def test_trig_invariance(angle: float, n: int):
"""Test that cos(θ), sin(θ) ≃ cos(θ + n*360°), sin(θ + n*360°)"""
r_cos, r_sin = Vector._trig(angle)
n_cos, n_sin = Vector._trig(angle + 360 * n)
note(f"δcos: {r_cos - n_cos}")
assert isclose(r_cos, n_cos, rel_to=[n / 1e9])
note(f"δsin: {r_sin - n_sin}")
assert isclose(r_sin, n_sin, rel_to=[n / 1e9])
def test_trig_stability(angle):
"""cos² + sin² == 1
We are testing that this equation holds, as otherwise rotations
would (slightly) change the length of vectors they are applied to.
"""
r_cos, r_sin = Vector._trig(angle)
# Don't use exponents here. Multiplication is generally more stable.
assert math.isclose(r_cos * r_cos + r_sin * r_sin, 1, rel_tol=1e-18)
def test_remarkable_angles(angle, trig):
"""Test that our table of remarkable angles agrees with Vector._trig.
This is useful both as a consistency test of the table,
and as a test of Vector._trig (which Vector.rotate uses).
"""
cos_t, sin_t = trig
cos_m, sin_m = Vector._trig(angle)
assert isclose(sin_t, sin_m, abs_tol=0, rel_tol=1e-14)
assert isclose(cos_t, cos_m, abs_tol=0, rel_tol=1e-14)
def test_dot_rotational_invariance(x: Vector, y: Vector, angle: float):
"""Test that rotating vectors doesn't change their dot product."""
t = x.angle(y)
cos_t, _ = Vector._trig(t)
note(f"θ: {t}")
note(f"cos θ: {cos_t}")
# Exclude near-orthogonal test inputs
assume(abs(cos_t) > 1e-6)
assert isclose(x * y,
x.rotate(angle) * y.rotate(angle),
rel_to=(x, y),
rel_exp=2)
def test_dot_from_angle(x: Vector, y: Vector):
"""Test x · y == |x| · |y| · cos(θ)"""
t = x.angle(y)
cos_t, _ = Vector._trig(t)
# Dismiss near-othogonal test inputs
assume(abs(cos_t) > 1e-6)
min_len, max_len = sorted((x.length, y.length))
geometric = min_len * (max_len * cos_t)
note(f"θ: {t}")
note(f"cos θ: {cos_t}")
note(f"algebraic: {x * y}")
note(f"geometric: {geometric}")
assert isclose(x * y, geometric, rel_to=(x, y), rel_exp=2)
def test_trig_stability(angle):
"""cos² + sin² == 1
We are testing that this equation holds, as otherwise rotations
would (slightly) change the length of vectors they are applied to.
Moreover, Vector._trig should get closer to fulfilling it than
math.{cos,sin}.
"""
r_cos, r_sin = Vector._trig(angle)
r_len = r_cos * r_cos + r_sin * r_sin
# Don't use exponents here. Multiplication is generally more stable.
assert math.isclose(r_len, 1, rel_tol=1e-18)
t_cos, t_sin = cos(radians(angle)), sin(radians(angle))
t_len = t_cos * t_cos + t_sin * t_sin
assert fabs(1 - r_len) <= fabs(1 - t_len)
