double altA = a.DistanceToLine(b, c.Minus(b)),

altB = b.DistanceToLine(c, a.Minus(c)),

altC = c.DistanceToLine(a, b.Minus(a));

return min(altA, min(altB, altC));

Vector n = Normal();

if(MinAltitude() < LENGTH_EPS) {

// shouldn't happen; zero-area triangle

return false;

}

return ContainsPointProjd(n.WithMagnitude(1), p);

Vector ab = b.Minus(a), bc = c.Minus(b), ca = a.Minus(c);

Vector no_ab = n.Cross(ab);

if(no_ab.Dot(p) < no_ab.Dot(a) - LENGTH_EPS) return false;

Vector no_bc = n.Cross(bc);

if(no_bc.Dot(p) < no_bc.Dot(b) - LENGTH_EPS) return false;

Vector no_ca = n.Cross(ca);

if(no_ca.Dot(p) < no_ca.Dot(c) - LENGTH_EPS) return false;

return true;

double *t, Vector *inters) const {

// Algorithm from: "Fast, Minimum Storage Ray/Triangle Intersection" by

// Tomas Moeller and Ben Trumbore.

// Find vectors for two edges sharing vertex A.

Vector edge1 = b.Minus(a);

Vector edge2 = c.Minus(a);

// Begin calculating determinant - also used to calculate U parameter.

Vector pvec = rayDir.Cross(edge2);

// If determinant is near zero, ray lies in plane of triangle.

// Also, cull back facing triangles here.

double det = edge1.Dot(pvec);

if(-det < LENGTH_EPS) return false;

double inv_det = 1.0f / det;

// Calculate distance from vertex A to ray origin.

Vector tvec = rayPoint.Minus(a);

// Calculate U parameter and test bounds.

double u = tvec.Dot(pvec) * inv_det;

if (u < 0.0f || u > 1.0f) return false;

// Prepare to test V parameter.

Vector qvec = tvec.Cross(edge1);

// Calculate V parameter and test bounds.

double v = rayDir.Dot(qvec) * inv_det;

if (v < 0.0f || u + v > 1.0f) return false;

// Calculate t, ray intersects triangle.

*t = edge2.Dot(qvec) * inv_det;

// Calculate intersection point.

if(inters != NULL) *inters = rayPoint.Plus(rayDir.ScaledBy(*t));

return true;

Vector ab = a.Minus(b);

Vector cb = c.Minus(b);

return ab.Cross(cb).Magnitude() / 2.0;

return a.OnLineSegment(b, c) ||

b.OnLineSegment(a, c) ||

c.OnLineSegment(a, b);

STriangle tr = *this;

tr.a = tr.a.ScaleOutOfCsys(u, v, n);

tr.an = tr.an.ScaleOutOfCsys(u, v, n);

tr.b = tr.b.ScaleOutOfCsys(u, v, n);

tr.bn = tr.bn.ScaleOutOfCsys(u, v, n);

tr.c = tr.c.ScaleOutOfCsys(u, v, n);

tr.cn = tr.cn.ScaleOutOfCsys(u, v, n);

return tr;

STriangle tr = {};

tr.meta = meta;

tr.a = a;

tr.b = b;

tr.c = c;

return tr;

SEdge se = {};

se.a = a;

se.b = b;

return se;

Vector d = eb.Minus(ea);

double t_eps = LENGTH_EPS/d.Magnitude();

double t, tthis;

bool skew;

Vector pi;

bool inOrEdge0, inOrEdge1;

Vector dthis = b.Minus(a);

double tthis_eps = LENGTH_EPS/dthis.Magnitude();

if((ea.Equals(a) && eb.Equals(b)) ||

(eb.Equals(a) && ea.Equals(b)))

{

if(ppi) *ppi = a;

if(spl) spl->Add(a);

return true;

}

// Can't just test if distance between d and a equals distance between d and b;

// they could be on opposite sides, since we don't have the sign.

double m = sqrt(d.Magnitude()*dthis.Magnitude());

if(sqrt(fabs(d.Dot(dthis))) > (m - LENGTH_EPS)) {

// The edges are parallel.

if(fabs(a.DistanceToLine(ea, d)) > LENGTH_EPS) {

// and not coincident, so can't be intersecting

return false;

}

// The edges are coincident. Make sure that neither endpoint lies

// on the other

bool inters = false;

double t;

t = a.Minus(ea).DivProjected(d);

if(t > t_eps && t < (1 - t_eps)) inters = true;

t = b.Minus(ea).DivProjected(d);

if(t > t_eps && t < (1 - t_eps)) inters = true;

t = ea.Minus(a).DivProjected(dthis);

if(t > tthis_eps && t < (1 - tthis_eps)) inters = true;

t = eb.Minus(a).DivProjected(dthis);

if(t > tthis_eps && t < (1 - tthis_eps)) inters = true;

if(inters) {

if(ppi) *ppi = a;

if(spl) spl->Add(a);

return true;

} else {

// So coincident but disjoint, okay.

return false;

}

}

// Lines are not parallel, so look for an intersection.

pi = Vector::AtIntersectionOfLines(ea, eb, a, b,

&skew,

&t, &tthis);

if(skew) return false;

inOrEdge0 = (t > -t_eps) && (t < (1 + t_eps));

inOrEdge1 = (tthis > -tthis_eps) && (tthis < (1 + tthis_eps));

if(inOrEdge0 && inOrEdge1) {

if(a.Equals(ea) || b.Equals(ea) ||

a.Equals(eb) || b.Equals(eb))

{

// Not an intersection if we share an endpoint with an edge

return false;

}

// But it's an intersection if a vertex of one edge lies on the

// inside of the other (or if they cross away from either's

// vertex).

if(ppi) *ppi = pi;

if(spl) spl->Add(pi);

return true;

}

return false;

SEdge e = {};

e.tag = tag;

e.a = a;

e.b = b;

e.auxA = auxA;

e.auxB = auxB;

l.Add(&e);

SEdge *errorAt, bool keepDir, int start) const

{

int i;

dest->AddPoint(first);

dest->AddPoint(last);

do {

for(i = start; i < l.n; i++) {

/// @todo fix const!

SEdge *se = const_cast<SEdge*>(&(l[i]));

if(se->tag) continue;

if(se->a.Equals(last)) {

dest->AddPoint(se->b);

last = se->b;

se->tag = 1;

break;

}

// Don't allow backwards edges if keepDir is true.

if(!keepDir && se->b.Equals(last)) {

dest->AddPoint(se->a);

last = se->a;

se->tag = 1;

break;

}

}

if(i >= l.n) {

// Couldn't assemble a closed contour; mark where.

if(errorAt) {

errorAt->a = first;

errorAt->b = last;

}

return false;

}

} while(!last.Equals(first));

return true;

dest->Clear();

bool allClosed = true;

Vector first = {};

Vector last = {};

int i;

for(i = 0; i < l.n; i++) {

if(!l[i].tag) {

first = l[i].a;

last = l[i].b;

/// @todo fix const!

const_cast<SEdge*>(&(l[i]))->tag = 1;

// Create a new empty contour in our polygon, and finish assembling

// into that contour.

dest->AddEmptyContour();

if(!AssembleContour(first, last, dest->l.Last(), errorAt, keepDir, i+1)) {

allClosed = false;

}

// But continue assembling, even if some of the contours are open

}

}

return allClosed;

auto cnt = std::count_if(l.begin(), l.end(),

[&](SEdge const &se) { return se.EdgeCrosses(a, b, ppi, spl); });

return static_cast<int>(cnt);

for(const SEdge *se = l.First(); se; se = l.NextAfter(se)) {

if(sel->ContainsEdge(se)) return true;

}

return false;

for(const SEdge *se = l.First(); se; se = l.NextAfter(se)) {

if((se->a).Equals(set->a) && (se->b).Equals(set->b)) return true;

if((se->b).Equals(set->a) && (se->a).Equals(set->b)) return true;

}

return false;

l.ClearTags();

for(int i = 0; i < l.n; i++) {

SEdge *se = &(l[i]);

for(int j = i + 1; j < l.n; j++) {

SEdge *set = &(l[j]);

if((set->a).Equals(se->a) && (set->b).Equals(se->b)) {

// Two parallel edges exist; so keep only the first one.

set->tag = 1;

}

if((set->a).Equals(se->b) && (set->b).Equals(se->a)) {

// Two anti-parallel edges exist; if both=true, keep neither,

// otherwise keep only one.

if (both) se->tag = 1;

set->tag = 1;

}

}

}

l.RemoveTagged();

Vector *pi, SPointList *spl) const

{

int inters = 0;

if(gt && lt) {

if(a.Element(which) < c + KDTREE_EPS ||

b.Element(which) < c + KDTREE_EPS)

{

inters += lt->AnyEdgeCrossings(a, b, cnt, pi, spl);

}

if(a.Element(which) > c - KDTREE_EPS ||

b.Element(which) > c - KDTREE_EPS)

{

inters += gt->AnyEdgeCrossings(a, b, cnt, pi, spl);

}

} else {

SEdgeLl *sell;

for(sell = edges; sell; sell = sell->next) {

SEdge *se = sell->se;

if(se->tag == cnt) continue;

if(se->EdgeCrosses(a, b, pi, spl)) {

inters++;

}

se->tag = cnt;

}

}

return inters;

const Vector lineStart = a;

const Vector lineDirection = b.Minus(a);

std::sort(l.begin(), l.end(), [&](const SEdge &a, const SEdge &b) {

double ta = (a.a.Minus(lineStart)).DivProjected(lineDirection);

double tb = (b.a.Minus(lineStart)).DivProjected(lineDirection);

return (ta < tb);

});

l.ClearTags();

SEdge *prev = nullptr;

for(auto &now : l) {

if(prev != nullptr) {

if((prev->b).Equals(now.a) && prev->auxA == now.auxA) {

// The previous segment joins up to us; so merge it into us.

prev->tag = 1;

now.a = prev->a;

}

}

prev = &now;

}

l.RemoveTagged();

int i;

for(i = 0; i < l.n; i++) {

const SPoint *p = &(l[i]);

if(pt.Equals(p->p)) {

return i;

}

}

// Not found, so return negative to indicate that.

return -1;

SPoint *p;

for(p = l.First(); p; p = l.NextAfter(p)) {

if(pt.Equals(p->p)) {

(p->tag)++;

return;

}

}

SPoint pa;

pa.p = pt;

pa.tag = 1;

l.Add(&pa);

SPoint p = {};

p.p = pt;

l.Add(&p);

SPoint sp;

sp.tag = 0;

sp.p = p;

l.Add(&sp);

int i;

for(i = 0; i < (l.n - 1); i++) {

el->AddEdge(l[i].p, l[i+1].p);

}

for(const SPoint *sp = l.First(); sp; sp = l.NextAfter(sp)) {

dest->AddPoint(sp->p);

}

xminPt = {VERY_POSITIVE, VERY_POSITIVE, VERY_POSITIVE};

for(const SPoint *sp = l.First(); sp; sp = l.NextAfter(sp)) {

if(sp->p.x < xminPt.x) {

xminPt = sp->p;

}

}

Vector n = {};

for(int i = 0; i < l.n - 2; i++) {

Vector u = (l[i+1].p).Minus(l[i+0].p).WithMagnitude(1);

Vector v = (l[i+2].p).Minus(l[i+1].p).WithMagnitude(1);

Vector nt = u.Cross(v);

if(nt.Magnitude() > n.Magnitude()) {

n = nt;

}

}

return n.WithMagnitude(1);

// Degenerate things might happen as we draw; doesn't really matter

// what we do then.

if(n.Magnitude() < 0.01) return true;

return (SignedAreaProjdToNormal(n) < 0);

// An arbitrary 2d coordinate system that has n as its normal

Vector u = n.Normal(0);

Vector v = n.Normal(1);

double area = 0;

for(int i = 0; i < (l.n - 1); i++) {

double u0 = (l[i ].p).Dot(u);

double v0 = (l[i ].p).Dot(v);

double u1 = (l[i+1].p).Dot(u);

double v1 = (l[i+1].p).Dot(v);

area += ((v0 + v1)/2)*(u1 - u0);

}

return area;

Vector u = n.Normal(0);

Vector v = n.Normal(1);

double up = p.Dot(u);

double vp = p.Dot(v);

bool inside = false;

for(int i = 0; i < (l.n - 1); i++) {

double ua = (l[i ].p).Dot(u);

double va = (l[i ].p).Dot(v);

// The curve needs to be exactly closed; approximation is death.

double ub = (l[(i+1)%(l.n-1)].p).Dot(u);

double vb = (l[(i+1)%(l.n-1)].p).Dot(v);

if ((((va <= vp) && (vp < vb)) ||

((vb <= vp) && (vp < va))) &&

(up < (ub - ua) * (vp - va) / (vb - va) + ua))

{

inside = !inside;

}

}

return inside;

int i;

for(i = 0; i < l.n; i++) {

(l[i]).l.Clear();

}

l.Clear();

int i;

for(i = 0; i < l.n; i++) {

(l[i]).MakeEdgesInto(el);

}

if(l.IsEmpty())

return {};

return (l[0]).ComputeNormal();

double area = 0;

// This returns the true area only if the contours are all oriented

// correctly, with the holes backwards from the outer contours.

for(const SContour *sc = l.First(); sc; sc = l.NextAfter(sc)) {

area += sc->SignedAreaProjdToNormal(normal);

}

return area;

auto winding = std::count_if(l.begin(), l.end(), [&](const SContour &sc) {

return sc.ContainsPointProjdToNormal(normal, p);

});

return winding;

// At output, the contour's tag will be 1 if we reversed it, else 0.

l.ClearTags();

// Outside curve looks counterclockwise, projected against our normal.

int i, j;

for(i = 0; i < l.n; i++) {

SContour *sc = &(l[i]);

if(sc->l.n < 2) continue;

// The contours may not intersect, but they may share vertices; so

// testing a vertex for point-in-polygon may fail, but the midpoint

// of an edge is okay.

Vector pt = (((sc->l[0]).p).Plus(sc->l[1].p)).ScaledBy(0.5);

sc->timesEnclosed = 0;

bool outer = true;

for(j = 0; j < l.n; j++) {

if(i == j) continue;

SContour *sct = &(l[j]);

if(sct->ContainsPointProjdToNormal(normal, pt)) {

outer = !outer;

(sc->timesEnclosed)++;

}

}

bool clockwise = sc->IsClockwiseProjdToNormal(normal);

if((clockwise && outer) || (!clockwise && !outer)) {

sc->Reverse();

sc->tag = 1;

}

}

if(l.IsEmpty() || l[0].l.IsEmpty())

return true;

return false;

SEdgeList el = {};

MakeEdgesInto(&el);

SKdNodeEdges *kdtree = SKdNodeEdges::From(&el);

int cnt = 1;

el.l.ClearTags();

bool ret = false;

SEdge *se;

for(se = el.l.First(); se; se = el.l.NextAfter(se)) {

int inters = kdtree->AnyEdgeCrossings(se->a, se->b, cnt, intersectsAt);

if(inters != 1) {

ret = true;

break;

}

cnt++;

}

el.Clear();

return ret;

for(const SContour &sc : l) {

SContour tsc = {};

tsc.timesEnclosed = sc.timesEnclosed;

for(const SPoint &sp : sc.l) {

tsc.AddPoint(sp.p.DotInToCsys(u, v, n));

}

sp->l.Add(&tsc);

}

int i;

dest->Clear();

for(i = 0; i < l.n; i++) {

const SContour *sc = &(l[i]);

dest->AddEmptyContour();

sc->OffsetInto(&(dest->l[dest->l.n-1]), r);

}

double x0B, double y0B, double dxB, double dyB,

double *xi, double *yi)

{

double A[2][2];

double b[2];

// The line is given to us in the form:

// (x(t), y(t)) = (x0, y0) + t*(dx, dy)

// so first rewrite it as

// (x - x0, y - y0) dot (dy, -dx) = 0

// x*dy - x0*dy - y*dx + y0*dx = 0

// x*dy - y*dx = (x0*dy - y0*dx)

// So write the matrix, pre-pivoted.

if(fabs(dyA) > fabs(dyB)) {

A[0][0] = dyA; A[0][1] = -dxA; b[0] = x0A*dyA - y0A*dxA;

A[1][0] = dyB; A[1][1] = -dxB; b[1] = x0B*dyB - y0B*dxB;

} else {

A[1][0] = dyA; A[1][1] = -dxA; b[1] = x0A*dyA - y0A*dxA;

A[0][0] = dyB; A[0][1] = -dxB; b[0] = x0B*dyB - y0B*dxB;

}

// Check the determinant; if it's zero then no solution.

if(fabs(A[0][0]*A[1][1] - A[0][1]*A[1][0]) < LENGTH_EPS) {

return false;

}

// Solve

double v = A[1][0] / A[0][0];

A[1][0] -= A[0][0]*v;

A[1][1] -= A[0][1]*v;

b[1] -= b[0]*v;

// Back-substitute.

*yi = b[1] / A[1][1];

*xi = (b[0] - A[0][1]*(*yi)) / A[0][0];

return true;

int i;

for(i = 0; i < l.n; i++) {

Vector a, b, c;

Vector dp, dn;

double thetan, thetap;

a = l[WRAP(i-1, (l.n-1))].p;

b = l[WRAP(i, (l.n-1))].p;

c = l[WRAP(i+1, (l.n-1))].p;

dp = a.Minus(b);

thetap = atan2(dp.y, dp.x);

dn = b.Minus(c);

thetan = atan2(dn.y, dn.x);

// A short line segment in a badly-generated polygon might look

// okay but screw up our sense of direction.

if(dp.Magnitude() < LENGTH_EPS || dn.Magnitude() < LENGTH_EPS) {

continue;

}

if(thetan > thetap && (thetan - thetap) > PI) {

thetap += 2*PI;

}

if(thetan < thetap && (thetap - thetan) > PI) {

thetan += 2*PI;

}

if(fabs(thetan - thetap) < (1*PI)/180) {

Vector p = { b.x - r*sin(thetap), b.y + r*cos(thetap), 0 };

dest->AddPoint(p);

} else if(thetan < thetap) {

// This is an inside corner. We have two edges, Ep and En. Move

// out from their intersection by radius, normal to En, and

// then draw a line parallel to En. Move out from their

// intersection by radius, normal to Ep, and then draw a second

// line parallel to Ep. The point that we want to generate is

// the intersection of these two lines--it removes as much

// material as we can without removing any that we shouldn't.

double px0, py0, pdx, pdy;

double nx0, ny0, ndx, ndy;

double x = 0.0, y = 0.0;

px0 = b.x - r*sin(thetap);

py0 = b.y + r*cos(thetap);

pdx = cos(thetap);

pdy = sin(thetap);

nx0 = b.x - r*sin(thetan);

ny0 = b.y + r*cos(thetan);

ndx = cos(thetan);

ndy = sin(thetan);

IntersectionOfLines(px0, py0, pdx, pdy,

nx0, ny0, ndx, ndy,

&x, &y);

dest->AddPoint({x, y, 0});

} else {

if(fabs(thetap - thetan) < (6*PI)/180) {

Vector pp = { b.x - r*sin(thetap),

b.y + r*cos(thetap), 0 };

dest->AddPoint(pp);

Vector pn = { b.x - r*sin(thetan),

b.y + r*cos(thetan), 0 };

dest->AddPoint(pn);

} else {

double theta;

for(theta = thetap; theta <= thetan; theta += (6*PI)/180) {

Vector p = { b.x - r*sin(theta),

b.y + r*cos(theta), 0 };

dest->AddPoint(p);

}

}

}

}