static void addcost(double *energy, double tweight, double tcost, int pr);
#define COST(weight, compute) addcost(&energy, (weight), (compute), printing)
+double density;
+
+void energy_init(void) {
+ density= sqrt(N);
+}
+
/*---------- main energy computation and subroutines ----------*/
double compute_energy(const struct Vertices *vs) {
if (printing) printf("cost > energy |");
- COST(3e2, edgewise_vertex_displacement_cost(vs->a));
+ COST(2.25e3, line_bending_adjcost(vs->a));
COST(1e3, edge_length_variation_cost(vs->a));
COST(0.2e3, rim_proximity_cost(vs->a));
// COST(1e2, graph_layout_cost(vs->a));
static void addcost(double *energy, double tweight, double tcost, int pr) {
double tenergy= tweight * tcost;
- if (pr) printf(" %# e > %# e* |", tcost, tenergy);
+ if (pr) printf(" %# e x %# e > %# e* |", tcost, tweight, tenergy);
*energy += tenergy;
}
void compute_edge_lengths(const Vertices vertices) {
int v1,e,v2;
-
+
FOR_EDGE(v1,e,v2)
edge_lengths[v1][e]= hypotD(vertices[v1],vertices[v2]);
}
// }
// xprod(av, e1v, e2v);
// total += magnD(av);
-
+
count++;
}
vertex_areas[v0]= total / count;
/*---------- Edgewise vertex displacement ----------*/
/*
+ * Definition:
+ *
+ * At each vertex Q, in each direction e:
+ *
+ * e
+ * Q ----->----- R
+ * _,-'\__/
+ * _,-' delta
+ * P '
*
+ * r
+ * cost = delta (we use r=3)
+ * Q,e
*
*
- * Q `-_
- * / | `-_
- * / | `-.
- * / M - - - - - S
- * / ' | _,-'
- * / ' | _,-'
- * / ' , P '
- * / ',-'
- * /,-'
- * /'
- * R
+ * Calculation:
*
- * Let delta = 180deg - angle RMS
+ * Let vector A = PQ
+ * B = QR
*
- * Let l = |PQ|
- * d = |RS|
+ * -1 A . B
+ * delta = tan -------
+ * | A x B |
*
- * Giving energy contribution:
+ * which is always in the range 0..pi because the denominator
+ * is nonnegative. We add epsilon to |AxB| to avoid division
+ * by zero.
*
- * 3
- * l delta
- * E = F . --------
- * vd, edge PQ vd d
+ * r
+ * cost = delta
+ * Q,e
*
+ * Normalisation:
*
- * (The dimensions of this are those of F_vd.)
+ * We want the minimum energy to remain unchanged with changes in
+ * triangle densitiy, when the vertices lie evenly spaced on
+ * circles, and we do this by normalising the force ie the
+ * derivative of the energy with respect to linear motions of the
+ * vertices.
*
- * We calculate delta as atan2(|AxB|, A.B)
- * where A = PQ, B = QR
+ * We consider only the force on Q due to PQR, wlog. (Forces on
+ * P qnd R due to PQR are equal and opposite so normalising
+ * forces on Q will normalise them too.)
+ *
+ * Force on Q is in the plnae PQR and normal to PR, so we can
+ * consider it only linearly in that dimension. WLOG let that be
+ * the x dimension. So with f' representing df'/dx_Q:
+ *
+ * , d
+ * F = cost = --
+ * Q,e Q,e err looks like we can only do
+ * this if we make some kind of
+ * assumption about delta or
+ * something give up
+ *
+ *
+ * Interposing M and N so that we have P-M-Q-N-R
+ * generates half as much delta for each vertex. So
*
- * In practice to avoid division by zero we'll add epsilon to d and
- * |AxB| and the huge energy ought then to be sufficient for the
- * model to avoid being close to R=S.
- */
-double edgewise_vertex_displacement_cost(const Vertices vertices) {
+ In that case the force on Q
+ * due to PQR
+ *
+ *Normalising for equal linear
+ * forces:
+ *
+ * d
+ * linear force on Q due to e = ------- cost
+ * d coord Q,e
+ * Q
+ *
+ * (we will consider only one e and one coord and hope
+ * that doesn't lead us astray.)
+ *
+ *
+ * , -r
+ * cost = D . cost
+ * Q,e Q,e
+ *
+ * where D is the linear density.
+ *
+ * , -r
+ * Sigma cost = N . D . Sigma cost
+ * Q,e Q,e Q,e Q,e
+ *
+ * */
+
+double line_bending_adjcost(const Vertices vertices) {
static const double axb_epsilon= 1e-6;
+ static const double exponent_r= 3;
- int pi,e,qi,ri, k; //,si
- double a[D3], b[D3], axb[D3]; //m[D3],
+ int pi,e,qi,ri, k;
+ double a[D3], b[D3], axb[D3];
double total_cost= 0;
FOR_EDGE(qi,e,ri) {
pi= EDGE_END2(qi,(e+3)%V6); if (pi<0) continue;
-// K m[k]= (vertices[pi][k] + vertices[qi][k]) * 0.5;
K a[k]= -vertices[pi][k] + vertices[qi][k];
K b[k]= -vertices[qi][k] + vertices[ri][k];
xprod(axb,a,b);
-
+
double delta= atan2(magnD(axb) + axb_epsilon, dotprod(a,b));
- double cost= pow(delta,3);
+ double cost= pow(delta,exponent_r);
if (!e && !(qi & YMASK))
cost *= 10;
total_cost += cost;
}
- return total_cost;
+ return total_cost / (N / density);
}
/*---------- edge length variation ----------*/
+ /*
+ * Definition:
+ *
+ * See the diagram above.
+ *
+ * cost =
+ * Q,e
+
double edge_length_variation_cost(const Vertices vertices) {
double diff, cost= 0;
- int v0, efwd,vfwd, eback;
+ int q, e,r, eback;
- FOR_EDGE(v0,efwd,vfwd) {
- eback= edge_reverse(v0,efwd);
- diff= edge_lengths[v0][efwd] - edge_lengths[v0][eback];
+ FOR_EDGE(q,e,r) {
+ eback= edge_reverse(q,e);
+ diff= edge_lengths[q][e] - edge_lengths[q][eback];
cost += diff*diff;
}
return cost;
-}
+}
/*---------- rim proximity cost ----------*/
FOR_RIM_VERTEX(vy,vx,v) {
find_nearest_oncircle(oncircle, vertices[v]);
-
+
double d2= hypotD2(vertices[v], oncircle);
cost += d2*d2;
}
~ian / moebius2.git / blobdiff