Solar & Acoustics

3 min read•1 page

Solar Projection Invariant:

1. Spherical Coordinate Transform: Altitude θ in [0, 90°] and Azimuth φ in [0, 360°]. 2. Direct Irradiance: Surface solar flux I = I_0 max(0, n · s_sun). 3. Right-to-Light Envelopes: Governs zoning regulations to prevent new skyscrapers from eclipsing public parks.

python

1import numpy as np
2
3def compute_solar_vector(altitude_deg, azimuth_deg):
4    """
5    Computes unit sun vector pointing towards the sun.
6    Altitude: angle above horizon [0, 90].
7    Azimuth: clockwise angle from North [0, 360].
8    """
9    alt = np.radians(altitude_deg)
10    az = np.radians(azimuth_deg)
11    
12    # Coordinates in East-North-Up frame:
13    sx = np.cos(alt) * np.sin(az)  # East
14    sy = np.cos(alt) * np.cos(az)  # North
15    sz = np.sin(alt)               # Up (Zenith)
16    return np.array([sx, sy, sz])
17
18def project_shadow_polygon(roof_polygon, sun_vec, ground_z=0.0):
19    """
20    Projects shadow of 3D roof vertices onto ground plane along solar vector.
21    P_ground = P - (P_z - ground_z) / sun_vec_z * sun_vec
22    """
23    shadow = []
24    for pt in roof_polygon:
25        t = (pt[2] - ground_z) / sun_vec[2]
26        shadow_pt = pt - t * sun_vec
27        shadow.append(shadow_pt)
28    return shadow

Solar Vector Calculations & Shadow Envelopes: In bioclimatic architecture, passive solar heating and daylighting require precise geometric tracking of the sun's path. Given geographic latitude and time of year, solar altitude θ and azimuth φ yield a unit ray vector. Projecting building roof cornices along this vector reveals shadow polygons cast onto neighboring structures and streets.

Solar Altitude (deg)

40.00

Solar Azimuth (deg)

135.00

2 min read•1 page

Acoustic Ray Invariant:

1. Specular Reflection Law: r = d - 2(d · n)n. 2. Ceiling Curvature Control: Convex reflectors diffuse sound energy; concave reflectors create focal echo concentrations. 3. Delay Gap & Intelligibility: First reflection must arrive within 50 ms of the direct sound path for speech clarity.

python

1import numpy as np
2
3def reflect_acoustic_ray(ray_origin, ray_dir, surface_normal):
4    """
5    Computes specular sound ray reflection vector (Snell's Law of Acoustics).
6    Angle of incidence theta_i equals angle of reflection theta_r.
7    r = d - 2 * (d . n) * n
8    """
9    n = surface_normal / np.linalg.norm(surface_normal)
10    d = ray_dir / np.linalg.norm(ray_dir)
11    r = d - 2.0 * np.dot(d, n) * n
12    return r

Architectural Acoustics & Specular Ray Tracing: In symphony halls and auditoriums, sound waves travel through air and reflect off curved ceiling canopies. Under high-frequency geometric acoustics (where wavelength is small relative to panel dimensions), sound behaves like optical rays: the angle of incidence equals the angle of reflection (θ_r = θ_i). Tuning ceiling curvature ensures even sound dispersion to rear balcony seats without acoustic flutter echoes or caustic focal hotspots.

Ceiling Curvature (+Convex / -Concave)

0.40

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