import torch
from typing import Union, Tuple, Optional

class ProjectiveUHG:
    """
    Universal Hyperbolic Geometry implementation based on projective geometry.
    
    This implementation strictly follows UHG principles:
    - Works directly with cross-ratios
    - Uses projective transformations
    - No differential geometry or manifold concepts
    - No curvature parameters
    - Pure projective operations
    
    References:
        - UHG.pdf Chapter 3: Projective Geometry
        - UHG.pdf Chapter 4: Cross-ratios and Invariants
        - UHG.pdf Chapter 5: The Fundamental Operations
    """
    
    def __init__(self):
        """Initialize the UHG projective geometry system."""
        pass
        
    def cross_ratio(self, a: torch.Tensor, b: torch.Tensor, c: torch.Tensor, d: torch.Tensor) -> torch.Tensor:
        """
        Compute the cross-ratio of four points.
        
        The cross-ratio is the fundamental invariant in projective geometry
        and the basis for all UHG calculations.
        
        Args:
            a, b, c, d: Points in projective space
            
        Returns:
            Cross-ratio value
        """
        # Cross-ratio formula from UHG.pdf Chapter 4
        ac = self.proj_dist(a, c)
        bd = self.proj_dist(b, d)
        ad = self.proj_dist(a, d)
        bc = self.proj_dist(b, c)
        return (ac * bd) / (ad * bc)
    
    def proj_dist(self, x: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
        """
        Compute the projective distance between points.
        
        This is NOT a metric distance, but rather a projective invariant
        based on the join and meet operations.
        
        Args:
            x, y: Points in projective space
            
        Returns:
            Projective distance value
        """
        # Join operation (wedge product)
        join = torch.cross(x, y)
        return torch.norm(join, dim=-1)
    
    def transform(self, points: torch.Tensor, matrix: torch.Tensor) -> torch.Tensor:
        """
        Apply a projective transformation to points.
        
        All transformations preserve cross-ratios as required by UHG.
        
        Args:
            points: Points to transform
            matrix: 3x3 projective transformation matrix
            
        Returns:
            Transformed points
        """
        # Apply projective transformation
        transformed = torch.matmul(points, matrix.T)
        # Normalize to preserve projective invariance
        return transformed / transformed[..., -1:]
    
    def join(self, a: torch.Tensor, b: torch.Tensor) -> torch.Tensor:
        """
        Compute the join (line through two points) in projective space.
        
        Args:
            a, b: Points to join
            
        Returns:
            Line in projective coordinates
        """
        return torch.cross(a, b)
    
    def meet(self, l1: torch.Tensor, l2: torch.Tensor) -> torch.Tensor:
        """
        Compute the meet (intersection) of two lines in projective space.
        
        Args:
            l1, l2: Lines to intersect
            
        Returns:
            Point of intersection in projective coordinates
        """
        return torch.cross(l1, l2)
    
    def absolute_polar(self, point: torch.Tensor) -> torch.Tensor:
        """
        Compute the absolute polar of a point.
        
        This is a fundamental operation in UHG that relates
        points to their polar lines with respect to the absolute.
        
        Args:
            point: Point to compute polar for
            
        Returns:
            Polar line in projective coordinates
        """
        # The absolute polar operation from UHG.pdf
        return torch.stack([
            point[..., 0],
            point[..., 1],
            -point[..., 2]
        ], dim=-1)
    
    def reflect(self, points: torch.Tensor, line: torch.Tensor) -> torch.Tensor:
        """
        Reflect points in a line using projective geometry.
        
        This preserves the cross-ratio and all hyperbolic properties.
        
        Args:
            points: Points to reflect
            line: Line to reflect in
            
        Returns:
            Reflected points
        """
        # Construct reflection matrix
        matrix = torch.eye(3, device=points.device)
        matrix -= 2 * torch.ger(line, line) / (line @ line)
        return self.transform(points, matrix) 