Figure 1
Flowchart of throwshed general method.
Figure 2
Graphical description of throwshed general method. The procedures shown are the construction of the trajectory in detail (A), the generation of the trajectory set (B), the selection of the target chunk (C), and the iterative search for the target cell’s intersecting trajectory (D).
Figure 3
Three scenarios of the intersection: target cell intersected and accessible (A), target cell intersected but inaccessible (B), and target cell out of reach (C). Note that the first and last trajectories of the set are also depicted.
Figure 4
Theoretical (green) and computed (red) envelopes of trajectories with omitted (grey) and accessible (yellow) areas depicted on the trajectory set (black).
Table 1
Input parameters for throwshed tool.
| MANDATORY | |
|---|---|
| Initial height h (m) | Height above the terrain from which the projectile starts its trajectory |
| Elevation θ (°) | Also known as shooting angle; it is set as a range in interval <–90, +90> |
| Gravitational acceleration g (m/s–2) | Causes projectile’s attraction vertically down, usually around 9.80 |
| Initial velocity v0 (m/s) | Or muzzle velocity; the speed at which the projectile leaves the weapon |
| Temperature T (°C) | Important to recalculate the temperature at different relative heights |
| Air density ρ (kg/m3) | One of the factors causing air resistance, a standard value of 1.225 |
| Mass m (kg) | Mass of the projectile |
| Area A (m2) (d (m)) | The area of the projectile perpendicular cross-section; can also be set as the diameter of the rotationally symmetric body |
| Drag coefficient CD | Nondimensional aerodynamic drag coefficient unique for projectile |
| Azimuth α (°) | Set as a range in interval <0, 360> |
| OPTIONAL | |
| Shooter’s eyes height hE (m) | Applicable for viewshed analysis |
| Target height hT (m) | |
| Wall height hW (m) | Applicable for burning artificial wall obstacles into DEM, negative wall height value is considered as a depth of artificial ditch |
| Wall width wW (m) | |
| Oscillation frequency f (Hz) | Frequency of oscillating/rotating projectile (e.g. wobbling arrow) |
| Oscillation distance dO (m) | The distance at which the oscillating effect ceases to be present |
| Peak drag coefficient | Drag coefficient of oscillating projectile at its max |
| Peak area AP (m2) | Area of oscillating projectile at its max |
Table 2
Values of input parameters of sling and rifle projectiles for throwshed demonstration.
| PARAMETER | SLING PROJECTILE | RIFLE PROJECTILE |
|---|---|---|
| Initial height (m) | 1.700 | 1.700 |
| Elevation (°) | <–90, +90> | <–90, +90> |
| Gravitational acceleration (m/s–2) | 9.810 | 9.810 |
| Initial velocity (m/s) | 70.000 | 808.000 |
| Temperature (°C) | 15.0 | 15.0 |
| Air density (kg/m3) | 1.058 | 1.058 |
| Mass (kg) | 0.2500 | 0.0109 |
| Diameter (m) | 0.0500 | 0.0078 |
| Drag coefficient | 0.47 | Drag-to-Mach data in Table 3 |
| Azimuth (°) | <0, 360> | <0, 360> |
Table 3
Drag-to-Mach coefficient values of 0.308 Sierra 168 gr bullet.
| Mach | 0.000 | 0.800 | 0.850 | 0.900 | 0.950 | 1.000 | 1.050 | 1.100 |
| CD | 0.140 | 0.140 | 0.142 | 0.160 | 0.240 | 0.430 | 0.449 | 0.447 |
| Mach | 1.200 | 1.400 | 1.600 | 1.800 | 2.000 | 2.200 | 2.500 | |
| CD | 0.434 | 0.410 | 0.385 | 0.365 | 0.350 | 0.339 | 0.320 |
Figure 5
DTM with shooting points, location for sling throwshed (A), and location for rifle throwshed (B).
Figure 6
Simple 1-point throwshed for sling (A) and rifle (B). Each smaller circle represents the shooting range of a weapon with a shooting angle of 0° in terrain shaped like a horizontal plane. The larger circles represent each weapon’s range with the shooting angle set to 45°.
Figure 7
Simple 3-point binary throwshed for sling (A) and rifle (B).
Figure 8
Simple 3-point cumulative throwshed for sling (A) and rifle (B).
Figure 9
Simple 1-point throwshed for sling (A) and rifle (B) clipped by viewshed.
Table 4
Standard deviations of input parameters and DTM for probable throwshed.
| STANDARD DEVIATION OF PARAMETER | SLING PROJECTILE | RIFLE PROJECTILE |
|---|---|---|
| Initial height (m) | 0.100 | 0.100 |
| Gravitational acceleration (m/s–2) | 0.010 | 0.010 |
| Initial velocity (m/s) | 2.500 | 5.000 |
| Temperature (°C) | 5.0 | 5.0 |
| Air density (kg/m3) | 0.050 | 0.050 |
| Mass (kg) | 0.0100 | 0.0005 |
| Diameter (m) | 0.0050 | 0.0001 |
| Drag coefficient | 0.05 | 0.00 |
| DTM elevation (m) | 0.160 | 0.160 |
Figure 10
Simple 1-point probable throwshed for sling (A) and rifle (B).
Figure 11
Classic and direct 1-point throwshed for sling (A) and rifle (B).
Figure 12
Classic and indirect 1-point throwshed for sling (A) and rifle (B).
Figure 13
Classic and direct 3-point cumulative throwshed for sling (A) and rifle (B).
Figure 14
Classic and direct 1-point throwshed using walls for sling (A) and rifle (B) with adjusted azimuth interval.
Figure 15
Classic and direct 1-point throwshed for sling (A) and rifle (B) with adjusted shooting angle range.
Figure 16
Classic and indirect 1-point throwshed for sling (A) and rifle (B) with adjusted shooting angle range.
Table 5
Deviations of experimental ranges and limiting parameters of the cannons.
| CANNON TYPE | CALIBRE (lb) | DISTANCE DEVIATION OF RANGE (%) | MIN–MAX ELEVATION OF THE REAR RING (zoll) | MIN–MAX ELEVATION ANGLE (°) |
|---|---|---|---|---|
| Field cannons | 6 | 11 | 0–10 | 0.6000–11.1000 |
| 12 | 8 | 0–18 | 0.6000–15.4167 | |
| 18 | 6 | 0–20 | 0.6000–15.0000 | |
| Battery cannons | 12 | 7 | 0–30 | 0.5667–16.2333 |
| 18 | 7 | 0–32 | 0.5833–15.8167 | |
| 24 | 8 | 0–34 | 0.6000–15.9500 | |
| Defence cannons | 6 | 9 | 0–20 | 0.4833–13.2333 |
| 12 | 10 | 0–22 | 0.4500–12.0667 | |
| 18 | 7 | 0–22 | 0.5000–11.5500 |
Table 6
Technical parameters of the cannons and cannonballs from the Handbook (Smola 1839).
| CANNON TYPE | CALIBRE (lb) | BORE DIAMETER (m) | BORE DEPTH (m) | CANNONBALL DIAMETER (m) | CANNONBALL MASS (kg) | GUNPOWDER CHARGE (kg) | INITIAL VELOCITY (m/s) |
|---|---|---|---|---|---|---|---|
| Field cannons | 6 | 0.0946 | 1.3568 | 0.0905 | 2.7475 | 0.8400 | 499.68 |
| 12 | 0.1183 | 1.7096 | 0.1139 | 5.4995 | 1.4000 | 471.81 | |
| 18 | 0.1348 | 1.9732 | 0.1305 | 8.2558 | 2.5200 | 497.61 | |
| Battery cannons | 12 | 0.1183 | 2.7424 | 0.1139 | 5.4995 | 2.2400 | 597.10 |
| 18 | 0.1348 | 3.0087 | 0.1305 | 8.2558 | 3.0801 | 575.42 | |
| 24 | 0.1480 | 3.1681 | 0.1436 | 11.0208 | 3.9201 | 561.56 | |
| Defence cannons | 6 | 0.0946 | 2.2572 | 0.0905 | 2.7475 | 1.1200 | 603.35 |
| 12 | 0.1183 | 2.7068 | 0.1139 | 5.4995 | 2.2400 | 595.56 | |
| 18 | 0.1348 | 2.8376 | 0.1305 | 8.2558 | 3.0801 | 568.84 |
Table 7
Simulated and experimental shooting ranges of the cannons according to elevation angles.
| PARAMETERS P: ELEVATION ANGLE EA (°), EXPERIMENTAL RANGE ER (m), SIMULATED RANGE SR (m) | ELEVATION OF REAR RING (ZOLL) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| CANNON TYPE | CAL. (lb) | P | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Field cannons | 6 | EA | 0.6000 | 1.7000 | 2.7833 | ||||||
| ER | 379 | 683 | 910 | ||||||||
| SR | 428 | 763 | 1012 | ||||||||
| 12 | EA | 0.6000 | 1.4667 | 2.3167 | 3.2000 | 4.0500 | |||||
| ER | 455 | 721 | 910 | 1138 | 1290 | ||||||
| SR | 423 | 715 | 944 | 1144 | 1312 | ||||||
| 18 | EA | 0.6000 | 1.3667 | 2.1167 | 2.8000 | 3.5333 | |||||
| ER | 455 | 759 | 986 | 1176 | 1366 | ||||||
| SR | 465 | 759 | 989 | 1167 | 1334 | ||||||
| Battery cannons | 12 | EA | 0.5667 | 1.1000 | 1.6500 | 2.2000 | 2.7333 | 3.2833 | 3.8333 | ||
| ER | 379 | 759 | 910 | 1100 | 1252 | 1396 | 1517 | ||||
| SR | 547 | 782 | 974 | 1135 | 1275 | 1272 | 1514 | ||||
| 18 | EA | 0.5833 | 1.0667 | 1.5667 | 2.0667 | 2.5667 | 3.0667 | 3.5500 | 4.0500 | ||
| ER | 379 | 683 | 936 | 1037 | 1214 | 1366 | 1487 | 1631 | |||
| SR | 551 | 774 | 964 | 1124 | 1266 | 1393 | 1504 | 1611 | |||
| 24 | EA | 0.6000 | 1.0667 | 1.5500 | 2.0167 | 2.4833 | 2.9667 | 3.4333 | 3.9000 | 4.3667 | |
| ER | 531 | 759 | 910 | 1062 | 1214 | 1366 | 1467 | 1669 | 1770 | ||
| SR | 556 | 778 | 969 | 1127 | 1268 | 1399 | 1514 | 1621 | 1721 | ||
| Defence cannons | 6 | EA | 0.4833 | 1.1500 | 1.8000 | 2.4667 | 3.1167 | ||||
| ER | 379 | 645 | 860 | 1062 | 1214 | ||||||
| SR | 478 | 738 | 931 | 1096 | 1235 | ||||||
| 12 | EA | 0.4500 | 1.0000 | 1.5333 | 2.0833 | 2.6167 | 3.1667 | 3.7167 | |||
| ER | 379 | 645 | 835 | 986 | 1163 | 1305 | 1517 | ||||
| SR | 486 | 740 | 934 | 1101 | 1241 | 1370 | 1488 | ||||
| 18 | EA | 0.5000 | 1.0333 | 1.5500 | 2.0667 | 2.5833 | 3.1000 | 3.6167 | 4.1333 | ||
| ER | 379 | 683 | 910 | 1100 | 1252 | 1404 | 1593 | 1644 | |||
| SR | 500 | 750 | 946 | 1112 | 1257 | 1387 | 1506 | 1614 | |||
Figure 17
1794 map of the Fortress of Luxembourg (Lefort 1905) with four forts of interest (A). DTM with the ditch and forts (B).
Table 8
Coordinates of shooting places in LUREF and NGL data and calibres used within forts (Waersegers).
| FORT | E (m) | N (m) | H (m) | CALIBRE (lb) |
|---|---|---|---|---|
| Rumigny | 78376.00 | 75163.00 | 307.64 | 6 |
| Rubamprez | 78186.00 | 75108.00 | 297.37 | 6, 12, 18 |
| Wallis | 77536.00 | 74403.00 | 305.00 | 6, 12, 18 |
| Neipperg | 78040.00 | 74418.00 | 312.12 | 6, 12 |
Table 9
Technical parameters of the cannons and recovered cannonballs. Mean technical parameters of the cannonballs were provided by Waersegers and Theis (unpublished work).
| CANNON TYPE | CALIBRE (lb) | BORE DIAMETER (m) | BORE DEPTH (m) | CANNONBALL DIAMETER (m) | CANNON-BALL MASS (kg) | GUNPOWDER CHARGE (kg) | INITIAL VELOCITY (m/s) | MIN–MAX ELEVATION ANGLE (°) |
|---|---|---|---|---|---|---|---|---|
| Field cannons | 6 | 0.0946 | 1.3568 | 0.0900 | 2.6426 | 0.8400 | 509.50 | 0.6000–11.1000 |
| 12 | 0.1192 | 1.7096 | 0.1125 | 5.3056 | 1.4000 | 481.78 | 0.6000–15.4167 | |
| 18 | 0.1364 | 1.9732 | 0.1305 | 8.1608 | 2.5200 | 503.00 | 0.6000–15.0000 | |
| Battery cannons | 12 | 0.1192 | 2.7424 | 0.1125 | 5.3056 | 2.2400 | 609.72 | 0.5667–16.2333 |
| 18 | 0.1348 | 3.0087 | 0.1305 | 8.1608 | 3.0801 | 581.41 | 0.5833–15.8167 | |
| Defence cannons | 6 | 0.0946 | 2.2572 | 0.0900 | 2.6426 | 1.1200 | 615.21 | 0.4833–13.2333 |
| 12 | 0.1192 | 2.7068 | 0.1125 | 5.3056 | 2.2400 | 608.16 | 0.4500–12.0667 | |
| 18 | 0.1364 | 2.8376 | 0.1305 | 8.1608 | 3.0801 | 574.82 | 0.5000–11.5500 |
Figure 18
Cumulative throwshed for the 6-pounder field (A) and defence cannons (B).
Figure 19
Cumulative throwshed for 12-pounder (A) and 18-pounder field cannons (B).
Figure 20
Cumulative throwshed for 12-pounder (A) and 18-pounder battery cannons (B).
Figure 21
Cumulative throwshed for 12-pounder (A) and 18-pounder defence cannons (B).