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add physics based roller coaster

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Marius Unsel 2026-03-17 16:23:08 +01:00
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# %%
"""
Physics-Based Roller Coaster Rail Generator
Creates smooth roller coaster rails by:
1. Detecting curvature discontinuities at segment junctions
2. Replacing them with G2-continuous quintic polynomial transitions
3. Computing physics-based binormals (hanging mass direction)
4. Generating parallel rails offset from centerline
The result is a track where a marble rolling along it will experience
smooth, continuous forces without abrupt banking changes.
"""
import numpy as np
from build123d import * # type: ignore
from ocp_vscode import * # type: ignore
# Physical constants
GRAVITY = 9.81 # m/s²
# ==============================================================================
# UTILITY FUNCTIONS
# ==============================================================================
def normalize(v):
"""
Normalize a vector to unit length.
Args:
v: Vector (numpy array or array-like)
Returns:
Normalized vector, or zero vector if input has negligible magnitude
"""
magnitude = np.linalg.norm(v)
if magnitude < 1e-12:
return np.zeros_like(v) if hasattr(v, "__len__") else 0
return v / magnitude
def sample_wire(wire, steps):
"""
Sample points along a build123d Wire at uniform parameter intervals.
Args:
wire: build123d Wire object
steps: Number of points to sample
Returns:
numpy array of shape (steps, 3) containing [x, y, z] coordinates
"""
pts = []
for i in range(steps):
u = i / (steps - 1)
p = wire.position_at(u)
pts.append([p.X, p.Y, p.Z])
return np.array(pts)
def arc_length(points):
"""
Compute cumulative arc length along a path.
Args:
points: Array of shape (n, 3) representing path points
Returns:
Array of shape (n,) with cumulative arc length at each point
"""
n = len(points)
s = np.zeros(n)
for i in range(1, n):
ds = np.linalg.norm(points[i] - points[i - 1])
s[i] = s[i - 1] + ds
return s
# ==============================================================================
# DIFFERENTIAL GEOMETRY
# ==============================================================================
def compute_geometry(points, s):
"""
Compute differential geometry properties along a path.
Uses central finite differences for numerical derivatives.
Args:
points: Array of shape (n, 3) representing path points
s: Array of shape (n,) with arc length at each point
Returns:
T: Tangent vectors (n, 3) - normalized first derivative
k: Curvature magnitudes (n,) - ||d²r/ds²||
k_vector: Curvature vectors (n, 3) - d²r/ds²
dk: Curvature derivative (n,) - dk/ds
"""
n = len(points)
# Tangent: dr/ds (first derivative w.r.t. arc length)
T = np.zeros_like(points)
for i in range(1, n - 1):
ds = s[i + 1] - s[i - 1]
if ds > 1e-12:
T[i] = (points[i + 1] - points[i - 1]) / ds
T[0], T[-1] = T[1], T[-2]
T = np.array([normalize(t) for t in T])
# Curvature vector: d²r/ds² (second derivative w.r.t. arc length)
k_vector = np.zeros_like(points)
for i in range(1, n - 1):
ds1 = s[i] - s[i - 1]
ds2 = s[i + 1] - s[i]
if ds1 > 1e-12 and ds2 > 1e-12:
k_vector[i] = (
2
* (
(points[i + 1] - points[i]) / ds2
- (points[i] - points[i - 1]) / ds1
)
/ (ds1 + ds2)
)
k_vector[0], k_vector[-1] = k_vector[1], k_vector[-2]
# Curvature magnitude
k = np.linalg.norm(k_vector, axis=1)
# Curvature derivative (rate of curvature change)
dk = np.gradient(k, s)
return T, k, k_vector, dk
# ==============================================================================
# JUNCTION DETECTION
# ==============================================================================
def detect_junctions(k, dk, threshold_percentile=88.0, min_separation=50):
"""
Detect curvature discontinuities (segment junctions) in a path.
Identifies points where curvature changes abruptly, indicating
a junction between two spline segments with different curvatures.
Args:
k: Curvature magnitudes along path
dk: Curvature derivatives (dk/ds)
threshold_percentile: Percentile for severity threshold (85-95 typical)
min_separation: Minimum point separation to prevent overlapping windows
Returns:
List of junction indices, sorted by position
"""
# Compute severity metric: combines curvature change and jerk
abs_dk = np.abs(dk)
d2k = np.abs(np.gradient(dk)) # Curvature jerk
severity = abs_dk + 2 * d2k
threshold = np.percentile(severity, threshold_percentile)
# Find local maxima above threshold
junctions = []
for i in range(10, len(severity) - 10):
if severity[i] > threshold:
if severity[i] > severity[i - 1] and severity[i] > severity[i + 1]:
junctions.append((i, severity[i]))
# Sort by severity (highest first) and enforce minimum separation
if len(junctions) > 0:
junctions.sort(key=lambda x: x[1], reverse=True)
merged = [junctions[0][0]] # Start with highest severity junction
for j_idx, j_severity in junctions[1:]:
# Only keep if far enough from all existing junctions
if all(abs(j_idx - m) > min_separation for m in merged):
merged.append(j_idx)
junctions = sorted(merged) # Sort by position along path
return junctions
# ==============================================================================
# G2-CONTINUOUS TRANSITIONS
# ==============================================================================
def quintic_hermite_blend(t, p0, v0, a0, p1, v1, a1):
"""
Quintic Hermite polynomial blending function.
Provides G2 continuity (continuous position, tangent, and curvature).
Args:
t: Parameter in [0, 1]
p0, p1: Start and end positions (scalars or vectors)
v0, v1: Start and end velocities (tangent * length)
a0, a1: Start and end accelerations (curvature * length²)
Returns:
Interpolated value at parameter t
"""
# Precompute powers of t
t2 = t * t
t3 = t2 * t
t4 = t3 * t
t5 = t4 * t
# Quintic Hermite basis functions
# These ensure G2 continuity by matching position, first, and second derivatives
h0 = 1 - 10 * t3 + 15 * t4 - 6 * t5
h1 = t - 6 * t3 + 8 * t4 - 3 * t5
h2 = 0.5 * t2 - 1.5 * t3 + 1.5 * t4 - 0.5 * t5
h3 = 10 * t3 - 15 * t4 + 6 * t5
h4 = -4 * t3 + 7 * t4 - 3 * t5
h5 = 0.5 * t3 - t4 + 0.5 * t5
return h0 * p0 + h1 * v0 + h2 * a0 + h3 * p1 + h4 * v1 + h5 * a1
def create_g2_transition(
p_start, T_start, k_vector_start, p_end, T_end, k_vector_end, n_points=100
):
"""
Create a G2-continuous transition curve between two points.
Uses quintic Hermite interpolation to match position, tangent,
and curvature at both endpoints.
Args:
p_start, p_end: Start and end positions (3D points)
T_start, T_end: Start and end tangent vectors (normalized)
k_vector_start, k_vector_end: Start and end curvature vectors
n_points: Number of points in transition curve
Returns:
Array of shape (n_points, 3) forming the transition curve
"""
L = np.linalg.norm(p_end - p_start)
t = np.linspace(0, 1, n_points)
# Scale derivatives appropriately
v0 = T_start * L # Velocity (tangent scaled by length)
v1 = T_end * L
a0 = k_vector_start * L * L # Acceleration (curvature scaled by length²)
a1 = k_vector_end * L * L
# Generate transition points using quintic interpolation
points = np.zeros((n_points, 3))
for i, t_i in enumerate(t):
points[i] = quintic_hermite_blend(t_i, p_start, v0, a0, p_end, v1, a1)
return points
def replace_junctions_with_g2_transitions(points, s, junctions, window_length=0.025):
"""
Replace detected junctions with G2-continuous transition curves.
For each junction, replaces a window of points around it with a smooth
quintic transition that matches position, tangent, and curvature at
both window boundaries.
Args:
points: Original path points (n, 3)
s: Arc length array (n,)
junctions: List of junction indices
window_length: Arc length of transition window (in meters)
Returns:
Modified path points with junctions replaced by smooth transitions
"""
if len(junctions) == 0:
return points
T, k, k_vector, dk = compute_geometry(points, s)
pts_new = points.copy()
# Calculate all window boundaries
windows = []
for junction_idx in junctions:
s_junction = s[junction_idx]
half_window = window_length / 2
start_idx = np.searchsorted(s, s_junction - half_window)
end_idx = np.searchsorted(s, s_junction + half_window)
start_idx = max(5, start_idx)
end_idx = min(len(points) - 5, end_idx)
if end_idx - start_idx >= 10:
windows.append((junction_idx, start_idx, end_idx))
# Check for overlapping windows
print(f"\nReplacing {len(windows)} junctions:")
overlaps_found = False
for i, (j1, s1, e1) in enumerate(windows):
for j2, s2, e2 in windows[i + 1 :]:
if not (e1 < s2 or e2 < s1): # Overlap detected
print(
f" ⚠️ WARNING: Junction {i} [{s1}:{e1}] overlaps with "
f"junction {i + 1} [{s2}:{e2}]"
)
overlaps_found = True
if overlaps_found:
print(
" 💡 Suggestion: increase junction_threshold or decrease junction_window_length"
)
# Replace junctions (skip overlapping ones)
replaced_ranges = []
for junction_idx, start_idx, end_idx in windows:
# Check for overlap with previously replaced ranges
overlap = False
for r_start, r_end in replaced_ranges:
if not (end_idx < r_start or r_end < start_idx):
overlap = True
print(
f" ⏭️ Skipping junction at idx {junction_idx} "
f"(overlaps with previous replacement)"
)
break
if overlap:
continue
# Extract boundary conditions
p_start = pts_new[start_idx]
T_start = T[start_idx]
k_vec_start = k_vector[start_idx]
p_end = pts_new[end_idx]
T_end = T[end_idx]
k_vec_end = k_vector[end_idx]
# Generate G2 transition
n_transition = end_idx - start_idx + 1
transition = create_g2_transition(
p_start,
T_start,
k_vec_start,
p_end,
T_end,
k_vec_end,
n_points=n_transition,
)
# Replace points
pts_new[start_idx : end_idx + 1] = transition
replaced_ranges.append((start_idx, end_idx))
# Print diagnostic info
s_junction = s[junction_idx]
arc_length_window = s[end_idx] - s[start_idx]
print(
f" ✓ Junction at s={s_junction:.3f}m (idx {junction_idx}): "
f"window [{start_idx}:{end_idx}] ({n_transition} pts, "
f"{arc_length_window * 1000:.1f}mm)"
)
return pts_new
# ==============================================================================
# PHYSICS SIMULATION
# ==============================================================================
def compute_velocity_profile(points, mass, v0, friction_coeff, g_scaled):
"""
Compute velocity along the path using energy conservation.
Energy equation:
KE_new = KE_old + PE_lost - friction_work
(1/2)mv² = (1/2)mv₀² + mg*Δh - μ*mg*Δs
Args:
points: Path points (n, 3)
mass: Mass of rolling object (kg)
v0: Initial velocity (m/s)
friction_coeff: Rolling resistance coefficient (dimensionless)
g_scaled: Gravitational acceleration (m/s²)
Returns:
Array of velocities at each point (m/s)
"""
n = len(points)
v = np.zeros(n)
v[0] = v0
for i in range(1, n):
# Distance traveled
d = np.linalg.norm(points[i] - points[i - 1])
# Height change (positive when going downhill)
dh = points[i - 1][2] - points[i][2]
# Energy lost to friction
friction_work = friction_coeff * mass * g_scaled * d
# Apply energy conservation
v2 = v[i - 1] ** 2 + 2 * g_scaled * dh - 2 * friction_work / mass
v[i] = np.sqrt(max(v2, 0))
return v
def compute_hanging_binormals(points, velocities, T, k, k_vector, g_scaled):
"""
Compute binormals representing the direction a hanging mass would point.
Physics model: In the reference frame of the cart, a hanging mass experiences
apparent forces from gravity and centrifugal effects. The binormal points in
the direction the mass hangs.
F_apparent = F_gravity + F_centrifugal
where F_centrifugal points outward (away from curve center)
Args:
points: Path points (n, 3)
velocities: Velocity at each point (n,)
T: Tangent vectors (n, 3)
k: Curvature magnitudes (n,)
k_vector: Curvature vectors (n, 3) - point toward curve center
g_scaled: Gravitational acceleration (m/s²)
Returns:
Binormal vectors (n, 3) - direction of hanging mass
"""
n = len(points)
gravity = np.array([0, 0, -g_scaled])
B = np.zeros((n, 3))
# First point: establish reference orientation
T0 = normalize(T[0])
if k[0] < 1e-9:
# Straight section: mass hangs straight down
apparent_force = gravity
else:
# Curved section: add centrifugal force
R = 1 / k[0]
N_inward = normalize(k_vector[0]) # Points toward curve center
centrifugal_mag = velocities[0] ** 2 / R
F_centrifugal = centrifugal_mag * (-N_inward) # Points outward
apparent_force = gravity + F_centrifugal
# Project onto plane perpendicular to tangent
apparent_force_perp = apparent_force - np.dot(apparent_force, T0) * T0
B[0] = normalize(apparent_force_perp)
# Ensure first binormal points generally downward
if B[0][2] > 0:
B[0] = -B[0]
# Compute remaining binormals with continuity enforcement
for i in range(1, n):
T_i = normalize(T[i])
if k[i] < 1e-9:
apparent_force = gravity
else:
R = 1 / k[i]
N_inward = normalize(k_vector[i])
centrifugal_mag = velocities[i] ** 2 / R
F_centrifugal = centrifugal_mag * (-N_inward)
apparent_force = gravity + F_centrifugal
apparent_force_perp = apparent_force - np.dot(apparent_force, T_i) * T_i
b = normalize(apparent_force_perp)
# Enforce continuity: prevent 180° flips
if np.dot(b, B[i - 1]) < 0:
b = -b
B[i] = b
return B
def smooth_binormals(B, tangents, iterations=5):
"""
Apply light smoothing to binormals to remove high-frequency jitter.
Uses weighted averaging while maintaining orthogonality to tangent.
Args:
B: Binormal vectors (n, 3)
tangents: Tangent vectors (n, 3)
iterations: Number of smoothing passes
Returns:
Smoothed binormal vectors (n, 3)
"""
B_smooth = B.copy()
n = len(B)
for _ in range(iterations):
B_new = B_smooth.copy()
for i in range(2, n - 2):
# Weighted average (favor current value)
avg = (B_smooth[i - 1] + 2 * B_smooth[i] + B_smooth[i + 1]) / 4
# Ensure orthogonality to tangent
T = normalize(tangents[i])
avg_perp = avg - np.dot(avg, T) * T
B_new[i] = normalize(avg_perp)
B_smooth = B_new
return B_smooth
# ==============================================================================
# MAIN PIPELINE
# ==============================================================================
def build_physics_based_roller_coaster_rails(
wire,
steps=5000,
rail_spacing=5.0,
mass=0.005,
initial_velocity=0.01,
friction_coeff=0.05,
junction_window_length=0.030,
junction_threshold=88.0,
binormal_smooth_iterations=5,
downsample_factor=10,
unit_scale=0.001,
):
"""
Generate physics-based roller coaster rails from a build123d Wire.
Creates two parallel rails that maintain proper banking angles based on
the physics of a rolling mass. Ensures smooth transitions at segment
junctions using G2-continuous quintic polynomial blending.
Args:
wire: build123d Wire defining the centerline path
steps: Number of points to sample from wire
rail_spacing: Distance between rails (in build123d units)
mass: Mass of rolling object (kg)
initial_velocity: Starting velocity (m/s)
friction_coeff: Rolling resistance coefficient
junction_window_length: Arc length of G2 transition windows (m)
junction_threshold: Percentile threshold for junction detection (85-95)
binormal_smooth_iterations: Number of binormal smoothing passes
downsample_factor: Factor to downsample output rails (1 = no downsampling)
unit_scale: Conversion from build123d units to meters (0.001 for mm)
Returns:
rail_1, rail_2: Two build123d Splines representing the rails
"""
# Sample wire and convert to meters
pts = sample_wire(wire, steps)
pts_m = pts * unit_scale
s = arc_length(pts_m)
print(f"Original track: {len(pts)} points, {s[-1]:.3f} m length")
# Compute initial geometry
T, k, k_vector, dk = compute_geometry(pts_m, s)
# Detect segment junctions
avg_point_spacing = s[-1] / len(pts_m)
min_separation_points = int((junction_window_length * 1.2) / avg_point_spacing)
junctions = detect_junctions(
k,
dk,
threshold_percentile=junction_threshold,
min_separation=min_separation_points,
)
print(f"Detected {len(junctions)} segment junctions")
# Replace junctions with G2 transitions
if len(junctions) > 0:
pts_m = replace_junctions_with_g2_transitions(
pts_m, s, junctions, window_length=junction_window_length
)
s = arc_length(pts_m)
print(f"After junction replacement: {s[-1]:.3f} m length")
# Recompute geometry after modifications
T, k, k_vector, dk = compute_geometry(pts_m, s)
# Compute velocity profile
velocities = compute_velocity_profile(
pts_m, mass, initial_velocity, friction_coeff, GRAVITY
)
# Compute physics-based binormals
B = compute_hanging_binormals(pts_m, velocities, T, k, k_vector, GRAVITY)
# Optional light smoothing
if binormal_smooth_iterations > 0:
B = smooth_binormals(B, T, iterations=binormal_smooth_iterations)
# Print diagnostics
print_diagnostics(pts_m, velocities, k, B)
# Convert back to build123d units
pts_out = pts_m / unit_scale
# Generate rail positions using cross product
# Rails are perpendicular to both tangent and binormal
rail_1_pts = [
Vector(*(p + rail_spacing * np.cross(b, t))) for p, b, t in zip(pts_out, B, T)
]
rail_2_pts = [
Vector(*(p - rail_spacing * np.cross(b, t))) for p, b, t in zip(pts_out, B, T)
]
# Downsample for final splines
rail_1_pts = [p for i, p in enumerate(rail_1_pts) if i % downsample_factor == 0]
rail_2_pts = [p for i, p in enumerate(rail_2_pts) if i % downsample_factor == 0]
return Spline(*rail_1_pts), Spline(*rail_2_pts)
def print_diagnostics(points, velocities, k, B):
"""Print diagnostic information about the track."""
print("\n=== Ride Diagnostics ===")
print(f"Height range: {np.min(points[:, 2]):.3f} - {np.max(points[:, 2]):.3f} m")
print(f"Velocity: {np.min(velocities):.3f} - {np.max(velocities):.3f} m/s")
# G-forces (filter unrealistic tight curves)
g_forces = []
max_g = 0
max_g_idx = 0
for i in range(len(points)):
if k[i] > 1e-6:
R = 1 / k[i]
if R > 0.002: # 2mm minimum radius
centripetal_accel = velocities[i] ** 2 / R
g_force = centripetal_accel / GRAVITY
g_forces.append(g_force)
if g_force > max_g:
max_g = g_force
max_g_idx = i
if g_forces:
print(f"G-forces: {np.min(g_forces):.2f}g - {np.max(g_forces):.2f}g")
print(
f" Max G at idx {max_g_idx}: "
f"R={1 / k[max_g_idx] * 1000:.2f}mm, "
f"v={velocities[max_g_idx]:.3f}m/s"
)
# Stall check
stall_idx = np.where(velocities < 0.01)[0]
if len(stall_idx) == 0:
print("✓ No stalls")
else:
print(f"⚠️ Stalls at {100 * stall_idx[0] / len(points):.1f}%")
# Binormal quality metrics
binormal_changes = np.array(
[np.linalg.norm(B[i] - B[i - 1]) for i in range(1, len(B))]
)
print(
f"\nBinormal variation: "
f"max={np.max(binormal_changes):.3f}, "
f"avg={np.mean(binormal_changes):.4f}"
)
down_pointing = np.sum(B[:, 2] < 0)
print(f"Downward binormal: {100 * down_pointing / len(B):.1f}%")
# ==============================================================================
# EXAMPLE USAGE
# ==============================================================================
if __name__ == "__main__":
# Define track segments
ln1 = Line((-20, 0), (0, 0))
powerup = Spline(
(39, 0, 71),
(40, 0, 70),
(100, 0, 0),
tangents=((1, 0, 0), (1, 0, 0)),
tangent_scalars=(0.5, 2),
)
corner = RadiusArc(powerup @ 1, (100, 60, 0), -30)
screw = Helix(75, 155, 15, center=(75, 40, 15), direction=(-1, 0, 0))
sp1 = Spline(corner @ 1, screw @ 0, tangents=(corner % 1, screw % 0))
sp2 = Spline(screw @ 1, (-100, 30, 10), ln1 @ 0, tangents=(screw % 1, powerup % 0))
# Assemble wire
track_centerline = Wire([powerup, corner, sp1, screw, sp2, ln1])
# Generate physics-based rails
rail_1, rail_2 = build_physics_based_roller_coaster_rails(
track_centerline,
steps=1000,
rail_spacing=1.0, # 1mm spacing
mass=0.002, # 2 grams
initial_velocity=0.01, # Nearly at rest
friction_coeff=0.05,
junction_window_length=0.08, # 80mm transition windows
junction_threshold=88,
binormal_smooth_iterations=100,
downsample_factor=10,
unit_scale=0.001, # mm to m
)
# Create rail geometry
sk1 = Location(rail_1 ^ 0) * Circle(0.4)
rail_1_solid = sweep(sections=sk1.face(), path=rail_1, binormal=rail_2)
sk2 = Location(rail_2 ^ 0) * Circle(0.4)
rail_2_solid = sweep(sections=sk2.face(), path=rail_2, binormal=rail_1)
show(rail_1_solid, rail_2_solid)