# [setup-1] from itertools import product import bpy import numpy as np import ddg ddg.blender.scene.clear() # [setup-1] ############################################# # Initial data ############################################# # [setup-2] p = np.array([0, 0, 0]) vs_tmp = np.array([[0, 0, 1], [0, -0.6, 1], [-2, 1, 1]]) vs = [v / np.linalg.norm(v) for v in vs_tmp] triangle_distances = np.array([[1, 1, 1], [2.1, 3.2, 2.8], [4, 8, 6]]) projection_plane_normal, projection_plane_level = [-1, 1, 0], 80 center_of_projection = [35, -27, 10] # [setup-2] ############################################# # Initial setup ############################################# # [setup-3] # Lines of perspectivity p_lines = [ ddg.geometry.subspace_from_affine_points_and_directions(p, vs[i]) for i in range(3) ] # [setup-3] ############################################# # Geometric construction ############################################# # [construction-1] # Three triangles in perspective affine = False if affine: # Implementation via affine coordinates triangles = [ [ ddg.geometry.subspace_from_affine_points(p + dists[i] * vs[i]) for i in range(3) ] for dists in triangle_distances ] else: # Implementation via homogeneous coordinates triangles = [ [ddg.geometry.Point([1, dists[i]]).embed(p_lines[i]) for i in range(3)] for dists in triangle_distances ] # [construction-1] # [construction-2] # Edges of triangles as ddg.geometry edges = [ [ddg.geometry.join(triangle[i], triangle[(i + 1) % 3]) for i in range(3)] for triangle in triangles ] # Calculate intersection points edges_intersections = [ [ ddg.geometry.intersect(line1, line2) for line1, line2 in zip(edges[i], edges[(i + 1) % 3]) ] for i in range(3) ] # Three lines formed by intersection points lines = [ddg.geometry.join(*points) for points in edges_intersections] # Calculate the second point of intersection x = ddg.geometry.intersect(*lines) # [construction-2] # [construction-3] # Prepare for visualization: Create lines/ddg.geometry that join # intersection points and edges of triangles def outer_points(*points): """Takes projective points on a line and computes, in affine coordinates, the 'outer' points to join their projective counterparts to a 1d subspace.""" affine_points = np.array([p.affine_point for p in points]) n = (affine_points[1] - affine_points[0]) / np.linalg.norm( affine_points[1] - affine_points[0] ) def signed_distance(pt): return np.dot(n, pt - affine_points[0]) dists = list(map(signed_distance, affine_points)) return ( points[np.argmin(dists)].affine_point, points[np.argmax(dists)].affine_point, ) triangles_line_segments = [ [ outer_points( edges_intersections[i - 1][j], edges_intersections[i][j], triangles[i][j], triangles[i][(j + 1) % 3], ) for j in range(3) ] for i in range(3) ] # [construction-3] ############################################# # Projection ############################################# # [construction-4] # Projection plane projection_plane = ddg.geometry.hyperplane_from_normal( projection_plane_normal, projection_plane_level ) projection_plane = projection_plane.orthonormalize() # [construction-4] ############################################# # Visualization ############################################# # [visualization-1] segment_sampling = 1 line_sampling = 1 point_radius = 0.1 segment_bevel = 0.03 line_bevel = 0.06 projection_plane_kwargs = {"bounding": 45, "sampling": 1} colors = [ ddg.blender.material.material("orange", (0.8, 0.1, 0.036), 0, 0), ddg.blender.material.material("blue", (0.019, 0.052, 0.445), 0, 0), ddg.blender.material.material("lightblue", (0.128, 0.59, 0.8), 0, 0), ] def draw_triangle(triangle, name=None, material=None): hds = ddg.halfedge.disc(3) for v in hds.verts: v.co = triangle[v.index].affine_point bobj = ddg.blender.convert(hds, name, material) return bobj # [visualization-1] # [visualization-2] for i in range(3): segment = ddg.arrays.line_segment_from_points( p, triangles[2][i], [-2 / triangle_distances[2][i], 1 + 2 / triangle_distances[2][i]], ) bobj = ddg.blender.convert(segment, f"p_line_{i}", material=colors[0]) bobj.data.bevel_depth = line_bevel bobj_triangles = [ draw_triangle(triangles[i], name=f"triangle_{i}", material=colors[0]) for i in range(3) ] for i, j in product(range(3), range(3)): bobj = ddg.blender.convert( ddg.arrays.line_segment_from_points(*triangles_line_segments[i][j]), f"segment_{i}_{j}", material=colors[2], ) bobj.data.bevel_depth = segment_bevel for i, j in product(range(3), range(3)): bobj = ddg.blender.vertices( edges_intersections[i][j], f"meet_{i}_{j}", material=colors[2], radius=0.1, ) for i in range(3): curve = ddg.arrays.from_1d_subspace(lines[i], 200) bobj = ddg.blender.convert(curve, f"line_{i}", material=colors[1]) bobj.data.bevel_depth = line_bevel bobj_x = ddg.blender.vertices(x, "x", material=colors[1], radius=0.1) ddg.blender.convert(projection_plane, "projection plane") # [visualization-2] # [visualization-3] bobj_camera = ddg.blender.camera.camera() bpy.context.scene.camera = bobj_camera bobj_light = ddg.blender.light.light( name="Light", type_="SPOT", location=center_of_projection, energy=500000 ) bobj_light.data.shadow_soft_size = 0 ddg.blender.light.look_at_point( bobj_light, ddg.math.projective.dehomogenize(projection_plane.points[0]) ) bobj_camera.location = (18.5, -17, 16.6) bobj_camera.data.clip_end = 200 bobj_camera.rotation_euler = np.array([74.2, 0, 46]) / 180 * np.pi bobj_light.data.spot_size = 0.404916 # [visualization-3] # [visualization-4] # rendering settings bpy.context.scene.render.resolution_x = 1000 bpy.context.scene.render.resolution_y = 1000 ddg.blender.render.setup_cycles_renderer(samples=8) ddg.blender.render.set_film_transparency() ddg.blender.render.set_world_background((0.184469, 0.184469, 0.184469, 1), 1.5) # [visualization-4]