), initial = p[0].at0();
const double ai = cross(final, initial);
atmp += ai;
centroid_tmp += (final + initial)*ai; // first moment.
area = atmp / 2;
if (atmp != 0) {
centroid = centroid_tmp / (3 * atmp);
return 0;
}
return 2;
}
Geom::Path Gear::path() {
Geom::Path pb;
// angle covered by a full tooth and fillet
double tooth_rotation = 2.0 * tooth_thickness_angle();
// angle covered by an involute
double involute_advance = involute_intersect_angle(outer_radius()) - involute_intersect_angle(root_radius());
// angle covered by the tooth tip
double tip_advance = tooth_thickness_angle() - (2 * (involute_intersect_angle(outer_radius()) - involute_intersect_angle(pitch_radius())));
// angle covered by the toothe root
double root_advance = (tooth_rotation - tip_advance) - (2.0 * involute_advance);
// begin drawing the involute at t if the root circle is larger than the base circle
double involute_t = involute_swath_angle(root_radius())/involute_swath_angle(outer_radius());
//rewind angle to start drawing from the leading edge of the tooth
double first_tooth_angle = _angle - ((0.5 * tip_advance) + involute_advance);
Geom::Point prev;
for (int i=0; i < _number_of_teeth; i++)
{
double cursor = first_tooth_angle + (i * tooth_rotation);
D2<SBasis> leading_I = compose(_involute(cursor, cursor + involute_swath_angle(outer_radius())), Linear(involute_t,1));
if(i != 0) makeContinuous(leading_I, prev);
pb.append(SBasisCurve(leading_I));
cursor += involute_advance;
prev = leading_I.at1();
D2<SBasis> tip = _arc(cursor, cursor+tip_advance, outer_radius());
makeContinuous(tip, prev);
pb.append(SBasisCurve(tip));
cursor += tip_advance;
prev = tip.at1();
cursor += involute_advance;
D2<SBasis> trailing_I = compose(_involute(cursor, cursor - involute_swath_angle(outer_radius())), Linear(1,involute_t));
makeContinuous(trailing_I, prev);
pb.append(SBasisCurve(trailing_I));
prev = trailing_I.at1();
if (base_radius() > root_radius()) {
Geom::Point leading_start = trailing_I.at1();
Geom::Point leading_end = (root_radius() * unit_vector(leading_start - _centre)) + _centre;
prev = leading_end;
pb.appendNew<LineSegment>(leading_end);
}
D2<SBasis> root = _arc(cursor, cursor+root_advance, root_radius());
makeContinuous(root, prev);
pb.append(SBasisCurve(root));
//cursor += root_advance;
prev = root.at1();
if (base_radius() > root_radius()) {
Geom::Point trailing_start = root.at1();
Geom::Point trailing_end = (base_radius() * unit_vector(trailing_start - _centre)) + _centre;
pb.appendNew<LineSegment>(trailing_end);
prev = trailing_end;
}
}
return pb;
}
/** Return a Piecewise<D2<SBasis> > which points in the same direction as V_in, but has unit_length.
\param V_in the original path.
\param tol the maximum error allowed.
\param order the maximum degree to use for approximation
unitVector(x,y) is computed as (b,-a) where a and b are solutions of:
ax+by=0 (eqn1) and a^2+b^2=1 (eqn2)
\relates Piecewise, D2
*/
Piecewise<D2<SBasis> >
Geom::unitVector(D2<SBasis> const &V_in, double tol, unsigned order){
//TODO: Handle vanishing vectors...
// -This approach is numerically bad. Find a stable way to rescale V_in to have non vanishing ends.
// -This done, unitVector will have jumps at zeros: fill the gaps with arcs of circles.
D2<SBasis> V = RescaleForNonVanishingEnds(V_in);
if (V[0].isZero(tol) && V[1].isZero(tol))
return Piecewise<D2<SBasis> >(D2<SBasis>(Linear(1),SBasis()));
SBasis x = V[0], y = V[1];
SBasis r_eqn1, r_eqn2;
Point v0 = unit_vector(V.at0());
Point v1 = unit_vector(V.at1());
SBasis a = SBasis(order+1, Linear(0.));
a[0] = Linear(-v0[1],-v1[1]);
SBasis b = SBasis(order+1, Linear(0.));
b[0] = Linear( v0[0], v1[0]);
r_eqn1 = -(a*x+b*y);
r_eqn2 = Linear(1.)-(a*a+b*b);
for (unsigned k=1; k<=order; k++){
double r0 = (k<r_eqn1.size())? r_eqn1.at(k).at0() : 0;
double r1 = (k<r_eqn1.size())? r_eqn1.at(k).at1() : 0;
double rr0 = (k<r_eqn2.size())? r_eqn2.at(k).at0() : 0;
double rr1 = (k<r_eqn2.size())? r_eqn2.at(k).at1() : 0;
double a0,a1,b0,b1;// coeffs in a[k] and b[k]
//the equations to solve at this point are:
// a0*x(0)+b0*y(0)=r0 & 2*a0*a(0)+2*b0*b(0)=rr0
//and
// a1*x(1)+b1*y(1)=r1 & 2*a1*a(1)+2*b1*b(1)=rr1
a0 = r0/dot(v0,V.at0())*v0[0]-rr0/2*v0[1];
b0 = r0/dot(v0,V.at0())*v0[1]+rr0/2*v0[0];
a1 = r1/dot(v1,V.at1())*v1[0]-rr1/2*v1[1];
b1 = r1/dot(v1,V.at1())*v1[1]+rr1/2*v1[0];
a[k] = Linear(a0,a1);
b[k] = Linear(b0,b1);
//TODO: use "incremental" rather than explicit formulas.
r_eqn1 = -(a*x+b*y);
r_eqn2 = Linear(1)-(a*a+b*b);
}
//our candidate is:
D2<SBasis> unitV;
unitV[0] = b;
unitV[1] = -a;
//is it good?
double rel_tol = std::max(1.,std::max(V_in[0].tailError(0),V_in[1].tailError(0)))*tol;
if (r_eqn1.tailError(order)>rel_tol || r_eqn2.tailError(order)>tol){
//if not: subdivide and concat results.
Piecewise<D2<SBasis> > unitV0, unitV1;
unitV0 = unitVector(compose(V,Linear(0,.5)),tol,order);
unitV1 = unitVector(compose(V,Linear(.5,1)),tol,order);
unitV0.setDomain(Interval(0.,.5));
unitV1.setDomain(Interval(.5,1.));
unitV0.concat(unitV1);
return(unitV0);
}else{
//if yes: return it as pw.
Piecewise<D2<SBasis> > result;
result=(Piecewise<D2<SBasis> >)unitV;
return result;
}
}
