map-projection

Map Projection Skill

Category theory of map projections: functors between manifolds with distortion analysis.

Trigger

• Map projection selection and analysis
• Distortion metrics (Tissot's indicatrix)
• Coordinate system transformations
• Cartographic design decisions

GF(3) Trit: +1 (Generator)

Generates projections from sphere to plane, creating new coordinate representations.

Category Theory of Projections

A map projection is a functor:

P: Sphere → Plane
   S² → ℝ²

Different projections preserve different properties:

• Conformal (angle-preserving): Mercator, Stereographic
• Equal-area: Albers, Lambert, Mollweide
• Equidistant: Azimuthal equidistant
• Compromise: Robinson, Winkel Tripel

Projection Functors

import math

class Projection:
    """Base projection functor."""
    
    def forward(self, lat, lon):
        """S² → ℝ²"""
        raise NotImplementedError
    
    def inverse(self, x, y):
        """ℝ² → S²"""
        raise NotImplementedError
    
    @property
    def distortion_type(self):
        raise NotImplementedError

class Mercator(Projection):
    """Conformal cylindrical projection."""
    
    def forward(self, lat, lon):
        x = math.radians(lon)
        y = math.log(math.tan(math.pi/4 + math.radians(lat)/2))
        return x, y
    
    def inverse(self, x, y):
        lon = math.degrees(x)
        lat = math.degrees(2 * math.atan(math.exp(y)) - math.pi/2)
        return lat, lon
    
    @property
    def distortion_type(self):
        return "conformal"  # Preserves angles

class LambertAzimuthal(Projection):
    """Equal-area azimuthal projection."""
    
    def __init__(self, lat0=0, lon0=0):
        self.lat0 = math.radians(lat0)
        self.lon0 = math.radians(lon0)
    
    def forward(self, lat, lon):
        phi = math.radians(lat)
        lam = math.radians(lon)
        
        k = math.sqrt(2 / (1 + math.sin(self.lat0)*math.sin(phi) + 
                          math.cos(self.lat0)*math.cos(phi)*math.cos(lam - self.lon0)))
        
        x = k * math.cos(phi) * math.sin(lam - self.lon0)
        y = k * (math.cos(self.lat0)*math.sin(phi) - 
                 math.sin(self.lat0)*math.cos(phi)*math.cos(lam - self.lon0))
        return x, y
    
    @property
    def distortion_type(self):
        return "equal-area"  # Preserves area

class Stereographic(Projection):
    """Conformal azimuthal projection."""
    
    def __init__(self, lat0=90, lon0=0):
        self.lat0 = math.radians(lat0)
        self.lon0 = math.radians(lon0)
    
    def forward(self, lat, lon):
        phi = math.radians(lat)
        lam = math.radians(lon)
        
        k = 2 / (1 + math.sin(self.lat0)*math.sin(phi) + 
                 math.cos(self.lat0)*math.cos(phi)*math.cos(lam - self.lon0))
        
        x = k * math.cos(phi) * math.sin(lam - self.lon0)
        y = k * (math.cos(self.lat0)*math.sin(phi) - 
                 math.sin(self.lat0)*math.cos(phi)*math.cos(lam - self.lon0))
        return x, y
    
    @property
    def distortion_type(self):
        return "conformal"