# Copyright (c) 2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
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"""
Primitives for 2D geometries
see https://www.iquilezles.org/www/articles/distfunctions/distfunctions.html
"""
import sys
from operator import mul
from sympy import Symbol, Abs, Max, Min, sqrt, sin, cos, acos, atan2, pi, Heaviside
from functools import reduce
pi = float(pi)
from sympy.vector import CoordSys3D
from .curve import SympyCurve
from .helper import _sympy_sdf_to_sdf
from .geometry import Geometry, csg_curve_naming
from .parameterization import Parameterization, Parameter, Bounds
[docs]class Line(Geometry):
"""
2D Line parallel to y-axis
Parameters
----------
point_1 : tuple with 2 ints or floats
lower bound point of line segment
point_2 : tuple with 2 ints or floats
upper bound point of line segment
normal : int or float
normal direction of line (+1 or -1)
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, point_1, point_2, normal=1, parameterization=Parameterization()):
assert point_1[0] == point_2[0], "Points must have same x-coordinate"
# make sympy symbols to use
l = Symbol(csg_curve_naming(0))
x = Symbol("x")
# curves for each side
curve_parameterization = Parameterization({l: (0, 1)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
dist_y = point_2[1] - point_1[1]
line_1 = SympyCurve(
functions={
"x": point_1[0],
"y": point_1[1] + l * dist_y,
"normal_x": 1e-10 + normal, # TODO rm 1e-10
"normal_y": 0,
},
parameterization=curve_parameterization,
area=dist_y,
)
curves = [line_1]
# calculate SDF
sdf = normal * (point_1[0] - x)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (point_1[0], point_2[0]),
Parameter("y"): (point_1[1], point_2[1]),
},
parameterization=parameterization,
)
# initialize Line
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Channel2D(Geometry):
"""
2D Channel (no bounding curves in x-direction)
Parameters
----------
point_1 : tuple with 2 ints or floats
lower bound point of channel
point_2 : tuple with 2 ints or floats
upper bound point of channel
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, point_1, point_2, parameterization=Parameterization()):
# make sympy symbols to use
l = Symbol(csg_curve_naming(0))
y = Symbol("y")
# curves for each side
curve_parameterization = Parameterization({l: (0, 1)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
dist_x = point_2[0] - point_1[0]
dist_y = point_2[1] - point_1[1]
line_1 = SympyCurve(
functions={
"x": l * dist_x + point_1[0],
"y": point_1[1],
"normal_x": 0,
"normal_y": -1,
},
parameterization=curve_parameterization,
area=dist_x,
)
line_2 = SympyCurve(
functions={
"x": l * dist_x + point_1[0],
"y": point_2[1],
"normal_x": 0,
"normal_y": 1,
},
parameterization=curve_parameterization,
area=dist_x,
)
curves = [line_1, line_2]
# calculate SDF
center_y = point_1[1] + (dist_y) / 2
y_diff = Abs(y - center_y) - (point_2[1] - center_y)
outside_distance = sqrt(Max(y_diff, 0) ** 2)
inside_distance = Min(y_diff, 0)
sdf = -(outside_distance + inside_distance)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (point_1[0], point_2[0]),
Parameter("y"): (point_1[1], point_2[1]),
},
parameterization=parameterization,
)
# initialize Channel2D
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Rectangle(Geometry):
"""
2D Rectangle
Parameters
----------
point_1 : tuple with 2 ints or floats
lower bound point of rectangle
point_2 : tuple with 2 ints or floats
upper bound point of rectangle
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, point_1, point_2, parameterization=Parameterization()):
# make sympy symbols to use
l = Symbol(csg_curve_naming(0))
x, y = Symbol("x"), Symbol("y")
# curves for each side
curve_parameterization = Parameterization({l: (0, 1)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
dist_x = point_2[0] - point_1[0]
dist_y = point_2[1] - point_1[1]
line_1 = SympyCurve(
functions={
"x": l * dist_x + point_1[0],
"y": point_1[1],
"normal_x": 0,
"normal_y": -1,
},
parameterization=curve_parameterization,
area=dist_x,
)
line_2 = SympyCurve(
functions={
"x": point_2[0],
"y": l * dist_y + point_1[1],
"normal_x": 1,
"normal_y": 0,
},
parameterization=curve_parameterization,
area=dist_y,
)
line_3 = SympyCurve(
functions={
"x": l * dist_x + point_1[0],
"y": point_2[1],
"normal_x": 0,
"normal_y": 1,
},
parameterization=curve_parameterization,
area=dist_x,
)
line_4 = SympyCurve(
functions={
"x": point_1[0],
"y": -l * dist_y + point_2[1],
"normal_x": -1,
"normal_y": 0,
},
parameterization=curve_parameterization,
area=dist_y,
)
curves = [line_1, line_2, line_3, line_4]
# calculate SDF
center_x = point_1[0] + (dist_x) / 2
center_y = point_1[1] + (dist_y) / 2
x_diff = Abs(x - center_x) - (point_2[0] - center_x)
y_diff = Abs(y - center_y) - (point_2[1] - center_y)
outside_distance = sqrt(Max(x_diff, 0) ** 2 + Max(y_diff, 0) ** 2)
inside_distance = Min(Max(x_diff, y_diff), 0)
sdf = -(outside_distance + inside_distance)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (point_1[0], point_2[0]),
Parameter("y"): (point_1[1], point_2[1]),
},
parameterization=parameterization,
)
# initialize Rectangle
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Circle(Geometry):
"""
2D Circle
Parameters
----------
center : tuple with 2 ints or floats
center point of circle
radius : int or float
radius of circle
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, center, radius, parameterization=Parameterization()):
# make sympy symbols to use
theta = Symbol(csg_curve_naming(0))
x, y = Symbol("x"), Symbol("y")
# curve for perimeter of the circle
curve_parameterization = Parameterization({theta: (0, 2 * pi)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
curve = SympyCurve(
functions={
"x": center[0] + radius * cos(theta),
"y": center[1] + radius * sin(theta),
"normal_x": 1 * cos(theta),
"normal_y": 1 * sin(theta),
},
parameterization=curve_parameterization,
area=2 * pi * radius,
)
curves = [curve]
# calculate SDF
sdf = radius - sqrt((x - center[0]) ** 2 + (y - center[1]) ** 2)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (center[0] - radius, center[0] + radius),
Parameter("y"): (center[1] - radius, center[1] + radius),
},
parameterization=parameterization,
)
# initialize Circle
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Triangle(Geometry):
"""
2D Isosceles Triangle
Symmetrical axis parallel to y-axis
Parameters
----------
center : tuple with 2 ints or floats
center of base of triangle
base : int or float
base of triangle
height : int or float
height of triangle
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, center, base, height, parameterization=Parameterization()):
# make sympy symbols to use
x, y = Symbol("x"), Symbol("y")
t, h = Symbol(csg_curve_naming(0)), Symbol(csg_curve_naming(1))
N = CoordSys3D("N")
P = x * N.i + y * N.j
O = center[0] * N.i + center[1] * N.j
H = center[0] * N.i + (center[1] + height) * N.j
B = (center[0] + base / 2) * N.i + center[1] * N.j
OP = P - O
OH = H - O
PH = OH - OP
angle = acos(PH.dot(OH) / sqrt(PH.dot(PH)) / sqrt(OH.dot(OH)))
apex_angle = atan2(base / 2, height)
hypo_sin = sqrt(height**2 + (base / 2) ** 2) * sin(apex_angle)
hypo_cos = sqrt(height**2 + (base / 2) ** 2) * cos(apex_angle)
dist = sqrt(PH.dot(PH)) * sin(Min(angle - apex_angle, pi / 2))
# curve for each side
curve_parameterization = Parameterization({t: (-1, 1), h: (0, 1)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
curve_1 = SympyCurve(
functions={
"x": center[0] + t * base / 2,
"y": center[1] + t * 0,
"normal_x": 0,
"normal_y": -1,
},
parameterization=curve_parameterization,
area=base,
)
curve_2 = SympyCurve(
functions={
"x": center[0] + h * hypo_sin,
"y": center[1] + height - h * hypo_cos,
"normal_x": 1 * cos(apex_angle),
"normal_y": 1 * sin(apex_angle),
},
parameterization=curve_parameterization,
area=sqrt(height**2 + (base / 2) ** 2),
)
curve_3 = SympyCurve(
functions={
"x": center[0] - h * hypo_sin,
"y": center[1] + height - h * hypo_cos,
"normal_x": -1 * cos(apex_angle),
"normal_y": 1 * sin(apex_angle),
},
parameterization=curve_parameterization,
area=sqrt(height**2 + (base / 2) ** 2),
)
curves = [curve_1, curve_2, curve_3]
# calculate SDF
outside_distance = 1 * sqrt(Max(0, dist) ** 2 + Max(0, center[1] - y) ** 2)
inside_distance = -1 * Min(Abs(Min(0, dist)), Abs(Min(0, center[1] - y)))
sdf = -(outside_distance + inside_distance)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (center[0] - base / 2, center[0] + base / 2),
Parameter("y"): (center[1], center[1] + height),
},
parameterization=parameterization,
)
# initialize Triangle
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Ellipse(Geometry):
"""
2D Ellipse
Parameters
----------
center : tuple with 2 ints or floats
center point of circle
radius : int or float
radius of circle
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, center, major, minor, parameterization=Parameterization()):
# make sympy symbols to use
theta = Symbol(csg_curve_naming(0))
x, y = Symbol("x"), Symbol("y")
mag = sqrt((minor * cos(theta)) ** 2 + (major * sin(theta)) ** 2)
area = pi * (
3 * (major + minor) - sqrt((3 * minor + major) * (3 * major + minor))
)
try:
area = float(area)
except:
pass
# curve for perimeter of the circle
curve_parameterization = Parameterization({theta: (0, 2 * pi)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
curve = SympyCurve(
functions={
"x": center[0] + major * cos(theta),
"y": center[1] + minor * sin(theta),
"normal_x": minor * cos(theta) / mag,
"normal_y": major * sin(theta) / mag,
},
parameterization=curve_parameterization,
area=area,
)
curves = [curve]
# calculate SDF
sdf = 1 - (((x - center[0]) / major) ** 2 + ((y - center[1]) / minor) ** 2)
# calculate bounds
bounds = Bounds(
{
Parameter("x"): (center[0] - major, center[0] + major),
Parameter("y"): (center[1] - minor, center[1] + minor),
},
parameterization=parameterization,
)
# initialize Ellipse
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
[docs]class Polygon(Geometry):
"""
2D Polygon
Parameters
----------
points : list of tuple with 2 ints or floats
lower bound point of line segment
parameterization : Parameterization
Parameterization of geometry.
"""
def __init__(self, points, parameterization=Parameterization()):
# make sympy symbols to use
s = Symbol(csg_curve_naming(0))
x = Symbol("x")
y = Symbol("y")
# wrap points
wrapted_points = points + [points[0]]
# curves for each side
curve_parameterization = Parameterization({s: (0, 1)})
curve_parameterization = Parameterization.combine(
curve_parameterization, parameterization
)
curves = []
for v1, v2 in zip(wrapted_points[:-1], wrapted_points[1:]):
# area
dx = v2[0] - v1[0]
dy = v2[1] - v1[1]
area = (dx**2 + dy**2) ** 0.5
# generate normals
normal_x = dy / area
normal_y = -dx / area
line = SympyCurve(
functions={
"x": dx * s + v1[0],
"y": dy * s + v1[1],
"normal_x": dy / area,
"normal_y": -dx / area,
},
parameterization=curve_parameterization,
area=area,
)
curves.append(line)
# calculate SDF
sdfs = [(x - wrapted_points[0][0]) ** 2 + (y - wrapted_points[0][1]) ** 2]
conds = []
for v1, v2 in zip(wrapted_points[:-1], wrapted_points[1:]):
# sdf calculation
dx = v1[0] - v2[0]
dy = v1[1] - v2[1]
px = x - v2[0]
py = y - v2[1]
d_dot_d = dx**2 + dy**2
p_dot_d = px * dx + py * dy
max_min = Max(Min(p_dot_d / d_dot_d, 1.0), 0.0)
vx = px - dx * max_min
vy = py - dy * max_min
sdf = vx**2 + vy**2
sdfs.append(sdf)
# winding calculation
cond_1 = Heaviside(y - v2[1])
cond_2 = Heaviside(v1[1] - y)
cond_3 = Heaviside((dx * py) - (dy * px))
all_cond = cond_1 * cond_2 * cond_3
none_cond = (1.0 - cond_1) * (1.0 - cond_2) * (1.0 - cond_3)
cond = 1.0 - 2.0 * Min(all_cond + none_cond, 1.0)
conds.append(cond)
# set inside outside
sdf = Min(*sdfs)
cond = reduce(mul, conds)
sdf = sqrt(sdf) * -cond
# calculate bounds
min_x = Min(*[p[0] for p in points])
if min_x.is_number:
min_x = float(min_x)
max_x = Max(*[p[0] for p in points])
if max_x.is_number:
max_x = float(max_x)
min_y = Min(*[p[1] for p in points])
if min_y.is_number:
min_y = float(min_y)
max_y = Max(*[p[1] for p in points])
if max_y.is_number:
max_y = float(max_y)
bounds = Bounds(
{
Parameter("x"): (min_x, max_x),
Parameter("y"): (min_y, max_y),
},
parameterization=parameterization,
)
# initialize Polygon
super().__init__(
curves,
_sympy_sdf_to_sdf(sdf),
dims=2,
bounds=bounds,
parameterization=parameterization,
)
