Area And Volume

1# Calculating Metrics of a Mesh
2def get_mesh_metrics(mesh):
3    """Calculates the total surface area and enclosed volume."""
4    
5    total_area = 0.0
6    total_volume = 0.0
7    
8    for face in mesh.faces:
9        v0, v1, v2 = get_vertices(mesh, face)
10        
11        # 1. Add triangle area
12        total_area += calculate_triangle_area(v0, v1, v2)
13        
14        # 2. Add signed tetrahedron volume
15        total_volume += calculate_signed_volume(v0, v1, v2)
16        
17    return total_area, abs(total_volume)

1# Calculating Triangle Area via Cross Product
2def calculate_triangle_area(v0, v1, v2):
3    """Returns the area of a 3D triangle."""
4    
5    # 1. Create two vectors originating from v0
6    edge1 = v1 - v0
7    edge2 = v2 - v0
8    
9    # 2. Compute the Cross Product
10    # The magnitude (length) of the cross product vector 
11    # equals the area of the Parallelogram formed by the two edges.
12    cross_prod = edge1.cross(edge2)
13    
14    # 3. The triangle is exactly half of that parallelogram
15    parallelogram_area = cross_prod.length()
16    triangle_area = parallelogram_area * 0.5
17    
18    return triangle_area

1# Calculating Signed Tetrahedron Volume
2def calculate_signed_volume(v0, v1, v2):
3    """Calculates volume from the Origin (0,0,0) to the triangle."""
4    
5    # The Scalar Triple Product: (v0 dot (v1 cross v2))
6    # This mathematically calculates the volume of a parallelepiped.
7    parallelepiped_vol = v0.dot(v1.cross(v2))
8    
9    # A tetrahedron is exactly 1/6th the volume of that parallelepiped!
10    tetrahedron_vol = parallelepiped_vol / 6.0
11    
12    # IMPORTANT: We do NOT use absolute value here.
13    # We want it to be negative if the triangle faces the origin,
14    # and positive if it faces away from the origin.
15    return tetrahedron_vol

1# Signed Volume Summation
2def test_divergence_theorem(mesh):
3    """How the math automatically hollows out the shape."""
4    total_volume = 0.0
5    
6    for face in mesh.faces:
7        a = mesh.vertices[face[0]]
8        b = mesh.vertices[face[1]]
9        c = mesh.vertices[face[2]]
10        
11        # Signed volume of tetrahedron from origin (0,0,0) to triangle (a, b, c).
12        # Front-facing triangles (towards origin) yield negative volume;
13        # Back-facing triangles (away from origin) yield positive volume.
14        signed_tet_vol = (a.dot(b.cross(c))) / 6.0
15        total_volume += signed_tet_vol
16        
17    # The space between the Origin and the Front of the mesh is subtracted,
18    # leaving ONLY the solid volume of the closed mesh itself!
19    return abs(total_volume)

1# Surface Area Aggregation
2def calculate_total_area(mesh):
3    """Sums the absolute area of all triangles."""
4    
5    total_area = 0.0
6    
7    for face in mesh.faces:
8        v0 = mesh.vertices[face[0]]
9        v1 = mesh.vertices[face[1]]
10        v2 = mesh.vertices[face[2]]
11        
12        # Cross product magnitude / 2
13        area = (v1 - v0).cross(v2 - v0).length() * 0.5
14        
15        # Area is always positive
16        total_area += area
17        
18    return total_area

1# Self-Intersection Warnings
2def has_self_intersections(mesh):
3    """Checks if the volume calculation might be corrupted."""
4    
5    # If a mesh intersects itself (e.g. pushed inside out),
6    # the Divergence Theorem breaks down because 'inside' and 'outside'
7    # become mathematically ambiguous.
8    
9    # We must run a Triangle-Triangle intersection test 
10    # across the entire mesh (often accelerated by a Bounding Volume Hierarchy).
11    
12    intersecting_faces = bvh_intersection_test(mesh.faces)
13    
14    if len(intersecting_faces) > 0:
15        print("WARNING: Mesh is self-intersecting! Volume calculation is invalid.")
16        return True
17    return False

1# Full Geometry Report
2def generate_fabrication_report(mesh):
3    """Outputs metrics for 3D printing or manufacturing."""
4    
5    if not is_watertight(mesh):
6        return "Error: Mesh has holes. Volume cannot be calculated."
7        
8    if has_self_intersections(mesh):
9        return "Error: Mesh intersects itself. Volume is corrupted."
10        
11    area, volume = get_mesh_metrics(mesh)
12    
13    print(f"Surface Area: {area:.2f} mm^2")
14    print(f"Enclosed Volume: {volume:.2f} mm^3")
15    
16    # E.g. PLA Plastic weighs ~1.24 grams per cubic centimeter
17    weight_grams = (volume / 1000.0) * 1.24
18    print(f"Estimated Weight (PLA): {weight_grams:.2f} g")