Correct mixed-up angles

The angle distortions at the second and third vertices in each triangle
seem to have been mixed up in the previous commit. I've checked, for a
few triangles, that the output of this commit is consistent with the law
of sines.

app-proto/main.css

@@ -227,6 +227,16 @@ details[open]:has(li) .element-switch::after {
   border-radius: 8px;
 }
 
+#distortion-bar {
+  display: flex;
+  margin-top: 8px;
+  gap: 8px;
+}
+
+#distortion-gauge {
+  flex-grow: 1;
+}
+
 /* display */
 
 #display {

app-proto/src/assembly.rs

@@ -3,6 +3,7 @@ use std::{
     cell::Cell,
     cmp::Ordering,
     collections::{BTreeMap, BTreeSet},
+    f64::consts::SQRT_2,
     fmt,
     fmt::{Debug, Formatter},
     hash::{Hash, Hasher},
@@ -122,6 +123,11 @@ pub trait Element: Serial + ProblemPoser + DisplayItem {
     // be used carefully to preserve invariant (1), described in the comment on
     // the `tangent` field of the `Assembly` structure
     fn set_column_index(&self, index: usize);
+    
+    /* KLUDGE */
+    fn is_point(&self) -> bool {
+        false
+    }
 }
 
 impl Debug for dyn Element {
@@ -334,6 +340,10 @@ impl Element for Point {
     fn set_column_index(&self, index: usize) {
         self.column_index.set(Some(index));
     }
+    
+    fn is_point(&self) -> bool {
+        true
+    }
 }
 
 impl Serial for Point {
@@ -357,6 +367,12 @@ pub trait Regulator: Serial + ProblemPoser + OutlineItem {
     fn subjects(&self) -> Vec<Rc<dyn Element>>;
     fn measurement(&self) -> ReadSignal<f64>;
     fn set_point(&self) -> Signal<SpecifiedValue>;
+    fn soft(&self) -> Option<Signal<bool>> {
+        None
+    }
+    fn distortion(&self) -> Option<ReadSignal<f64>> { /* KLUDGE */
+        None
+    }
 }
 
 impl Hash for dyn Regulator {
@@ -389,6 +405,8 @@ pub struct InversiveDistanceRegulator {
     pub subjects: [Rc<dyn Element>; 2],
     pub measurement: ReadSignal<f64>,
     pub set_point: Signal<SpecifiedValue>,
+    pub soft: Signal<bool>,
+    distortion: Option<ReadSignal<f64>>, /* KLUDGE */
     serial: u64,
 }
 
@@ -404,9 +422,24 @@ impl InversiveDistanceRegulator {
         });
         
         let set_point = create_signal(SpecifiedValue::from_empty_spec());
+        let distortion = if subjects.iter().all(|subj| subj.is_point()) {
+            Some(create_memo(move || {
+                let set_point_opt = set_point.with(|set_pt| set_pt.value);
+                let measurement_val = measurement.get();
+                match set_point_opt {
+                    None => 0.0,
+                    Some(set_point_val) => SQRT_2 * (
+                        (-measurement_val).sqrt() - (-set_point_val).sqrt()
+                    ),
+                }
+            }))
+        } else {
+            None
+        };
+        let soft = create_signal(false);
         let serial = Self::next_serial();
         
-        Self { subjects, measurement, set_point, serial }
+        Self { subjects, measurement, set_point, soft, distortion, serial }
     }
 }
 
@@ -422,6 +455,14 @@ impl Regulator for InversiveDistanceRegulator {
     fn set_point(&self) -> Signal<SpecifiedValue> {
         self.set_point
     }
+    
+    fn soft(&self) -> Option<Signal<bool>> {
+        Some(self.soft)
+    }
+    
+    fn distortion(&self) -> Option<ReadSignal<f64>> {
+        self.distortion
+    }
 }
 
 impl Serial for InversiveDistanceRegulator {
@@ -432,6 +473,7 @@ impl Serial for InversiveDistanceRegulator {
 
 impl ProblemPoser for InversiveDistanceRegulator {
     fn pose(&self, problem: &mut ConstraintProblem) {
+        let soft = self.soft.get_untracked();
         self.set_point.with_untracked(|set_pt| {
             if let Some(val) = set_pt.value {
                 let [row, col] = self.subjects.each_ref().map(
@@ -439,7 +481,11 @@ impl ProblemPoser for InversiveDistanceRegulator {
                         "Subjects should be indexed before inversive distance regulator writes problem data"
                     )
                 );
-                problem.gram.push_sym(row, col, val);
+                if soft {
+                    problem.soft.push_sym(row, col, val);
+                } else {
+                    problem.gram.push_sym(row, col, val);
+                }
             }
         });
     }
@@ -705,7 +751,7 @@ impl Assembly {
         
         // look for a configuration with the given Gram matrix
         let Realization { result, history } = realize_gram(
-            &problem, 1.0e-12, 0.5, 0.9, 1.1, 200, 110
+            &problem, 1.0e-20, 0.5, 0.9, 1.1, 400, 110
         );
         
         /* DEBUG */

app-proto/src/components/diagnostics.rs

@@ -111,6 +111,94 @@ fn StepInput() -> View {
     }
 }
 
+#[component]
+fn DistortionGauge() -> View {
+    let state = use_context::<AppState>();
+    let total_distortion = create_memo(move || {
+        state.assembly.regulators.with(|regs| {
+            let mut total = 0.0;
+            for reg in regs {
+                if let Some(distortion) = reg.distortion() {
+                    total += distortion.get().abs();
+                }
+            }
+            total
+        })
+    });
+    
+    view! {
+        div(id = "distortion-gauge") {
+            "Distortion: " (total_distortion.with(|distort| distort.to_string()))
+        }
+    }
+}
+
+#[component]
+fn PrintButton() -> View {
+    view! {
+        button(
+            on:click = |_| {
+                let state = use_context::<AppState>();
+                
+                // print the edge length distortions
+                let mut hard_distortion_table = String::new();
+                let mut soft_distortion_table = String::new();
+                let mut highest_distortion = f64::NEG_INFINITY;
+                let mut lowest_distortion = f64::INFINITY;
+                let mut largest_hard_distortion = f64::NEG_INFINITY;
+                state.assembly.regulators.with_untracked(|regs| {
+                    for reg in regs {
+                        if let Some(distortion) = reg.distortion() {
+                            let distortion_val = distortion.get();
+                            let subjects = reg.subjects();
+                            let distortion_line = format!(
+                                "{}, {}: {distortion_val}\n",
+                                subjects[0].id(),
+                                subjects[1].id(),
+                            );
+                            match reg.soft() {
+                                Some(soft) if soft.get() => {
+                                    soft_distortion_table += &distortion_line;
+                                    highest_distortion = highest_distortion.max(distortion_val);
+                                    lowest_distortion = lowest_distortion.min(distortion_val);
+                                },
+                                _ => {
+                                    hard_distortion_table += &distortion_line;
+                                    largest_hard_distortion = largest_hard_distortion.max(distortion_val.abs());
+                                }
+                            };
+                        }
+                    }
+                });
+                console_log!("\
+                    === Distortions of flexible edges (for labels) ===\n\n\
+                    --- Range ---\n\n\
+                    Highest: {highest_distortion}\n\
+                    Lowest: {lowest_distortion}\n\n\
+                    --- Table ---\n\n{soft_distortion_table}\n\
+                    === Distortions of rigid edges (for validation) ===\n\n\
+                    These values should be small relative to the ones for the flexible edges\n\n\
+                    --- Range ---\n\n\
+                    Largest absolute: {largest_hard_distortion}\n\n\
+                    --- Table ---\n\n{hard_distortion_table}\
+                ");
+                
+                // print the vertex coordinates
+                let mut coords_table = String::new();
+                state.assembly.elements.with_untracked(|elts| {
+                    for elt in elts.iter().filter(|elt| elt.is_point()) {
+                        let (x, y, z) = elt.representation().with(
+                            |rep| (rep[0], rep[1], rep[2])
+                        );
+                        coords_table += &format!("{}: {x}, {y}, {z}\n", elt.id());
+                    }
+                });
+                console_log!("=== Vertex coordinates ===\n\n{coords_table}");
+            },
+        ) { "Print" }
+    }
+}
+
 fn into_log10_time_point((step, value): (usize, f64)) -> Vec<Option<f64>> {
     vec![
         Some(step as f64),
@@ -315,6 +403,10 @@ pub fn Diagnostics() -> View {
             }
             DiagnosticsPanel(name = "loss") { LossHistory {} }
             DiagnosticsPanel(name = "spectrum") { SpectrumHistory {} }
+            div(id = "distortion-bar") {
+                DistortionGauge {}
+                PrintButton {}
+            }
         }
     }
 }

app-proto/src/components/display.rs

@@ -597,16 +597,16 @@ pub fn Display() -> View {
                 |status| status.is_ok()
             );
             let step_val = state.assembly.step.with_untracked(|step| step.value);
-            let on_init_step = step_val.is_some_and(|n| n == 0.0);
             let on_last_step = step_val.is_some_and(
                 |n| state.assembly.descent_history.with_untracked(
                     |history| n as usize + 1 == history.config.len().max(1)
                 )
             );
-            let on_manipulable_step =
-                !realization_successful && on_init_step
-                || realization_successful && on_last_step;
-            if on_manipulable_step && state.selection.with(|sel| sel.len() == 1) {
+            if
+                state.selection.with(|sel| sel.len() == 1)
+                && realization_successful
+                && on_last_step
+            {
                 let sel = state.selection.with(
                     |sel| sel.into_iter().next().unwrap().clone()
                 );

app-proto/src/engine.rs

@@ -1,6 +1,7 @@
 use lazy_static::lazy_static;
 use nalgebra::{Const, DMatrix, DVector, DVectorView, Dyn, SymmetricEigen};
 use std::fmt::{Display, Error, Formatter};
+use sycamore::prelude::console_log; /* DEBUG */
 
 // --- elements ---
 
@@ -240,6 +241,7 @@ impl DescentHistory {
 
 pub struct ConstraintProblem {
     pub gram: PartialMatrix,
+    pub soft: PartialMatrix,
     pub frozen: PartialMatrix,
     pub guess: DMatrix<f64>,
 }
@@ -249,6 +251,7 @@ impl ConstraintProblem {
         const ELEMENT_DIM: usize = 5;
         Self {
             gram: PartialMatrix::new(),
+            soft: PartialMatrix::new(),
             frozen: PartialMatrix::new(),
             guess: DMatrix::<f64>::zeros(ELEMENT_DIM, element_count),
         }
@@ -258,6 +261,7 @@ impl ConstraintProblem {
     pub fn from_guess(guess_columns: &[DVector<f64>]) -> Self {
         Self {
             gram: PartialMatrix::new(),
+            soft: PartialMatrix::new(),
             frozen: PartialMatrix::new(),
             guess: DMatrix::from_columns(guess_columns),
         }
@@ -280,14 +284,18 @@ lazy_static! {
 struct SearchState {
     config: DMatrix<f64>,
     err_proj: DMatrix<f64>,
+    loss_hard: f64,
     loss: f64,
 }
 
 impl SearchState {
-    fn from_config(gram: &PartialMatrix, config: DMatrix<f64>) -> Self {
-        let err_proj = gram.sub_proj(&(config.tr_mul(&*Q) * &config));
+    fn from_config(gram: &PartialMatrix, soft: &PartialMatrix, softness: f64, config: DMatrix<f64>) -> Self {
+        let config_gram = &(config.tr_mul(&*Q) * &config);
+        let err_proj_hard = gram.sub_proj(config_gram);
+        let err_proj = &err_proj_hard + softness * soft.sub_proj(config_gram);
+        let loss_hard = err_proj_hard.norm_squared();
         let loss = err_proj.norm_squared();
-        Self { config, err_proj, loss }
+        Self { config, err_proj, loss_hard, loss }
     }
 }
 
@@ -331,6 +339,8 @@ pub fn local_unif_to_std(v: DVectorView<f64>) -> DMatrix<f64> {
 // use backtracking line search to find a better configuration
 fn seek_better_config(
     gram: &PartialMatrix,
+    soft: &PartialMatrix,
+    softness: f64,
     state: &SearchState,
     base_step: &DMatrix<f64>,
     base_target_improvement: f64,
@@ -341,7 +351,7 @@ fn seek_better_config(
     let mut rate = 1.0;
     for backoff_steps in 0..max_backoff_steps {
         let trial_config = &state.config + rate * base_step;
-        let trial_state = SearchState::from_config(gram, trial_config);
+        let trial_state = SearchState::from_config(gram, soft, softness, trial_config);
         let improvement = state.loss - trial_state.loss;
         if improvement >= min_efficiency * rate * base_target_improvement {
             return Some((trial_state, backoff_steps));
@@ -376,7 +386,7 @@ pub fn realize_gram(
     max_backoff_steps: i32,
 ) -> Realization {
     // destructure the problem data
-    let ConstraintProblem { gram, guess, frozen } = problem;
+    let ConstraintProblem { gram, soft, guess, frozen } = problem;
     
     // start the descent history
     let mut history = DescentHistory::new();
@@ -403,13 +413,18 @@ pub fn realize_gram(
     let scale_adjustment = (gram.0.len() as f64).sqrt();
     let tol = scale_adjustment * scaled_tol;
     
+    // set up constants and variables related to minimizing the soft loss
+    const GRAD_TOL: f64 = 1e-12;
+    let mut grad_size = f64::INFINITY;
+    let mut softness = 1.0;
+    
     // convert the frozen indices to stacked format
     let frozen_stacked: Vec<usize> = frozen.into_iter().map(
         |MatrixEntry { index: (row, col), .. }| col*element_dim + row
     ).collect();
     
     // use a regularized Newton's method with backtracking
-    let mut state = SearchState::from_config(gram, frozen.freeze(guess));
+    let mut state = SearchState::from_config(gram, soft, softness, frozen.freeze(guess));
     let mut hess = DMatrix::zeros(element_dim, assembly_dim);
     for _ in 0..max_descent_steps {
         // find the negative gradient of the loss function
@@ -426,7 +441,7 @@ pub fn realize_gram(
                 let neg_d_err =
                     basis_mat.tr_mul(&*Q) * &state.config
                     + state.config.tr_mul(&*Q) * &basis_mat;
-                let neg_d_err_proj = gram.proj(&neg_d_err);
+                let neg_d_err_proj = gram.proj(&neg_d_err) + softness * soft.proj(&neg_d_err);
                 let deriv_grad = 4.0 * &*Q * (
                     -&basis_mat * &state.err_proj
                     + &state.config * &neg_d_err_proj
@@ -455,10 +470,13 @@ pub fn realize_gram(
             hess[(k, k)] = 1.0;
         }
         
-        // stop if the loss is tolerably low
+        // stop if the hard loss is tolerably low and the total loss is close to
+        // stationary. we use `neg_grad_stacked` to measure the size of the
+        // gradient because it's been projected onto the frozen subspace
         history.config.push(state.config.clone());
-        history.scaled_loss.push(state.loss / scale_adjustment);
-        if state.loss < tol { break; }
+        history.scaled_loss.push(state.loss_hard / scale_adjustment);
+        grad_size = neg_grad_stacked.norm_squared();
+        if state.loss_hard < tol && grad_size < softness * GRAD_TOL { break; }
         
         // compute the Newton step
         /* TO DO */
@@ -482,7 +500,7 @@ pub fn realize_gram(
         
         // use backtracking line search to find a better configuration
         if let Some((better_state, backoff_steps)) = seek_better_config(
-            gram, &state, &base_step, neg_grad.dot(&base_step),
+            gram, soft, softness, &state, &base_step, neg_grad.dot(&base_step),
             min_efficiency, backoff, max_backoff_steps,
         ) {
             state = better_state;
@@ -493,8 +511,18 @@ pub fn realize_gram(
                 history,
             };
         }
+        
+        // if we're near a minimum of the total loss, but the hard loss still
+        // isn't tolerably low, make the soft constraints softer
+        const SOFTNESS_BACKOFF_THRESHOLD: f64 = 1e-6;
+        const SOFTNESS_BACKOFF: f64 = 0.95;
+        if state.loss_hard >= tol && grad_size < softness * SOFTNESS_BACKOFF_THRESHOLD {
+            softness *= SOFTNESS_BACKOFF;
+            state = SearchState::from_config(gram, soft, softness, state.config);
+            console_log!("Softness decreased to {softness}");
+        }
     }
-    let result = if state.loss < tol {
+    let result = if state.loss_hard < tol && grad_size < softness * GRAD_TOL {
         // express the uniform basis in the standard basis
         const UNIFORM_DIM: usize = 4;
         let total_dim_unif = UNIFORM_DIM * assembly_dim;

impossolid/angle-distortion.py

@@ -0,0 +1,59 @@
+import collections
+import math
+import sys
+
+def read_edge_distortions(filename):
+  vertices = set()
+  distortions = {}
+  with open(filename, 'r') as edge_file:
+    while edge_line := edge_file.readline():
+      line_parts = edge_line.rstrip().split(': ')
+      endpoints = tuple(sorted(line_parts[0].split(', ')))
+      vertices.update(endpoints)
+      if len(line_parts) > 1:
+        distortions[endpoints] = float(line_parts[1])
+      else:
+        distortions[endpoints] = 0
+  return (vertices, distortions)
+
+def find_triangles(vertices, edges):
+  triangles = []
+  for e in sorted(edges):
+    for v in sorted(vertices):
+      if e[1] < v:
+        if (e[0], v) in edges and (e[1], v) in edges:
+          triangles.append((e[0], e[1], v))
+  return triangles
+
+# use the law of cosines to get the angle distortion
+def angle_distortion(edge_distortions):
+  a, b, c = list(edge_distortions)
+  cos_angle_a = (1 + 2*(b + c - a) + b*b + c*c - a*a) / (2*(1 + b)*(1 + c))
+  return math.degrees(math.acos(cos_angle_a)) - 60
+
+if __name__ == '__main__':
+  if len(sys.argv) <= 1:
+    print('Pass the path to the file that lists the edge distortions')
+  else:
+    vertices, distortions = read_edge_distortions(sys.argv[1])
+    triangles = find_triangles(vertices, distortions.keys())
+    total_angle_distortion = 0
+    highest_angle_distortion = -math.inf
+    lowest_angle_distortion = math.inf
+    print('{} triangles\n'.format(len(triangles)))
+    for t in triangles:
+      print('Triangle {0}, {1}, {2}'.format(t[0], t[1], t[2]))
+      edge_distortions = collections.deque(
+        [distortions[(t[j], t[k])] for (j, k) in [(1, 2), (0, 1), (0, 2)]]
+      )
+      for k in range(3):
+        ang_distort = angle_distortion(edge_distortions)
+        total_angle_distortion += abs(ang_distort)
+        highest_angle_distortion = max(highest_angle_distortion, ang_distort)
+        lowest_angle_distortion = min(lowest_angle_distortion, ang_distort)
+        print('  {0}: {1}°'.format(t[k], ang_distort))
+        edge_distortions.rotate()
+      print()
+    print('Total angle distortion: {}°'.format(total_angle_distortion))
+    print('Highest angle distortion: {}°'.format(highest_angle_distortion))
+    print('Lowest angle distortion: {}°'.format(lowest_angle_distortion))

impossolid/twice-augmented

@@ -0,0 +1,150 @@
+=== Distortions of flexible edges (for labels) ===
+
+--- Range ---
+
+Highest: 0.000039581305103782685
+Lowest: -0.00003959022952266363
+
+--- Table ---
+
+a_SE, b_SE: 0.000015249448583054653
+b_SW, c_S: 0.000006918255000202575
+b_SE, c_S: 0.00001022525369434997
+b_SE, c_E: 0.000010225282824746383
+b_NE, c_E: 0.000006918450075555058
+z_S, a_SW: -0.0000219346808875234
+z_S, a_SE: 0.00002021118621352795
+z_E, a_SE: 0.00002021122888534378
+z_E, a_NE: -0.00002193465480671757
+z_S, b_SW: 0.000015276737515369292
+z_S, b_SE: -0.000010771908316892743
+z_E, b_SE: -0.00001077200999765046
+z_E, b_NE: 0.000015276631362360217
+z_S, c_S: -0.0000027853351723499016
+z_E, c_E: -0.000002785228607191555
+c_N, d_NE: -0.000007632775579087326
+c_N, d_NW: 0.00001402754980516112
+c_W, d_NW: 0.000014028001282264439
+c_W, d_SW: -0.00000763316944431703
+c_S, d_SW: 0.000006566974972180328
+c_S, d_SE: -0.0000033161321458747414
+c_E, d_SE: -0.00000331586236646513
+c_E, d_NE: 0.000006566624318564243
+y_NE, b_NE: -0.0000017608392620486498
+y_NW, b_NW: -0.00003959022952266363
+y_SW, b_SW: -0.0000017608153313274316
+y_SE, b_SE: 0.000005637007821097203
+y_NE, c_N: 0.000001760310254250749
+y_NW, c_N: 0.00003958128591112948
+y_NW, c_W: 0.000039581305103782685
+y_SW, c_W: 0.0000017603065237110672
+y_SW, c_S: 0.0000017602552603493003
+y_SE, c_S: -0.00000563497752606038
+y_SE, c_E: -0.0000056349905677193694
+y_NE, c_E: 0.0000017602528935919282
+y_NE, d_NE: -0.0000017608481082635805
+y_NW, d_NW: -0.0000395891283971962
+y_SW, d_SW: -0.0000017608381747596222
+y_SE, d_SE: 0.0000056371661583272735
+d_NE, e_N: 0.000007303314580670295
+d_NW, e_N: -0.000016255656816635924
+d_NW, e_W: -0.000016256069869023512
+d_SW, e_W: 0.00000730369455466398
+d_SW, e_S: -0.000006000901856373712
+d_SE, e_S: 0.000003879307405619109
+d_SE, e_E: 0.000003879053499059208
+d_NE, e_E: -0.00000600059436381427
+c_N, e_N: 0.00000899767908293228
+c_W, e_W: 0.000008997328779446816
+c_S, e_S: 0.0000042757526759396545
+c_E, e_E: 0.000004275945999382972
+e_N, f_NE: 0.0000026971057195376348
+e_N, f_NW: 0.000006498321003978716
+e_W, f_NW: 0.000006498333167484994
+e_W, f_SW: 0.000002697065747495785
+e_S, f_SW: 0.000001359196116957704
+e_S, f_SE: 0.0000010550443264554641
+e_E, f_SE: 0.0000010550540852081317
+e_E, f_NE: 0.000001359177866673992
+d_NE, f_NE: -0.0000029405886467824407
+d_NW, f_NW: -0.000009421855439304757
+d_SW, f_SW: -0.000002940615976753832
+d_SE, f_SE: -0.000001529705558996715
+f_NE, g_ENE: -0.00000011649787286240574
+f_NE, g_NNE: -0.000002135681696645076
+f_NW, g_NNW: -0.000003607774033318148
+f_NW, g_WNW: -0.000003607785721714447
+f_SW, g_WSW: -0.0000021356459140809334
+f_SW, g_SSW: -0.00000011652747271646458
+f_SE, g_SSE: -0.0000005858121892925972
+f_SE, g_ESE: -0.0000005858113552594831
+e_N, g_NNE: -0.0000034483169191011346
+e_N, g_NNW: 0.0000034059972640161084
+e_W, g_WNW: 0.000003405934362501001
+e_W, g_WSW: -0.00000344858668736309
+e_S, g_SSW: 0.0000029689592297656908
+e_S, g_SSE: -0.0000006279732347365118
+e_E, g_ESE: -0.0000006278110244023648
+e_E, g_ENE: 0.0000029689801689897473
+
+=== Distortions of rigid edges (for validation) ===
+
+These values should be small relative to the ones for the flexible edges
+
+--- Range ---
+
+Largest absolute: 0.00000000002916368237382178
+
+--- Table ---
+
+a_NE, a_NW: 0.000000000024039999653399018
+a_NW, a_SW: 0.00000000002451793579782236
+a_SW, a_SE: -0.000000000028570972470668657
+a_SE, a_NE: -0.00000000002916368237382178
+a_NE, b_NE: -0.000000000022712800498073622
+a_NW, b_NW: 0.0000000000010783395006240168
+a_SW, b_SW: -0.00000000002146677512286218
+b_NE, c_N: 0.00000000000907905964233893
+b_NW, c_N: -0.00000000002003437977280497
+b_NW, c_W: -0.000000000019747523880603447
+b_SW, c_W: 0.000000000009824853560213723
+g_NNE, g_NNW: 0.0000000000003400820265509057
+g_NNW, g_WNW: -0.0000000000004886127731423908
+g_WNW, g_WSW: 0.00000000000005762239323369454
+g_WSW, g_SSW: -0.00000000000020348398264541725
+g_SSW, g_SSE: -0.0000000000010258984125039788
+g_SSE, g_ESE: -0.0000000000001306316925624901
+g_ESE, g_ENE: -0.0000000000011334497459238174
+g_ENE, g_NNE: -0.00000000000007913265991766227
+a_SE, a_NW: 0.00000000001299251509560824
+a_SW, a_NE: -0.000000000019852249047597653
+a_NW, c_N: 0.00000000000757318396521532
+b_NW, b_NE: -0.000000000015613313427643198
+c_N, a_NE: -0.000000000013250324277324116
+a_NW, b_NE: 0.0000000000145795645528458
+b_NW, a_NE: 0.000000000004191204809210469
+a_SW, c_W: -0.000000000012779924576702457
+b_SW, b_NW: -0.000000000015326928563179278
+c_W, a_NW: 0.00000000000786836134744787
+a_SW, b_NW: 0.000000000004368311238549999
+b_SW, a_NW: 0.00000000001505750069726914
+g_NNE, g_WNW: 0.0000000000005718276734526309
+g_NNW, g_WSW: 0.0000000000004788781998985515
+g_WNW, g_SSW: -0.00000000000035703902510469045
+g_WSW, g_SSE: 0.0000000000002254652770669901
+g_SSW, g_ESE: -0.0000000000002515288118811408
+g_SSE, g_ENE: -0.00000000000030679606642680965
+g_NNE, g_WSW: 0.0000000000003206128800632269
+g_NNW, g_SSW: 0.00000000000032846334235664577
+g_WNW, g_SSE: 0.00000000000012278123026907122
+g_WSW, g_ESE: 0.00000000000014507654318238083
+g_SSW, g_ENE: 0.000000000000033599978615832786
+g_NNE, g_SSW: 0.0000000000001815026882238444
+g_NNW, g_SSE: 0.0000000000002631474960754007
+g_WNW, g_ESE: 0.00000000000031684465816238584
+g_WSW, g_ENE: 0.0000000000002348858318190928
+g_NNE, g_SSE: 0.00000000000007913265991766227
+g_NNW, g_ESE: 0.0000000000001438204692154338
+g_WNW, g_ENE: 0.0000000000004248670193198296
+g_NNE, g_ESE: 0.00000000000019783164979415566
+g_NNW, g_ENE: -0.0000000000002364559242777765

impossolid/twice-augmented_angle-distortions

@@ -0,0 +1,295 @@
+58 triangles
+
+Triangle a_NE, a_SE, z_E
+  a_NE: 0.0020627780860564826°
+  a_SE: -0.0021197445448706276°
+  z_E: 5.696645881414497e-05°
+
+Triangle a_NE, b_NE, z_E
+  a_NE: 0.0017363010366011622°
+  b_NE: -0.001956513180971342°
+  z_E: 0.00022021214437017989°
+
+Triangle a_SE, a_SW, z_S
+  a_SE: -0.0021197448587955137°
+  a_SW: 0.0020627761256122312°
+  z_S: 5.696873318328244e-05°
+
+Triangle a_SE, b_SE, z_E
+  a_SE: -0.0018856584331388149°
+  b_SE: 0.0011890357645540917°
+  z_E: 0.0006966226686131449°
+
+Triangle a_SE, b_SE, z_S
+  a_SE: -0.0018856502946889009°
+  b_SE: 0.0011890295778727022°
+  z_S: 0.0006966207168233041°
+
+Triangle a_SW, b_SW, z_S
+  a_SW: 0.0017363089225312933°
+  b_SW: -0.001956518417863151°
+  z_S: 0.00022020949535317413°
+
+Triangle b_NE, c_E, y_NE
+  b_NE: -5.4156720231901545e-05°
+  c_E: -0.00040358386964811643°
+  y_NE: 0.00045774058987291255°
+
+Triangle b_NE, c_E, z_E
+  b_NE: -0.0009184660407299816°
+  c_E: 0.0008739657117544652°
+  z_E: 4.450032897551637e-05°
+
+Triangle b_NE, c_N, y_NE
+  b_NE: 0.00017470943667774463°
+  c_N: -0.0001747266285079263°
+  y_NE: 1.7191837287100498e-08°
+
+Triangle b_NW, c_N, y_NW
+  b_NW: 0.003928388803437599°
+  c_N: -0.0039285291447015425°
+  y_NW: 1.4034125683792809e-07°
+
+Triangle b_NW, c_W, y_NW
+  b_NW: 0.0039283900732698385°
+  c_W: -0.0039285297795643714°
+  y_NW: 1.3970630163839814e-07°
+
+Triangle b_SE, c_E, y_SE
+  b_SE: -0.000897519700536975°
+  c_E: 0.0002210891647536073°
+  y_SE: 0.0006764305357691569°
+
+Triangle b_SE, c_E, z_E
+  b_SE: -0.0001661945662831954°
+  c_E: -0.0009587837744717831°
+  z_E: 0.0011249783407549785°
+
+Triangle b_SE, c_S, y_SE
+  b_SE: -0.0008975178741081891°
+  c_S: 0.00022108969701406522°
+  y_SE: 0.0006764281771083347°
+
+Triangle b_SE, c_S, z_S
+  b_SE: -0.00016620401637368332°
+  c_S: -0.0009587725587678619°
+  z_S: 0.0011249765751415453°
+
+Triangle b_SW, c_S, y_SW
+  b_SW: -5.415090214455631e-05°
+  c_S: -0.00040357591168316276°
+  y_SW: 0.00045772681384903535°
+
+Triangle b_SW, c_S, z_S
+  b_SW: -0.0009184701495854597°
+  c_S: 0.00087398271318051°
+  z_S: 4.448743641916053e-05°
+
+Triangle b_SW, c_W, y_SW
+  b_SW: 0.00017470839824795803°
+  c_W: -0.0001747249218624347°
+  y_SW: 1.6523628687536984e-08°
+
+Triangle c_E, d_NE, e_E
+  c_E: -0.0007556600615572506°
+  d_NE: 0.0002641663727089849°
+  e_E: 0.0004914936888269494°
+
+Triangle c_E, d_NE, y_NE
+  c_E: -0.0003919462083246117°
+  d_NE: -4.2518017124848484e-05°
+  y_NE: 0.00043446422546367103°
+
+Triangle c_E, d_SE, e_E
+  c_E: 0.00022487539229842923°
+  d_SE: 0.000264262914654978°
+  e_E: -0.0004891383069605126°
+
+Triangle c_E, d_SE, y_SE
+  c_E: 0.0006690477311224186°
+  d_SE: -0.0004495970440387964°
+  y_SE: -0.000219450687097833°
+
+Triangle c_N, d_NE, e_N
+  c_N: 0.00043802608161769285°
+  d_NE: 0.0006061756293931353°
+  e_N: -0.0010442017109895119°
+
+Triangle c_N, d_NE, y_NE
+  c_N: 7.77608707309696e-05°
+  d_NE: 0.0004272013700656885°
+  y_NE: -0.0005049622408037635°
+
+Triangle c_N, d_NW, e_N
+  c_N: -0.0018371050143599632°
+  d_NW: 0.0006689659636691658°
+  e_N: 0.0011681390506907974°
+
+Triangle c_N, d_NW, y_NW
+  c_N: -0.004392411525167006°
+  d_NW: 0.0034642502022776966°
+  y_NW: 0.0009281613228964147°
+
+Triangle c_S, d_SE, e_S
+  c_S: 0.00022490750998827025°
+  d_SE: 0.00026425064935864384°
+  e_S: -0.0004891581593469141°
+
+Triangle c_S, d_SE, y_SE
+  c_S: 0.0006690562241331577°
+  d_SE: -0.000449587257172368°
+  y_SE: -0.00021946896695368423°
+
+Triangle c_S, d_SW, e_S
+  c_S: -0.0007556856092989506°
+  d_SW: 0.0002641521543509384°
+  e_S: 0.0004915334549480121°
+
+Triangle c_S, d_SW, y_SW
+  c_S: -0.0003919572288921813°
+  d_SW: -4.252978876451152e-05°
+  y_SW: 0.00043448701768511455°
+
+Triangle c_W, d_NW, e_W
+  c_W: -0.0018371356879498535°
+  d_NW: 0.0006689415152791867°
+  e_W: 0.0011681941726706668°
+
+Triangle c_W, d_NW, y_NW
+  c_W: -0.00439242709246912°
+  d_NW: 0.00346423653491712°
+  y_NW: 0.000928190557552°
+
+Triangle c_W, d_SW, e_W
+  c_W: 0.0004380758375859273°
+  d_SW: 0.0006061529123826404°
+  e_W: -0.0010442287499685676°
+
+Triangle c_W, d_SW, y_SW
+  c_W: 7.777468017877709e-05°
+  d_SW: 0.00042721382367005845°
+  y_SW: -0.0005049885038488355°
+
+Triangle d_NE, e_E, f_NE
+  d_NE: 0.00038569630375207°
+  e_E: -4.1012391825745453e-05°
+  f_NE: -0.0003446839119192191°
+
+Triangle d_NE, e_N, f_NE
+  d_NE: 3.4118590889420375e-05°
+  e_N: -0.0005253562237612641°
+  f_NE: 0.0004912376328576329°
+
+Triangle d_NW, e_N, f_NW
+  d_NW: 0.0012793501149417352°
+  e_N: -0.0003005889518448157°
+  f_NW: -0.0009787611630969195°
+
+Triangle d_NW, e_W, f_NW
+  d_NW: 0.0012793645837376744°
+  e_W: -0.00030057569104258164°
+  f_NW: -0.0009787888926879873°
+
+Triangle d_SE, e_E, f_SE
+  d_SE: -7.914704212907964e-06°
+  e_E: -0.00026442283958516555°
+  f_SE: 0.0002723375437980735°
+
+Triangle d_SE, e_S, f_SE
+  d_SE: -7.92374907376825e-06°
+  e_S: -0.00026443091589101186°
+  f_SE: 0.0002723546649647801°
+
+Triangle d_SW, e_S, f_SW
+  d_SW: 0.0003857085871175059°
+  e_S: -4.10046320098445e-05°
+  f_SW: -0.00034470395510055596°
+
+Triangle d_SW, e_W, f_SW
+  d_SW: 3.4104280807412124e-05°
+  e_W: -0.0005253692789679576°
+  f_SW: 0.0004912649981676509°
+
+Triangle e_E, f_NE, g_ENE
+  e_E: -0.00015088143299379908°
+  f_NE: 0.0001553185337925811°
+  g_ENE: -4.437100798782012e-06°
+
+Triangle e_E, f_SE, g_ESE
+  e_E: -5.289010589137888e-05°
+  f_SE: -5.705811280876105e-05°
+  g_ESE: 0.00010994821870013993°
+
+Triangle e_E, g_ENE, g_ESE
+  e_E: -7.744546709176348e-05°
+  g_ENE: -0.000139748678492424°
+  g_ESE: 0.0002171941456126092°
+
+Triangle e_N, f_NE, g_NNE
+  e_N: -0.00011644664380128233°
+  f_NE: -0.0002467109250190447°
+  g_NNE: 0.000363157568820327°
+
+Triangle e_N, f_NW, g_NNW
+  e_N: -0.0005663172445693476°
+  f_NW: 0.00012971776202874707°
+  g_NNW: 0.000436599482547706°
+
+Triangle e_N, g_NNE, g_NNW
+  e_N: 1.3987573481699656e-06°
+  g_NNE: 0.0003394089509995979°
+  g_NNW: -0.00034080770834776786°
+
+Triangle e_S, f_SE, g_SSE
+  e_S: -5.28844724030364e-05°
+  f_SE: -5.706849415076931e-05°
+  g_SSE: 0.00010995296653959485°
+
+Triangle e_S, f_SW, g_SSW
+  e_S: -0.00015088330234647174°
+  f_SW: 0.00015531752390529618°
+  g_SSW: -4.434221558824447e-06°
+
+Triangle e_S, g_SSE, g_SSW
+  e_S: -7.743940860649445e-05°
+  g_SSE: 0.00021719812617959633°
+  g_SSW: -0.00013975871755889102°
+
+Triangle e_W, f_NW, g_WNW
+  e_W: -0.0005663163395084325°
+  f_NW: 0.00012971358477642525°
+  g_WNW: 0.00043660275473200727°
+
+Triangle e_W, f_SW, g_WSW
+  e_W: -0.00011643403040295652°
+  f_SW: -0.000246728634124338°
+  g_WSW: 0.0003631626645272945°
+
+Triangle e_W, g_WNW, g_WSW
+  e_W: 1.4097619143171869e-06°
+  g_WNW: -0.00034082347526265266°
+  g_WSW: 0.0003394137133483355°
+
+Triangle f_NE, g_ENE, g_NNE
+  f_NE: 7.450149718835064e-05°
+  g_ENE: -0.00013744180584041032°
+  g_NNE: 6.294030865205968e-05°
+
+Triangle f_NW, g_NNW, g_WNW
+  f_NW: 0.00023868980004237983°
+  g_NNW: -0.00011934547999459255°
+  g_WNW: -0.00011934432004778728°
+
+Triangle f_SE, g_ESE, g_SSE
+  f_SE: 3.87570213717936e-05°
+  g_ESE: -1.9378552060800303e-05°
+  g_SSE: -1.937846930388787e-05°
+
+Triangle f_SW, g_SSW, g_WSW
+  f_SW: 7.450129267994043e-05°
+  g_SSW: -0.00013743845931912801°
+  g_WSW: 6.293716665339844e-05°
+
+Total angle distortion: 0.11233054610809035°
+Highest angle distortion: 0.0039283900732698385°
+Lowest angle distortion: -0.00439242709246912°

impossolid/twice-augmented_edge-distortions

@@ -0,0 +1,98 @@
+a_SE, b_SE: 0.000015249448583054653
+b_SW, c_S: 0.000006918255000202575
+b_SE, c_S: 0.00001022525369434997
+b_SE, c_E: 0.000010225282824746383
+b_NE, c_E: 0.000006918450075555058
+z_S, a_SW: -0.0000219346808875234
+z_S, a_SE: 0.00002021118621352795
+z_E, a_SE: 0.00002021122888534378
+z_E, a_NE: -0.00002193465480671757
+z_S, b_SW: 0.000015276737515369292
+z_S, b_SE: -0.000010771908316892743
+z_E, b_SE: -0.00001077200999765046
+z_E, b_NE: 0.000015276631362360217
+z_S, c_S: -0.0000027853351723499016
+z_E, c_E: -0.000002785228607191555
+c_N, d_NE: -0.000007632775579087326
+c_N, d_NW: 0.00001402754980516112
+c_W, d_NW: 0.000014028001282264439
+c_W, d_SW: -0.00000763316944431703
+c_S, d_SW: 0.000006566974972180328
+c_S, d_SE: -0.0000033161321458747414
+c_E, d_SE: -0.00000331586236646513
+c_E, d_NE: 0.000006566624318564243
+y_NE, b_NE: -0.0000017608392620486498
+y_NW, b_NW: -0.00003959022952266363
+y_SW, b_SW: -0.0000017608153313274316
+y_SE, b_SE: 0.000005637007821097203
+y_NE, c_N: 0.000001760310254250749
+y_NW, c_N: 0.00003958128591112948
+y_NW, c_W: 0.000039581305103782685
+y_SW, c_W: 0.0000017603065237110672
+y_SW, c_S: 0.0000017602552603493003
+y_SE, c_S: -0.00000563497752606038
+y_SE, c_E: -0.0000056349905677193694
+y_NE, c_E: 0.0000017602528935919282
+y_NE, d_NE: -0.0000017608481082635805
+y_NW, d_NW: -0.0000395891283971962
+y_SW, d_SW: -0.0000017608381747596222
+y_SE, d_SE: 0.0000056371661583272735
+d_NE, e_N: 0.000007303314580670295
+d_NW, e_N: -0.000016255656816635924
+d_NW, e_W: -0.000016256069869023512
+d_SW, e_W: 0.00000730369455466398
+d_SW, e_S: -0.000006000901856373712
+d_SE, e_S: 0.000003879307405619109
+d_SE, e_E: 0.000003879053499059208
+d_NE, e_E: -0.00000600059436381427
+c_N, e_N: 0.00000899767908293228
+c_W, e_W: 0.000008997328779446816
+c_S, e_S: 0.0000042757526759396545
+c_E, e_E: 0.000004275945999382972
+e_N, f_NE: 0.0000026971057195376348
+e_N, f_NW: 0.000006498321003978716
+e_W, f_NW: 0.000006498333167484994
+e_W, f_SW: 0.000002697065747495785
+e_S, f_SW: 0.000001359196116957704
+e_S, f_SE: 0.0000010550443264554641
+e_E, f_SE: 0.0000010550540852081317
+e_E, f_NE: 0.000001359177866673992
+d_NE, f_NE: -0.0000029405886467824407
+d_NW, f_NW: -0.000009421855439304757
+d_SW, f_SW: -0.000002940615976753832
+d_SE, f_SE: -0.000001529705558996715
+f_NE, g_ENE: -0.00000011649787286240574
+f_NE, g_NNE: -0.000002135681696645076
+f_NW, g_NNW: -0.000003607774033318148
+f_NW, g_WNW: -0.000003607785721714447
+f_SW, g_WSW: -0.0000021356459140809334
+f_SW, g_SSW: -0.00000011652747271646458
+f_SE, g_SSE: -0.0000005858121892925972
+f_SE, g_ESE: -0.0000005858113552594831
+e_N, g_NNE: -0.0000034483169191011346
+e_N, g_NNW: 0.0000034059972640161084
+e_W, g_WNW: 0.000003405934362501001
+e_W, g_WSW: -0.00000344858668736309
+e_S, g_SSW: 0.0000029689592297656908
+e_S, g_SSE: -0.0000006279732347365118
+e_E, g_ESE: -0.0000006278110244023648
+e_E, g_ENE: 0.0000029689801689897473
+a_NE, a_NW
+a_NW, a_SW
+a_SW, a_SE
+a_SE, a_NE
+a_NE, b_NE
+a_NW, b_NW
+a_SW, b_SW
+b_NE, c_N
+b_NW, c_N
+b_NW, c_W
+b_SW, c_W
+g_NNE, g_NNW
+g_NNW, g_WNW
+g_WNW, g_WSW
+g_WSW, g_SSW
+g_SSW, g_SSE
+g_SSE, g_ESE
+g_ESE, g_ENE
+g_ENE, g_NNE

impossolid/twice-augmented_vertices

@@ -0,0 +1,38 @@
+a_NE: 0.4974660839869507, 0.5025284618118182, -0.21962917587407324
+a_NW: -0.5025160085784002, 0.5025084403663106, -0.21364467368968995
+a_SW: -0.5025317897634418, -0.4974736643225916, -0.21962716242129268
+a_SE: 0.49745030267915796, -0.4974536427543331, -0.22561166444858374
+z_S: 0.0023385661484333744, -0.22870402619169428, 0.60063190420276
+z_E: 0.22870024445875012, -0.0023413854336990325, 0.6006314485234188
+b_NE: 0.8118652156106724, 0.8061651522670251, 0.6797917189944392
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