Basic Poincare Disk Renderer

main.py

@@ -3,82 +3,129 @@ from kingdon import Algebra
 import math
 
 ERROR = 0.00001
+DISK_RADIUS = 250
+WIDTH, HEIGHT = 800, 600
+CENTER_X, CENTER_Y = 400, 300
 
 alg = Algebra(2,1,0)
 locals().update(alg.blades)
-b = 2*e12
-print(b)
 
-def motor(generator):
+def point(x,y):
-    bivector = generator.grade(2)
+    #Construct a point.
-    sqr = (bivector*bivector).grade(0).e
+    #Hyperbolic space is modeled as a hyperboloid embedded in 3D Minkowski space.
-    mag = bivector.norm().e
+    #Find w such that x,y,w is a point on the hyperboloid.
-    generator = bivector.normalized()
+    w_sqr = x*x + y*y + 1
-    if mag < ERROR:
+    w = math.sqrt(w_sqr)
-            #Zero Norm
+    return x*e23 -y*e13 + w*e12
-            return 1+bivector
+
-    elif sqr < 0:
+def draw_point(point):
-        return math.cos(mag) + math.sin(mag)*generator
+    #Normalize the point
+    x_0 = point.e12
+    x_1 = -point.e23
+    x_2 = point.e13
+    norm = math.sqrt(x_0*x_0 - x_1*x_1 - x_2*x_2)
+    x_0 = x_0/norm
+    x_1 = x_1/norm
+    x_2 = x_2/norm
+    #Transform it into the Poincare Space
+    u = x_1/(1+x_0)
+    v = x_2/(1+x_0)
+
+    pygame.draw.circle(screen, RED, (int(u*DISK_RADIUS+CENTER_X), int(v*DISK_RADIUS+CENTER_Y)), 5)
+
+def draw_line(line):
+    delta = line.e3
+    if abs(delta) < ERROR:
+        # The geodesic is a line through the origin.
+        if abs(line.e2) < ERROR:
+            pygame.draw.line(screen, BLUE, (CENTER_X, CENTER_Y+DISK_RADIUS), (CENTER_X, CENTER_Y-DISK_RADIUS), 1)
+        else:
+            slope = line.e1/line.e2
+            pygame.draw.line(screen, BLUE, (CENTER_X-DISK_RADIUS, int(CENTER_Y-slope*DISK_RADIUS)), (CENTER_X+DISK_RADIUS, int(CENTER_Y+slope*DISK_RADIUS)), 1)
     else:
-        return math.cosh(mag) + math.sinh(mag)*generator
+        #The geodesic is a circle
-
+        a = line.e1
-motor(1*e12)
+        b = line.e2
-motor(1*e13)
+        sqr = a*a + b*b
-motor(1*e23)
+        norm = math.sqrt(sqr)
+        x_c = a/delta
+        y_c = b/delta
+        r = math.sqrt(x_c*x_c+y_c*y_c - 1)
+        c_x = int(CENTER_X + x_c*DISK_RADIUS)
+        c_y = int(CENTER_Y + y_c*DISK_RADIUS)
+        pygame.draw.circle(screen, BLUE, (c_x, c_y), int(r*DISK_RADIUS), 1)
 
 
 
 
+y_axis = e1 #x = 0 line
+x_axis = e2 #y = 0 line
+
+tx_generator = e13 #A translation in the x direction
+ty_generator = e23 #A translation in the y direction
+origin = e12 #The origin, and a rotation about the origin
 
 
-
-'''
 # Initialize Pygame
 pygame.init()
 
-# Set up the screen
-WIDTH, HEIGHT = 800, 600
 screen = pygame.display.set_mode((WIDTH, HEIGHT))
-pygame.display.set_caption("Clifford and Pygame Example")
+pygame.display.set_caption("Pygame Example")
 
 # Colors
 BLACK = (0, 0, 0)
 WHITE = (255, 255, 255)
 RED = (255, 0, 0)
+BLUE = (0, 0, 255)
 
-# Initial point in GA (e.g., a vector representing a point)
-# We'll represent it as a multivector with a vector component
-initial_point_ga = 100 * e1 + 50 * e2
-
-# Create a rotor for rotation (e.g., 45 degrees counter-clockwise)
-angle = math.pi / 4 # 45 degrees
-rotor = math.e**(angle * e12)
 
 # Game loop
 running = True
+t = 0
+
 while running:
+    t += 0.0002
     for event in pygame.event.get():
         if event.type == pygame.QUIT:
             running = False
 
     # Clear the screen
-    screen.fill(BLACK)
+    screen.fill(WHITE)
 
-    # Apply the rotation to the point
+    #Poincare Disk Boundary
-    rotated_point_ga = rotor * initial_point_ga * ~rotor
+    pygame.draw.circle(screen, BLACK, (CENTER_X, CENTER_Y), DISK_RADIUS, 1)
 
-    # Extract the vector components for Pygame display
+    #Generate a transformation for demonstration
-    # We assume the point is a vector, so we access its e1 and e2 components
+    transform = math.sin(t)*tx_generator + math.sin(math.sqrt(2)*t)*ty_generator + math.sin(math.sqrt(3)*t)*origin
-    x_coord = rotated_point_ga.value[e1.index] + WIDTH // 2  # Adjust for screen center
+    transform = transform
-    y_coord = -rotated_point_ga.value[e2.index] + HEIGHT // 2 # Invert y for Pygame coordinates
+    motor = transform.exp()
 
-    # Draw the point
+    #Apply the transformation
-    pygame.draw.circle(screen, RED, (int(x_coord), int(y_coord)), 5)
+    y_tick = motor*y_axis*motor.reverse()
+    x_tick = motor*x_axis*motor.reverse()
+
+    #Construct a point as the intersection of the two lines
+    point = y_tick^x_tick
+
+    #Draw everything
+    draw_line(y_tick)
+    draw_line(x_tick)
+    draw_point(point)
+
+    tx = tx_generator.exp()
+    ty = ty_generator.exp()
+    point_a = tx*ty*origin*ty.reverse()*tx.reverse()
+    point_b = ty*tx*origin*tx.reverse()*ty.reverse()
+    draw_line(x_axis)
+    draw_line(y_axis)
+    draw_point(origin)
+    draw_point(point_a)
+    draw_point(point_b)
 
     # Update the display
     pygame.display.flip()
 
+
+
 # Quit Pygame
 pygame.quit()
-'''
-