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calculate surface angles using unit normal #2702

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calculate surface angles using unit normal
cwhanse Feb 28, 2026
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cwhanse Feb 28, 2026
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cwhanse Feb 28, 2026
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94 changes: 70 additions & 24 deletions pvlib/tracking.py
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Original file line number Diff line number Diff line change
Expand Up @@ -43,8 +43,10 @@ def singleaxis(apparent_zenith, solar_azimuth,

axis_tilt : float, default 0
The tilt of the axis of rotation (i.e, the y-axis defined by
``axis_azimuth``) with respect to horizontal.
``axis_tilt`` must be >= 0 and <= 90. [degrees]
``axis_azimuth``) with respect to horizontal (degrees). Positive
``axis_tilt`` is *downward* in the direction of ``axis_azimuth``. For
example, for a tracker with ``axis_azimuth``=180 and ``axis_tilt``=10,
the north end is higher than the south end of the axis.

axis_azimuth : float, default 0
A value denoting the compass direction along which the axis of
Expand All @@ -62,9 +64,7 @@ def singleaxis(apparent_zenith, solar_azimuth,
y-axis of the tracker coordinate system. For example, for a tracker
with ``axis_azimuth`` oriented to the south, a rotation to
``max_angle`` is towards the west, and a rotation toward ``-max_angle``
is in the opposite direction, toward the east. Hence, a ``max_angle``
of 180 degrees (equivalent to max_angle = (-180, 180)) allows the
tracker to achieve its full rotation capability.
is in the opposite direction, toward the east.

backtrack : bool, default True
Controls whether the tracker has the capability to "backtrack"
Expand All @@ -84,7 +84,7 @@ def singleaxis(apparent_zenith, solar_azimuth,
intersection between the slope containing the tracker axes and a plane
perpendicular to the tracker axes. The cross-axis tilt should be
specified using a right-handed convention. For example, trackers with
axis azimuth of 180 degrees (heading south) will have a negative
``axis_azimuth`` of 180 degrees (heading south) will have a negative
cross-axis tilt if the tracker axes plane slopes down to the east and
positive cross-axis tilt if the tracker axes plane slopes down to the
west. Use :func:`~pvlib.tracking.calc_cross_axis_tilt` to calculate
Expand Down Expand Up @@ -210,6 +210,50 @@ def singleaxis(apparent_zenith, solar_azimuth,
return out


def _unit_normal(axis_azimuth, axis_tilt, theta):
"""
Unit normal to rotated tracker surface, in global E-N-Up coordinates,
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@adriesse adriesse Apr 1, 2026

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I don't know whether there is a precedent somewhere, but I have personally been using a left-handed N-E-Up system in my toolbox to match the usual azimuth angle convention. Just for info; not recommending a change.

given by R*(0, 0, 1)^T, where:

R = Rz(-axis_azimuth) Rx(-axis_tilt) Ry(theta) *

Rz is a rotation by -axis_azimuth about the z-axis (axis_azimuth
is negated to convert from an azimuth angle to a rotation angle). Rx is a
rotation by -axis_tilt about the x-axis, where axis_tilt is negated
because pvlib's convention is that the positive y-axis is tilted
downwards. Ry is a rotation by theta
about the y-axis. theta is negated so that a negative.

Parameters
----------
axis_azimuth : scalar
axis_tilt : scalar
theta : scalar or array-like

Returns
-------
ndarray
Shape (3,) if theta scalar
Shape (N,3) if theta has length N
"""

theta = np.asarray(theta)

cA, sA = cosd(-axis_azimuth), sind(-axis_azimuth)
cT, sT = cosd(-axis_tilt), sind(-axis_tilt)

cTh = cosd(theta)
sTh = sind(theta)

x = sA * sT * cTh + cA * sTh
y = sA * sTh - cA * sT * cTh
z = cT * cTh

result = np.column_stack((x, y, z))

return result


def calc_surface_orientation(tracker_theta, axis_tilt=0, axis_azimuth=0):
"""
Calculate the surface tilt and azimuth angles for a given tracker rotation.
Expand All @@ -223,8 +267,7 @@ def calc_surface_orientation(tracker_theta, axis_tilt=0, axis_azimuth=0):
results in ``surface_azimuth`` to the West while ``tracker_theta < 0``
results in ``surface_azimuth`` to the East. [degree]
axis_tilt : float, default 0
The tilt of the axis of rotation with respect to horizontal.
``axis_tilt`` must be >= 0 and <= 90. [degree]
The tilt of the axis of rotation with respect to horizontal. [degree]
axis_azimuth : float, default 0
A value denoting the compass direction along which the axis of
rotation lies. Measured east of north. [degree]
Expand All @@ -234,7 +277,9 @@ def calc_surface_orientation(tracker_theta, axis_tilt=0, axis_azimuth=0):
dict or DataFrame
Contains keys ``'surface_tilt'`` and ``'surface_azimuth'`` representing
the module orientation accounting for tracker rotation and axis
orientation. [degree]
orientation (degree).
Where ``surface_tilt``=0, ``surface_azimuth`` is set equal to
``axis_azimuth``.

References
----------
Expand All @@ -245,16 +290,18 @@ def calc_surface_orientation(tracker_theta, axis_tilt=0, axis_azimuth=0):
with np.errstate(invalid='ignore', divide='ignore'):
surface_tilt = acosd(cosd(tracker_theta) * cosd(axis_tilt))

# clip(..., -1, +1) to prevent arcsin(1 + epsilon) issues:
azimuth_delta = asind(np.clip(sind(tracker_theta) / sind(surface_tilt),
a_min=-1, a_max=1))
# Combine Eqs 2, 3, and 4:
azimuth_delta = np.where(abs(tracker_theta) < 90,
azimuth_delta,
-azimuth_delta + np.sign(tracker_theta) * 180)
# handle surface_tilt=0 case:
azimuth_delta = np.where(sind(surface_tilt) != 0, azimuth_delta, 90)
surface_azimuth = (axis_azimuth + azimuth_delta) % 360
# unit normal to rotated tracker surface
unit_normal = _unit_normal(axis_azimuth, axis_tilt, tracker_theta)

# deviate from [1] to allow for negative tilt.
# project unit_normal to x-y plane to calculate azimuth
surface_azimuth = np.degrees(
np.arctan2(unit_normal[:, 0], unit_normal[:, 1]))
Comment on lines +298 to +299
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@adriesse adriesse Apr 1, 2026

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I would include this line in the function that calculates the unit normal so that it is self-contained w.r.t. its conventions.

# constrain angles to [0, 360)
surface_azimuth = np.mod(surface_azimuth, 360.0)

surface_azimuth = np.where(surface_tilt == 0., axis_azimuth,
surface_azimuth)

out = {
'surface_tilt': surface_tilt,
Expand Down Expand Up @@ -378,15 +425,14 @@ def calc_cross_axis_tilt(
----------
slope_azimuth : float
direction of the normal to the slope containing the tracker axes, when
projected on the horizontal [degrees]
projected on the horizontal. [degrees]
slope_tilt : float
angle of the slope containing the tracker axes, relative to horizontal
angle of the slope containing the tracker axes, relative to horizontal.
[degrees]
axis_azimuth : float
direction of tracker axes projected on the horizontal [degrees]
direction of tracker axes projected on the horizontal. [degrees]
axis_tilt : float
tilt of trackers relative to horizontal. ``axis_tilt`` must be >= 0
and <= 90. [degree]
tilt of trackers relative to horizontal. [degree]

Returns
-------
Expand Down
86 changes: 76 additions & 10 deletions tests/test_tracking.py
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Original file line number Diff line number Diff line change
Expand Up @@ -23,7 +23,7 @@ def test_solar_noon():
gcr=2.0/7.0)

expect = pd.DataFrame({'tracker_theta': 0, 'aoi': 10,
'surface_azimuth': 90, 'surface_tilt': 0},
'surface_azimuth': 0, 'surface_tilt': 0},
index=index, dtype=np.float64)
expect = expect[SINGLEAXIS_COL_ORDER]

Expand All @@ -38,7 +38,7 @@ def test_scalars():
max_angle=90, backtrack=True,
gcr=2.0/7.0)
assert isinstance(tracker_data, dict)
expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 90,
expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 0,
'surface_tilt': 0}
for k, v in expect.items():
assert np.isclose(tracker_data[k], v)
Expand All @@ -52,7 +52,7 @@ def test_arrays():
max_angle=90, backtrack=True,
gcr=2.0/7.0)
assert isinstance(tracker_data, dict)
expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 90,
expect = {'tracker_theta': 0, 'aoi': 10, 'surface_azimuth': 0,
'surface_tilt': 0}
for k, v in expect.items():
assert_allclose(tracker_data[k], v, atol=1e-7)
Expand All @@ -68,7 +68,7 @@ def test_nans():
gcr=2.0/7.0)
expect = {'tracker_theta': np.array([0, nan, nan]),
'aoi': np.array([10, nan, nan]),
'surface_azimuth': np.array([90, nan, nan]),
'surface_azimuth': np.array([0, nan, nan]),
'surface_tilt': np.array([0, nan, nan])}
for k, v in expect.items():
assert_allclose(tracker_data[k], v, atol=1e-7)
Expand All @@ -82,7 +82,7 @@ def test_nans():
max_angle=90, backtrack=True,
gcr=2.0/7.0)
expect = pd.DataFrame(np.array(
[[ 0., 10., 90., 0.],
[[ 0., 10., 0., 0.],
[nan, nan, nan, nan],
[nan, nan, nan, nan]]),
columns=['tracker_theta', 'aoi', 'surface_azimuth', 'surface_tilt'])
Expand Down Expand Up @@ -195,6 +195,54 @@ def test_backtrack():
assert_frame_equal(expect, tracker_data)


def test__unit_normal():
# with scalar input
unorm = tracking._unit_normal(180., 45., 45.)
np.allclose(unorm, np.array([-np.sqrt(2)/2, -0.5, -0.5]))
# with vector input
az = np.array([0., 90., 180., 270.,
0., 90., 180., 270.,
180., 180., 180, 180.,
180., 180., 180., 180,
0., 90., 180., 270.,
])
tilt = np.array([30., 30., 30., 30.,
0., 0., 0., 0.,
-30., -90., 90., 180.,
0., 0., 0., 0.,
30., 30., 30., 30,
])
theta = np.array([0., 0., 0., 0.,
0., 0., 0., 0.,
0., 0., 0., 0.,
-30., 30., -90., 90.,
30., 30., 30., 30.,
])
expected = np.array(
[[ 0., 0.5, 0.8660254],
[ 0.5, 0., 0.8660254],
[ 0., -0.5, 0.8660254],
[-0.5, -0., 0.8660254],
[ 0., 0., 1.],
[ 0., 0., 1.],
[ 0., -0., 1.],
[-0., 0., 1.],
[-0., 0.5, 0.8660254],
[-0., 1., 0.],
[ 0., -1., 0.],
[ 0., -0., -1.],
[ 0.5, 0., 0.8660254],
[-0.5, -0., 0.8660254],
[ 1., 0., 0.],
[-1., -0., 0.],
[ 0.5, 0.4330127, 0.75],
[ 0.4330127, -0.5, 0.75],
[-0.5, -0.4330127, 0.75],
[-0.4330127, 0.5, 0.75]])
unorms = tracking._unit_normal(az, tilt, theta)
assert np.allclose(unorms, expected)


def test_axis_tilt():
apparent_zenith = pd.Series([30])
apparent_azimuth = pd.Series([135])
Expand Down Expand Up @@ -226,6 +274,7 @@ def test_axis_tilt():


def test_axis_azimuth():
# sun to the east, horizontal east-oriented tracker
apparent_zenith = pd.Series([30])
apparent_azimuth = pd.Series([90])

Expand All @@ -234,13 +283,30 @@ def test_axis_azimuth():
max_angle=90, backtrack=True,
gcr=2.0/7.0)

expect = pd.DataFrame({'aoi': 30, 'surface_azimuth': 180,
expect = pd.DataFrame({'aoi': 30, 'surface_azimuth': 90,
'surface_tilt': 0, 'tracker_theta': 0},
index=[0], dtype=np.float64)
expect = expect[SINGLEAXIS_COL_ORDER]

assert_frame_equal(expect, tracker_data)

# sun to the east, horizontal south-oriented tracker
apparent_zenith = pd.Series([30])
apparent_azimuth = pd.Series([90])

tracker_data = tracking.singleaxis(apparent_zenith, apparent_azimuth,
axis_tilt=0, axis_azimuth=180,
max_angle=90, backtrack=True,
gcr=2.0/7.0)

expect = pd.DataFrame({'aoi': 0, 'surface_azimuth': 90,
'surface_tilt': 30, 'tracker_theta': -30},
index=[0], dtype=np.float64)
expect = expect[SINGLEAXIS_COL_ORDER]

assert_frame_equal(expect, tracker_data)

# sun to the south, horizontal east-oriented tracker
apparent_zenith = pd.Series([30])
apparent_azimuth = pd.Series([180])

Expand Down Expand Up @@ -269,7 +335,7 @@ def test_horizon_flat():
axis_azimuth=180, backtrack=False, max_angle=180)
expected = pd.DataFrame(np.array(
[[ nan, nan, nan, nan],
[ 0., 45., 270., 0.],
[ 0., 45., 180., 0.],
[ nan, nan, nan, nan]]),
columns=['tracker_theta', 'aoi', 'surface_azimuth', 'surface_tilt'])
assert_frame_equal(out, expected)
Expand Down Expand Up @@ -408,7 +474,7 @@ def test_calc_surface_orientation_types():
# numpy arrays
rotations = np.array([-10, 0, 10])
expected_tilts = np.array([10, 0, 10], dtype=float)
expected_azimuths = np.array([270, 90, 90], dtype=float)
expected_azimuths = np.array([270, 0, 90], dtype=float)
out = tracking.calc_surface_orientation(tracker_theta=rotations)
np.testing.assert_allclose(expected_tilts, out['surface_tilt'])
np.testing.assert_allclose(expected_azimuths, out['surface_azimuth'])
Expand Down Expand Up @@ -445,7 +511,7 @@ def test_calc_surface_orientation_special():
# special cases for rotations
rotations = np.array([-180, -90, -0, 0, 90, 180])
expected_tilts = np.array([180, 90, 0, 0, 90, 180], dtype=float)
expected_azimuths = [270, 270, 90, 90, 90, 90]
expected_azimuths = [270, 270, 0, 0, 90, 90]
out = tracking.calc_surface_orientation(rotations)
np.testing.assert_allclose(out['surface_tilt'], expected_tilts)
np.testing.assert_allclose(out['surface_azimuth'], expected_azimuths)
Expand All @@ -461,7 +527,7 @@ def test_calc_surface_orientation_special():
# special cases for axis_azimuth
rotations = np.array([-10, 0, 10])
expected_tilts = np.array([10, 0, 10], dtype=float)
expected_azimuth_offsets = np.array([-90, 90, 90], dtype=float)
expected_azimuth_offsets = np.array([-90, 0, 90], dtype=float)
for axis_azimuth in [0, 90, 180, 270, 360]:
expected_azimuths = (axis_azimuth + expected_azimuth_offsets) % 360
out = tracking.calc_surface_orientation(rotations,
Expand Down