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algebra.result

Alegbraic results for the geometry of sundials and the analemma

dialface_shadowbivector_angle_cos(dialface, shadow_bivector) cached

Form the cosine of the angle \(\Psi\) between the dial face and shadow plane

The cosine of the angle between \(S\) and \(G\) is given by

\[\cos(\Psi) = \frac{S\cdot G}{\sqrt{-S^2}\sqrt{-G^2}}\]

Given that \(S^2 = -\sin^2(\Xi)\) and \(G^2 = -1\), we have

$$ \cos(\Psi) = \frac{S\cdot G}{\sin(\Xi)}$$ as implemented by this function

See Normalization

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Expression

Cosine of angle \(\Psi\) between dial face and shadow plane
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def dialface_shadowbivector_angle_cos(
    dialface: mv.Mv, shadow_bivector: mv.Mv
) -> Expression:
    r"""
    Form the cosine of the angle $\Psi$ between the dial face and shadow plane

    The cosine of the angle between $S$ and $G$ is given by

    $$\cos(\Psi) = \frac{S\cdot G}{\sqrt{-S^2}\sqrt{-G^2}}$$

    Given that $S^2 = -\sin^2(\Xi)$ and $G^2 = -1$, we have

    $$ \cos(\Psi) = \frac{S\cdot G}{\sin(\Xi)}$$ as implemented by this function

    See [Normalization](../nb/shadow_angle.md#normalization)

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        Cosine of angle $\Psi$ between dial face and shadow plane
    """
    return sp.trigsimp(((dialface | shadow_bivector) / sin(Xi)).obj)

dialface_shadowbivector_intersection(dialface, shadow_bivector) cached

Form a vector along the intersection of the shadow plane and dial face

Given the dialface \(G\) and shadow_bivector \(S\), this is given by

\[u = \langle I G S \rangle_1\]

where \(I = e_1 \wedge e_2 \wedge e_3\) is the pseudo-scalar.

\(u\) has length \(\sin(\Xi)\sin(Psi)\) (see Normalization and dialface_shadowbivector_angle_cos).

See Intersecting the Shadow Plane and Dial Face.

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Mv

The vector \(u\) along the intersection between the shadow plane and dial face
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def dialface_shadowbivector_intersection(
    dialface: mv.Mv, shadow_bivector: mv.Mv
) -> mv.Mv:
    r"""
    Form a vector along the intersection of the shadow plane and dial face

    Given the `dialface` $G$ and `shadow_bivector` $S$, this is given by

    $$u = \langle I G S \rangle_1$$

    where $I = e_1 \wedge e_2 \wedge e_3$ is the pseudo-scalar.

    $u$ has length $\sin(\Xi)\sin(Psi)$
    (see [Normalization](../nb/shadow_angle.md#normalization) and
    [dialface_shadowbivector_angle_cos][analemma.algebra.result.dialface_shadowbivector_angle_cos]).

    See [Intersecting the Shadow Plane and Dial Face](../nb/shadow_angle.md#intersecting-the-shadow-plane-and-dial-face).

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        The vector $u$ along the intersection between the shadow plane and dial face
    """
    n1, n2, n3 = frame.base("n")
    return ((n1 ^ n2 ^ n3) * dialface * shadow_bivector).get_grade(1).trigsimp()

gnomon_dialface_angle_sin(dial_incl_symbol=i, dial_decl_symbol=d, gnomon_incl_symbol=iota)

The sine of the angle between the gnomon and the dial face

In Rohr's 1996 book SUNDIALS History, Theory and Practice, he refers to the substyle, which is the projection of the style onto the dial face. Here, we allow the gnomon to have an independent inclination \(\iota\) which matches a style when \(\iota = \theta\), the (\(90^\circ\) minus) latitude angle.

We can readily project the gnomon \(g\) onto the dial face \(G\). The vector \(\bar{b} = g\cdot G\) is parallel to \(G\) (ie lies in the dial face) and is rotated with the orientation of \(G\) (anticlockwise as seen from above the dial) by \(\pi/2\) relative to the subgnomon \(b = R \bar{b} \tilde{R}\) where \(R = \exp(\frac{\pi}{4} \, G)\) undoes that \(90^\circ\) rotation (notice the angle in \(R\) is positive in contrast to typical rotations).

Let \(A\) denote the angle between the (unit length) gnomon \(g\) and the subgnomon \(b\) (or equivalently the dial face \(G\)). Then, \(\cos(A) = \sqrt{b^2}\), and \(1-\cos^2(A)\) factorizes nicely to give \(\sin(A)\) as returned by this function.

See also Comparison with Rohr's Book and analemma.tests.test_results.test_gnomon_dialface_angle_pythagoras_identity.

Parameters:

Name Type Description Default
dial_incl_symbol Symbol

The inclination angle of the dial face relative to the surface frame

 i
dial_decl_symbol Symbol

The declination angle of the dial face relative to the surface frame

 d
gnomon_incl_symbol Symbol

The inclination angle of the gnomon relative to the surface frame

 iota

Returns:

Type Description
Expression

\(\sin(A)\), the angle between gnomon and subgnomon (or, equivalently, the dial face)
Source code in src/analemma/algebra/result.py 
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def gnomon_dialface_angle_sin(
    dial_incl_symbol: sp.Symbol = i,
    dial_decl_symbol: sp.Symbol = d,
    gnomon_incl_symbol: sp.Symbol = iota,
) -> Expression:
    r"""
    The sine of the angle between the gnomon and the dial face

    In Rohr's 1996 book *SUNDIALS History, Theory and Practice*, he refers to the substyle, which is the projection of
    the style onto the dial face. Here, we allow the gnomon to have an independent inclination $\iota$ which matches a
    style when $\iota = \theta$, the ($90^\circ$ minus) latitude angle.

    We can readily project the gnomon $g$ onto the dial face $G$. The vector $\bar{b} = g\cdot G$ is parallel to $G$ (ie
    lies in the dial face) and is rotated with the orientation of $G$ (anticlockwise as seen from above the dial) by
    $\pi/2$ relative to the subgnomon $b = R \bar{b} \tilde{R}$ where $R = \exp(\frac{\pi}{4} \, G)$ undoes that $90^\circ$
    rotation (notice the angle in $R$ is positive in contrast to typical [rotations][analemma.algebra.util.rotate]).

    Let $A$ denote the angle between the (unit length) gnomon $g$ and the subgnomon $b$ (or
    equivalently the dial face $G$). Then, $\cos(A) = \sqrt{b^2}$, and $1-\cos^2(A)$ factorizes nicely to give $\sin(A)$
    as returned by this function.

    See also [Comparison with Rohr's Book](../nb/rohr_comparison.md) and
    [analemma.tests.test_results.test_gnomon_dialface_angle_pythagoras_identity][].

    Parameters:
        dial_incl_symbol: The inclination angle of the dial face relative to the surface frame
        dial_decl_symbol: The declination angle of the dial face relative to the surface frame
        gnomon_incl_symbol: The inclination angle of the gnomon relative to the surface frame

    Returns:
        $\sin(A)$, the angle between gnomon and subgnomon (or, equivalently, the dial face)
    """
    i = dial_incl_symbol
    d = dial_decl_symbol
    iota = gnomon_incl_symbol
    return cos(i) * cos(iota) + sin(i) * sin(iota) * cos(d)

gnomon_shadow_angle_sincos(dial_incl_symbol=i, dial_decl_symbol=d, gnomon_incl_symbol=iota, hour_angle_symbol=mu, dialface_shadowbivector_angle_symbol=Psi) cached

The sine and cosine of the angle \(\beta\) between the gnomon and the shadow

\(\cos(\beta)\) is readily available via \(g \cdot \hat{w}\) and \(\sin(\beta)\) is related to \(g \wedge \hat{w}\). This function provides explicit forms for these results.

See Angle between Gnomon and Shadow and shadow_triangle_solution

Parameters:

Name Type Description Default
dial_incl_symbol Symbol

The inclination angle of the dial face relative to the surface frame

 i
dial_decl_symbol Symbol

The declination angle of the dial face relative to the surface frame

 d
gnomon_incl_symbol Symbol

The inclination angle of the gnomon relative to the surface frame

 iota
hour_angle_symbol Symbol

The hour angle of the sun

 mu
dialface_shadowbivector_angle_symbol Symbol

The angle between the dial face and shadow bivector (see dialface_shadowbivector_angle_cos)

 Psi

Returns:

Type Description
Tuple[Expression]

The 2-tuple (\(\sin(\beta)\),\(\cos(\beta)\)), where \(\beta\) is the gnomon-shadow angle
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def gnomon_shadow_angle_sincos(
    dial_incl_symbol: sp.Symbol = i,
    dial_decl_symbol: sp.Symbol = d,
    gnomon_incl_symbol: sp.Symbol = iota,
    hour_angle_symbol: sp.Symbol = mu,
    dialface_shadowbivector_angle_symbol: sp.Symbol = Psi,
) -> Tuple[Expression]:
    r"""
    The sine and cosine of the angle $\beta$ between the gnomon and the shadow

    $\cos(\beta)$ is readily available via $g \cdot \hat{w}$ and $\sin(\beta)$ is related to $g \wedge \hat{w}$.
    This function provides explicit forms for these results.

    See [Angle between Gnomon and Shadow](../nb/shadow_length.md#angle-between-gnomon-and-shadow) and
    [shadow_triangle_solution][analemma.algebra.result.shadow_triangle_solution]

    Parameters:
        dial_incl_symbol: The inclination angle of the dial face relative to the surface frame
        dial_decl_symbol: The declination angle of the dial face relative to the surface frame
        gnomon_incl_symbol: The inclination angle of the gnomon relative to the surface frame
        hour_angle_symbol: The [hour angle][analemma.algebra.result.hour_angle_tan] of the sun
        dialface_shadowbivector_angle_symbol: The angle between the dial face and shadow bivector (see [dialface_shadowbivector_angle_cos][analemma.algebra.result.dialface_shadowbivector_angle_cos])

    Returns:
        The 2-tuple `(`$\sin(\beta)$`, `$\cos(\beta)$`)`, where $\beta$ is the gnomon-shadow angle
    """
    i = dial_incl_symbol
    d = dial_decl_symbol
    iota = gnomon_incl_symbol
    mu = hour_angle_symbol
    Psi = dialface_shadowbivector_angle_symbol

    sin_beta = (sin(i) * sin(iota) * cos(d) + cos(i) * cos(iota)) / sin(Psi)
    cos_beta = (
        (sin(i) * cos(d) * cos(iota) - sin(iota) * cos(i)) * cos(mu)
        - sin(d) * sin(i) * sin(mu)
    ) / sin(Psi)
    return sin_beta, cos_beta

hour_angle_sincos(shadow_plane, meridian_plane, zero_decl=True)

Explicit expressions for the sine and cosine of the hour angle \(\mu\)

The hour angle is measured between the meridian plane and the shadow plane (containing sun ray and gnomon). It is given by

\[\cos(\mu) = \frac{-S\cdot M}{\sqrt{-S^2}\sqrt{-M^2}} = \frac{-S\cdot M}{\sin(\Xi)}\]

given that \(M^2 = -1\). \(\sin(\mu) = \sqrt{1-\cos^2(\mu)}\) can be factorized when the gnomon declination \(\delta = 0\).

See The Hour Angle

Parameters:

Name Type Description Default
shadow_plane Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required
meridian_plane Mv

The bivector \(M = n_1\wedge n_3\) (see analemma.algebra.frame.meridian_plane)

 required
zero_decl bool

Flag controlling assumption of zero gnomon declination. If False, None is returned for \(\sin(\mu)\)

 True

Returns:

Type Description
Tuple[Expression]

A 2-tuple of explicit expressions for (\(\sin(\mu)\),\(\cos(\mu)\))
Source code in src/analemma/algebra/result.py 
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def hour_angle_sincos(
    shadow_plane: mv.Mv, meridian_plane: mv.Mv, zero_decl: bool = True
) -> Tuple[Expression]:
    r"""
    Explicit expressions for the sine and cosine of the hour angle $\mu$

    The hour angle is measured between the meridian plane and the [shadow plane][analemma.algebra.result.shadow_bivector]
    (containing sun ray and gnomon). It is given by

    $$\cos(\mu) = \frac{-S\cdot M}{\sqrt{-S^2}\sqrt{-M^2}} = \frac{-S\cdot M}{\sin(\Xi)}$$

    given that $M^2 = -1$. $\sin(\mu) = \sqrt{1-\cos^2(\mu)}$ can be factorized when the gnomon declination $\delta =
    0$.

    See [The Hour Angle](../nb/hour_angle.md#the-hour-angle_1)

    Parameters:
        shadow_plane: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$
        meridian_plane: The bivector $M = n_1\wedge n_3$ (see [analemma.algebra.frame.meridian_plane][])
        zero_decl: Flag controlling assumption of zero gnomon declination. If False, None is returned for $\sin(\mu)$

    Returns:
        A 2-tuple of explicit expressions for `(`$\sin(\mu)$`, `$\cos(\mu)$`)`
    """
    sinXi_sin_mu_delta = (
        cos(alpha) * sin(psi) * cos(sigma) - cos(psi) * sin(sigma)
        if zero_decl
        else None
    )
    sinXi_cos_mu_delta = sp.trigsimp(-(shadow_plane | meridian_plane).obj)
    return sinXi_sin_mu_delta, sinXi_cos_mu_delta

hour_angle_tan(shadow_plane, meridian_plane, zero_decl=True) cached

Explicit expression for the tangent of the hour angle \(\mu\)

The hour angle is measured between the meridian plane and the shadow plane (containing sun ray and gnomon).

\(\tan(\mu)\) is formed as the ratio of \(\sin(\Xi)\sin(\mu)\) and \(\sin(\Xi)\cos(\mu)\) as returned by hour_angle_sincos in which the shadow plane magnitude factor of \(\sin(\Xi)\) cancels.

See The Hour Angle.

Parameters:

Name Type Description Default
shadow_plane Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required
meridian_plane Mv

The bivector \(M = n_1\wedge n_3\) (see analemma.algebra.frame.meridian_plane)

 required
zero_decl bool

Flag controlling assumption of zero gnomon declination. Currently, supplying False reuslts in an error

 True

Returns:

Type Description
Expression

An explicit expression for \(\tan(\mu)\)
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def hour_angle_tan(
    shadow_plane: mv.Mv, meridian_plane: mv.Mv, zero_decl: bool = True
) -> Expression:
    r"""
    Explicit expression for the tangent of the hour angle $\mu$

    The hour angle is measured between the meridian plane and the [shadow
    plane][analemma.algebra.result.shadow_bivector] (containing sun ray and gnomon).

    $\tan(\mu)$ is formed as the ratio of $\sin(\Xi)\sin(\mu)$ and $\sin(\Xi)\cos(\mu)$ as returned by
    [hour_angle_sincos][analemma.algebra.result.hour_angle_sincos] in which the
    [shadow plane magnitude][analemma.algebra.result.shadow_bivector_magnitude_angle_cos] factor of $\sin(\Xi)$ cancels.

    See [The Hour Angle](../nb/hour_angle.md#the-hour-angle_1).

    Parameters:
        shadow_plane: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$
        meridian_plane: The bivector $M = n_1\wedge n_3$ (see [analemma.algebra.frame.meridian_plane][])
        zero_decl: Flag controlling assumption of zero gnomon declination. Currently, supplying False reuslts in an error

    Returns:
        An explicit expression for $\tan(\mu)$
    """
    if not zero_decl:
        raise Exception(
            "A nice factorization for sin(mu) remains elusive for non-zero gnomon declination"
        )
    sinXi_sin_mu, sinXi_cos_mu = hour_angle_sincos(shadow_plane, meridian_plane)
    mu_ratio = sinXi_sin_mu / sp.collect(sinXi_cos_mu, cos(iota - theta))
    return sp.simplify(mu_ratio.subs(sin(sigma), tan(sigma) * cos(sigma)))

noon_angle_sincos(dialface, shadow_bivector) cached

The sine and cosine of the angle \(\zeta\) between the shadow at an arbitrary time and the shadow at noon

\(\cos(\zeta)\) is calculated as \(\hat{w}(\mu) \cdot \hat{w}(0)\) where \(\hat{w}(\mu)\) is the unit shadow at an arbitrary time, or equivalently hour angle, and \(\hat{w}(0)\) is the unit shadow at noon, when \(\mu = 0\).

Instead of attempting to factorize \(\sqrt{1-\cos^2(\zeta)}\), we can use the fact that the shadow lives in the plane of the dial face, so \(\hat{w}(\mu) \wedge \hat{w}(0) = \sin(\zeta) G\), and \(G^2 = -1\).

See The Shadow Angle Relative to Noon

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Tuple[Mv]

The 2-tuple (\(\sin(\zeta)\),\(\cos(\zeta)\)), where \(\zeta\) is the noon shadow angle
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def noon_angle_sincos(dialface: mv.Mv, shadow_bivector: mv.Mv) -> Tuple[mv.Mv]:
    r"""
    The sine and cosine of the angle $\zeta$ between the shadow at an arbitrary time and the shadow at noon

    $\cos(\zeta)$ is calculated as $\hat{w}(\mu) \cdot \hat{w}(0)$ where $\hat{w}(\mu)$ is the unit shadow at
    an arbitrary time, or equivalently hour angle, and $\hat{w}(0)$ is the unit shadow at noon, when $\mu = 0$.

    Instead of attempting to factorize $\sqrt{1-\cos^2(\zeta)}$, we can use the fact that the shadow lives in
    the plane of the dial face, so $\hat{w}(\mu) \wedge \hat{w}(0) = \sin(\zeta) G$, and $G^2 = -1$.

    See [The Shadow Angle Relative to Noon](../nb/shadow_angle.md#the-shadow-angle-relative-to-noon)

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        The 2-tuple `(`$\sin(\zeta)$`, `$\cos(\zeta)$`)`, where $\zeta$ is the noon shadow angle
    """
    w_hat = unit_shadow(dialface, shadow_bivector)
    noon = unit_noon_shadow(dialface, shadow_bivector)
    # TODO bug here if other names are chosen for angles
    cos_zeta = sp.collect(
        sp.collect(sp.trigsimp((w_hat | noon).obj), sin(mu) * sin(i) * sin(d)), cos(mu)
    )

    sin_zeta = -(w_hat ^ noon) | frame.dialface()
    return sin_zeta, cos_zeta

noon_angle_tan(dialface, shadow_bivector) cached

The tangent of the angle \(\zeta\) between the shadow at an arbitrary time and the shadow at noon

Formed as the ratio of \(\sin(\zeta)\) and \(\cos(\zeta)\) as returned by noon_angle_sincos

See The Shadow Angle Relative to Noon

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Mv

$\tan(\zeta), where \(\zeta\) is the noon shadow angle
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def noon_angle_tan(dialface: mv.Mv, shadow_bivector: mv.Mv) -> mv.Mv:
    r"""
    The tangent of the angle $\zeta$ between the shadow at an arbitrary time and the shadow at noon

    Formed as the ratio of $\sin(\zeta)$ and $\cos(\zeta)$ as returned by
    [noon_angle_sincos][analemma.algebra.result.noon_angle_sincos]

    See [The Shadow Angle Relative to Noon](../nb/shadow_angle.md#the-shadow-angle-relative-to-noon)

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        $\tan(\zeta), where $\zeta$ is the noon shadow angle
    """
    sin_zeta, cos_zeta = noon_angle_sincos(dialface, shadow_bivector)
    return (sin_zeta / cos_zeta).subs(sin(mu), tan(mu) * cos(mu))

shadow_bivector(sunray=frame.sunray(), gnomon=frame.gnomon('e'))

Form the bivector encoding the plane containing the sun ray, the gnomon, and (therefore) the shadow

Given the sunray \(s\) and gnomon \(g\), form \(S = s\wedge g\).

See also Setup and Definitions.

Parameters:

Name Type Description Default
sunray Mv

The vector encoding the sun ray

 sunray()
gnomon Mv

The vector encoding the gnomon

 gnomon('e')

Returns:

Type Description
Mv

The shadow bivector \(S\)
Source code in src/analemma/algebra/result.py 
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def shadow_bivector(
    sunray: mv.Mv = frame.sunray(), gnomon: mv.Mv = frame.gnomon("e")
) -> mv.Mv:
    r"""
    Form the bivector encoding the plane containing the sun ray, the gnomon, and (therefore) the shadow

    Given the `sunray` $s$ and `gnomon` $g$, form $S = s\wedge g$.

    See also [Setup and Definitions](../nb/sundial_setup.md).

    Parameters:
        sunray: The vector encoding the sun ray
        gnomon: The vector encoding the gnomon

    Returns:
        The shadow bivector $S$
    """
    return (sunray ^ gnomon).trigsimp()

shadow_bivector_explicit(hour_angle_symbol=mu, shadow_plane_magnitude_symbol=Xi, gnomon_incl_symbol=iota)

An explicit expression for the bivector encoding the plane containing the sun ray, the gnomon, and (therefore) the shadow

See The Shadow Plane.

Parameters:

Name Type Description Default
hour_angle_symbol Symbol

The hour angle, \(\mu\)

 mu
shadow_plane_magnitude_symbol Symbol

The magnitude of the shadow bivector (see angle between sun ray and gnomon)

 Xi
gnomon_incl_symbol Symbol

The gnomon inclination, \(\iota\)

 iota

Returns:

Type Description
Mv

An explicit expression for the shadow bivector \(S\) under the assumption of zero gnomon declination (\(\delta=0\))
Source code in src/analemma/algebra/result.py 
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def shadow_bivector_explicit(
    hour_angle_symbol: sp.Symbol = mu,
    shadow_plane_magnitude_symbol: sp.Symbol = Xi,
    gnomon_incl_symbol: sp.Symbol = iota,
) -> mv.Mv:
    r"""
    An explicit expression for the bivector encoding the plane containing the sun ray, the gnomon, and (therefore) the shadow

    See [The Shadow Plane](../nb/hour_angle.md#the-shadow-plane).

    Parameters:
        hour_angle_symbol: The hour angle, $\mu$
        shadow_plane_magnitude_symbol: The magnitude of the shadow bivector (see [angle between sun ray and gnomon][analemma.algebra.result.shadow_bivector_magnitude_angle_cos])
        gnomon_incl_symbol: The gnomon inclination, $\iota$

    Returns:
        An explicit expression for the shadow bivector $S$ under the assumption of zero gnomon declination ($\delta=0$)
    """
    n12, n13, n23 = frame.base_bivec("n")
    mu = hour_angle_symbol
    Xi = shadow_plane_magnitude_symbol
    iota = gnomon_incl_symbol
    return mv.Mv(
        sin(Xi) * (-sin(mu) * (sin(iota) * n12 + cos(iota) * n23) + cos(mu) * n13),
        ga=n12.Ga,
    )

shadow_bivector_magnitude_angle_cos(sunray, gnomon)

Cosine of the angle between sun ray and gnomon, \(\cos(\Xi)\)

Given the sunray \(s\) and gnomon \(g\), form \(\cos(\Xi) = s\cdot g\). Note also that the magnitude of the shadow bivector \(S\) is \(\sin(\Xi)\) because \(S^2 = -\sin^2(\Xi)\).

See The Shadow Plane and Normalization.

Parameters:

Name Type Description Default
sunray Mv

The vector encoding the sun ray

 required
gnomon Mv

The vector encoding the gnomon

 required

Returns:

Type Description
Expression

\(s\cdot g = \cos(\Xi)\)
Source code in src/analemma/algebra/result.py 
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def shadow_bivector_magnitude_angle_cos(sunray: mv.Mv, gnomon: mv.Mv) -> Expression:
    r"""
    Cosine of the angle between sun ray and gnomon, $\cos(\Xi)$

    Given the `sunray` $s$ and `gnomon` $g$, form $\cos(\Xi) = s\cdot g$. Note also that the magnitude of the shadow
    bivector $S$ is $\sin(\Xi)$ because $S^2 = -\sin^2(\Xi)$.

    See [The Shadow Plane](../nb/hour_angle.md#the-shadow-plane) and [Normalization](../nb/shadow_angle.md#normalization).

    Parameters:
        sunray: The vector encoding the sun ray
        gnomon: The vector encoding the gnomon

    Returns:
        $s\cdot g = \cos(\Xi)$
    """
    return sp.trigsimp((sunray | gnomon).obj)

shadow_coords_xy(dial_incl_symbol=i, dial_decl_symbol=d, gnomon_incl_symbol=iota, hour_angle_symbol=mu, shadow_plane_magnitude_symbol=Xi)

The Cartesian coordinates of the analemma on the face of a sundial

Given the shadow vector \(w = L\hat{w}\) (constructed by scaling the unit shadow by its length), the coordinates of the tip of the shadow on the dial face can be found via

\(x = m_1 \cdot w\)

\(y = m_2 \cdot w\)

where \(m_1\) and \(m_2\) are the first two vectors in the dial frame.

\(z = 0\) because the shadow is parallel to the face of the dial by definition.

This function provides \(x\) and \(y\) in closed form.

See The Analemma

Parameters:

Name Type Description Default
dial_incl_symbol Symbol

The inclination angle of the dial face relative to the surface frame

 i
dial_decl_symbol Symbol

The declination angle of the dial face relative to the surface frame

 d
gnomon_incl_symbol Symbol

The inclination angle of the gnomon relative to the surface frame

 iota
hour_angle_symbol Symbol

The hour angle of the sun

 mu
shadow_plane_magnitude_symbol Symbol

The angle between the sun ray and gnomon in the shadow bivector

 Xi

Returns:

Type Description
Tuple[Expression]

2-tuple of (\(x\),\(y\))
Source code in src/analemma/algebra/result.py 
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def shadow_coords_xy(
    dial_incl_symbol: sp.Symbol = i,
    dial_decl_symbol: sp.Symbol = d,
    gnomon_incl_symbol: sp.Symbol = iota,
    hour_angle_symbol: sp.Symbol = mu,
    shadow_plane_magnitude_symbol: sp.Symbol = Xi,
) -> Tuple[Expression]:
    r"""
    The Cartesian coordinates of the analemma on the face of a sundial

    Given the shadow vector $w = L\hat{w}$ (constructed by scaling the
    [unit shadow][analemma.algebra.result.unit_shadow] by its [length][analemma.algebra.result.shadow_length]),
    the coordinates of the tip of the shadow on the dial face can be found via

    $x = m_1 \cdot w$

    $y = m_2 \cdot w$

    where $m_1$ and $m_2$ are the first two vectors in the [dial frame][analemma.algebra.frame.dial].

    $z = 0$ because the shadow is parallel to the face of the dial by definition.

    This function provides $x$ and $y$ in closed form.

    See [The Analemma](../nb/analemma.md)

    Parameters:
        dial_incl_symbol: The inclination angle of the dial face relative to the surface frame
        dial_decl_symbol: The declination angle of the dial face relative to the surface frame
        gnomon_incl_symbol: The inclination angle of the gnomon relative to the surface frame
        hour_angle_symbol: The [hour angle][analemma.algebra.result.hour_angle_tan] of the sun
        shadow_plane_magnitude_symbol: The angle between the sun ray and gnomon in the [shadow bivector][analemma.algebra.result.shadow_bivector]

    Returns:
        2-tuple of `(`$x$`, `$y$`)`
    """
    i = dial_incl_symbol
    d = dial_decl_symbol
    iota = gnomon_incl_symbol
    mu = hour_angle_symbol
    Xi = shadow_plane_magnitude_symbol

    x = sin(Xi) * (cos(mu) * cos(d) - sin(mu) * sin(d) * cos(iota)) / D
    y = -sin(Xi) * (
        (
            cos(mu) * sin(d) * cos(i)
            + sin(mu) * (sin(i) * sin(iota) + cos(d) * cos(i) * cos(iota))
        )
        / D
    )
    return x, y

shadow_coords_xy_explicit()

Explicit expressions for the Cartesian coordinates of the analemma in terms of fundamental angles

shadow_coords_xy expresses the analemma in terms of a mixture of the angles found in and the following derived quantities:

  1. \(\mu\), the hour angle
  2. \(\Xi\), the shadow bivector magnitude angle
  3. \(D\), the denominator of \(\lambda\), the solution to the shadow triangle problem

This function eliminates these quantities in the expressions for \(x\) and \(y\), the coordinates of the analemma, leaving only the fundamental angles that determine the location, orientation and geometry of the sundial as a function of date and time.

See The Analemma

Returns:

Type Description
Tuple[Expression]

2-tuple of (\(x\),\(y\)) in terms of fundamental angles
Source code in src/analemma/algebra/result.py 
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def shadow_coords_xy_explicit() -> Tuple[Expression]:
    r"""
    Explicit expressions for the Cartesian coordinates of the analemma in terms of fundamental angles

    [shadow_coords_xy][analemma.algebra.result.shadow_coords_xy] expresses the analemma in terms of a mixture
    of the angles found in [][analemma.algebra.frame] and the following derived quantities:

     1. $\mu$, the [hour angle][analemma.algebra.result.hour_angle_tan]
     1. $\Xi$, the [shadow bivector magnitude angle][analemma.algebra.result.shadow_bivector_magnitude_angle_cos]
     1. $D$, the denominator of $\lambda$, the [solution][analemma.algebra.result.shadow_triangle_solution] to the shadow triangle problem

    This function eliminates these quantities in the expressions for $x$ and $y$, the coordinates of the analemma,
    leaving only the fundamental angles that determine the location, orientation and geometry of the sundial
    as a function of date and time.

    See [The Analemma](../nb/analemma.md)

    Returns:
        2-tuple of `(`$x$`, `$y$`)` in terms of fundamental angles
    """

    x, y = shadow_coords_xy()

    S = shadow_bivector()
    M = frame.meridian_plane()
    sinXi_sin_mu, sinXi_cos_mu = hour_angle_sincos(S, M)

    Xi = sp.Symbol(r"\Xi")

    def _subs_numer(term):
        term = (
            (term / sin(Xi))
            .subs(sin(mu), sinXi_sin_mu)
            .subs(cos(mu), sinXi_cos_mu)
            .simplify()
        )
        term = term.subs(sin(sigma), tan(sigma) * cos(sigma)).simplify()
        return sp.collect(sp.collect(sp.collect(term, cos(psi)), sin(psi)), tan(sigma))

    mu_s = sp.Symbol(r"\mu_s")
    Xi_s = sp.Symbol(r"\Xi_s")
    Xi = sp.Symbol(r"\Xi")
    D_soln = shadow_triangle_solution()

    D_explicit = sp.simplify(
        D_soln.subs(mu_s, mu)
        .subs(Xi_s, Xi)
        .subs(sin(mu), sinXi_sin_mu / sin(Xi))
        .subs(cos(mu), sinXi_cos_mu / sin(Xi))
        .subs(iota, theta)
    )
    D_simple = sp.denom(D_explicit).subs(sin(sigma), tan(sigma) * cos(sigma)).simplify()

    xe = _subs_numer(x * D) / D_simple
    ye = _subs_numer(y * D) / D_simple
    return xe, ye

shadow_length(denom_symbol=D, shadow_plane_magnitude_symbol=Xi, dialface_shadowbivector_angle_symbol=Psi)

An expression for \(L\), the length of the shadow

\(N\), the numerator of \(\lambda\) in the solution to the shadow triangle is equal to \(\sin(\beta)\), which given the form of \(\sin(\beta)\) from gnomon_shadow_angle_sincos means that

\[\sin(\beta) = \frac{N}{\sin(\Psi)}\]

and so, given the form of \(L\) given in Solving the Shadow Triangle, we have

\[L = \frac{\sin(\Xi)\sin(\Psi)}{D}\]

See also The Shadow Vector

Parameters:

Name Type Description Default
denom_symbol Symbol

The denominator \(D\) of the shadow triangle solution \(\lambda = \frac{N}{D}\)

 D
shadow_plane_magnitude_symbol Symbol

The magnitude angle of the shadow bivector (see angle between sun ray and gnomon)

 Xi
dialface_shadowbivector_angle_symbol Symbol

The angle between the dial face and shadow bivector (see dialface_shadowbivector_angle_cos)

 Psi

Returns:

Type Description
Expression

\(L = \sin(\Xi)\sin(\Psi)/D\)
Source code in src/analemma/algebra/result.py 
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def shadow_length(
    denom_symbol: sp.Symbol = D,
    shadow_plane_magnitude_symbol: sp.Symbol = Xi,
    dialface_shadowbivector_angle_symbol: sp.Symbol = Psi,
) -> Expression:
    r"""
    An expression for $L$, the length of the shadow

    $N$, the numerator of $\lambda$ in the solution to the shadow triangle is equal to
    $\sin(\beta)$, which given the form of $\sin(\beta)$ from
    [gnomon_shadow_angle_sincos][analemma.algebra.result.gnomon_shadow_angle_sincos] means that

    $$\sin(\beta) = \frac{N}{\sin(\Psi)}$$

    and so, given the form of $L$ given in
    [Solving the Shadow Triangle](../nb/shadow_length.md#solving-the-shadow-triangle), we have

    $$L = \frac{\sin(\Xi)\sin(\Psi)}{D}$$

    See also [The Shadow Vector](../nb/shadow_length.md#the-shadow-vector)

    Parameters:
        denom_symbol: The denominator $D$ of the shadow triangle solution $\lambda = \frac{N}{D}$
        shadow_plane_magnitude_symbol: The magnitude angle of the shadow bivector (see [angle between sun ray and gnomon][analemma.algebra.result.shadow_bivector_magnitude_angle_cos])
        dialface_shadowbivector_angle_symbol: The angle between the dial face and shadow bivector (see [dialface_shadowbivector_angle_cos][analemma.algebra.result.dialface_shadowbivector_angle_cos])

    Returns:
        $L = \sin(\Xi)\sin(\Psi)/D$
    """
    D = denom_symbol
    Xi = shadow_plane_magnitude_symbol
    Psi = dialface_shadowbivector_angle_symbol

    return sin(Xi) * sin(Psi) / D

shadow_triangle_solution(latitude_symbol=theta, dial_incl_symbol=i, dial_decl_symbol=d, gnomon_incl_symbol=iota, hour_angle_symbol=mu_s, shadow_plane_magnitude_symbol=Xi_s) cached

The scale factor \(\lambda\) on the unit sun ray \(s\) that solves the shadow triangle

Given the triangle \(w(\lambda) = g + \lambda s\) where \(g\) is the gnomon and \(s\) is the sun ray, enforce that \(w(\lambda)\) lies in the dial face by solving \(w(\lambda) \wedge G = 0\) for \(\lambda\). The resulting solution is given by this function.

See Solving the Shadow Triangle

.

Parameters:

Name Type Description Default
latitude_symbol Symbol

\(90^\circ\) minus the latitude at which the frame is embedded

 theta
dial_incl_symbol Symbol

The inclination angle of the dial face relative to the surface frame

 i
dial_decl_symbol Symbol

The declination angle of the dial face relative to the surface frame

 d
gnomon_incl_symbol Symbol

The inclination angle of the gnomon relative to the surface frame

 iota
hour_angle_symbol Symbol

The hour angle of the sun

 mu_s
shadow_plane_magnitude_symbol Symbol

The angle between the sun ray and gnomon in the shadow bivector

 Xi_s

Returns:

Type Description
Expression

An expression for \(\lambda\) that ensures that \(w(\lambda) = g + \lambda s\) is the shadow cast by the gnomon on the dial face
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def shadow_triangle_solution(
    latitude_symbol: sp.Symbol = theta,
    dial_incl_symbol: sp.Symbol = i,
    dial_decl_symbol: sp.Symbol = d,
    gnomon_incl_symbol: sp.Symbol = iota,
    hour_angle_symbol: sp.Symbol = mu_s,
    shadow_plane_magnitude_symbol: sp.Symbol = Xi_s,
) -> Expression:
    r"""
    The scale factor $\lambda$ on the unit sun ray $s$ that solves the shadow triangle

    Given the triangle $w(\lambda) = g + \lambda s$ where $g$ is the gnomon and $s$ is the sun ray,
    enforce that $w(\lambda)$ lies in the dial face by solving $w(\lambda) \wedge G = 0$ for $\lambda$.
    The resulting solution is given by this function.

    See [Solving the Shadow Triangle](../nb/shadow_length.md#solving-the-shadow-triangle)

    ![](https://raw.githubusercontent.com/russellgoyder/analemma/main/docs/paper/figs/ShadowTriangle.png "The shadow triangle.").

    Parameters:
        latitude_symbol: $90^\circ$ minus the latitude at which the frame is embedded
        dial_incl_symbol: The inclination angle of the dial face relative to the surface frame
        dial_decl_symbol: The declination angle of the dial face relative to the surface frame
        gnomon_incl_symbol: The inclination angle of the gnomon relative to the surface frame
        hour_angle_symbol: The [hour angle][analemma.algebra.result.hour_angle_tan] of the sun
        shadow_plane_magnitude_symbol: The angle between the sun ray and gnomon in the [shadow bivector][analemma.algebra.result.shadow_bivector]

    Returns:
        An expression for $\lambda$ that ensures that $w(\lambda) = g + \lambda s$ is the shadow cast by the gnomon on the dial face
    """
    theta = latitude_symbol
    i = dial_incl_symbol
    d = dial_decl_symbol
    iota = gnomon_incl_symbol
    mu_s = hour_angle_symbol
    Xi_s = shadow_plane_magnitude_symbol

    numerator = sin(i) * sin(iota) * cos(d) + cos(i) * cos(iota)
    denominator = sin(Xi_s) * (
        (sin(i) * cos(d) * cos(theta) - cos(i) * sin(theta)) * cos(mu_s)
        - sin(i) * sin(d) * sin(mu_s)
    ) + sin(alpha) * cos(sigma) * (sin(i) * sin(theta) * cos(d) + cos(i) * cos(theta))
    return numerator / denominator

sunray_dialface_angle_sin()

The sine of the angle \(\gamma\) between the sun ray and the dial face

It can be important (when plotting analemmas for example) to determine an effective sunrise and sunset for a given sundial, defined as the points in the sun's journey across the sky when the angle between the sun ray \(s\) and the dial face \(G\) is zero (or \(\pi\)). This angle is given by

\(\sin(\gamma) = -(G \wedge s) \cdot I\)

where \(I = e_1 \wedge e_2 \wedge e_3\).

This function returns an explicit expression for \(\sin(\gamma)\) in terms of fundamental angles.

Source code in src/analemma/algebra/result.py 
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def sunray_dialface_angle_sin() -> Expression:
    r"""
    The sine of the angle $\gamma$ between the sun ray and the dial face

    It can be important (when plotting analemmas for example) to determine an effective sunrise and sunset
    for a given sundial, defined as the points in the sun's journey across the sky when the angle between
    the sun ray $s$ and the dial face $G$ is zero (or $\pi$). This angle is given by

    $\sin(\gamma) = -(G \wedge s) \cdot I$

    where $I = e_1 \wedge e_2 \wedge e_3$.

    This function returns an explicit expression for $\sin(\gamma)$ in terms of fundamental angles.
    """
    return (
        -sin(alpha) * sin(i) * sin(theta) * cos(d) * cos(sigma)
        - sin(alpha) * cos(i) * cos(sigma) * cos(theta)
        + sin(d) * sin(i) * sin(psi) * cos(alpha) * cos(sigma)
        - sin(d) * sin(i) * sin(sigma) * cos(psi)
        - sin(i) * sin(psi) * sin(sigma) * cos(d) * cos(theta)
        - sin(i) * cos(alpha) * cos(d) * cos(psi) * cos(sigma) * cos(theta)
        + sin(psi) * sin(sigma) * sin(theta) * cos(i)
        + sin(theta) * cos(alpha) * cos(i) * cos(psi) * cos(sigma)
    )

unit_noon_shadow(dialface, shadow_bivector) cached

Evalute the unit shadow vector at noon

Noon occurs when the hour angle \(\mu\) is zero, which is when the shadow plane is parallel to the meridian plane.

See The Shadow Angle Relative to Noon

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Mv

The unit shadow vector at noon when \(\mu = 0\)
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def unit_noon_shadow(dialface: mv.Mv, shadow_bivector: mv.Mv) -> mv.Mv:
    r"""
    Evalute the unit shadow vector at noon

    Noon occurs when the hour angle $\mu$ is zero, which is when the shadow plane is parallel to the meridian plane.

    See [The Shadow Angle Relative to Noon](../nb/shadow_angle.md#the-shadow-angle-relative-to-noon)

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        The unit shadow vector at noon when $\mu = 0$
    """
    w_hat = unit_shadow(dialface, shadow_bivector)

    cosPsi = dialface_shadowbivector_angle_cos(dialface, shadow_bivector)
    sinPsi_noon = sp.sqrt(1 - cosPsi**2).subs(mu, 0)
    return w_hat.subs(sin(Psi), sinPsi_noon).subs(mu, 0)

unit_shadow(dialface, shadow_bivector) cached

Form \(\hat{w}\), a unit vector parallel to the shadow

The Normalization section of the shadow angle notebook shows that the length of \(u\), the vector formed as the intersection of the shadow plane and dial face is \(\sin(\Xi)\sin(\psi)\). Here, we divide \(u\) by its length to form a unit vector parallel to the shadow:

\[\hat{w} = \frac{w}{L} = \frac{u}{|u|} = \frac{u}{\sin(\Xi)\sin(\psi)}\]

See Normalization

Parameters:

Name Type Description Default
dialface Mv

Unit bivector encoding the face of the sundial see analemma.algebra.frame.dialface)

 required
shadow_bivector Mv

The bivector \(S = s\wedge g\) for sun ray \(s\) and gnomon \(g\)

 required

Returns:

Type Description
Mv

Unit vector parallel to the shadow, \(\hat{w}\).
Source code in src/analemma/algebra/result.py 
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@lru_cache(maxsize=_cache_max_size)
def unit_shadow(dialface: mv.Mv, shadow_bivector: mv.Mv) -> mv.Mv:
    r"""
    Form $\hat{w}$, a unit vector parallel to the shadow

    The [Normalization](../nb/shadow_angle.md#normalization) section of the [shadow angle notebook](../nb/shadow_angle.md)
    shows that the length of $u$, the vector formed as the
    [intersection of the shadow plane and dial face][analemma.algebra.result.dialface_shadowbivector_intersection]
    is $\sin(\Xi)\sin(\psi)$. Here, we divide $u$ by its length to form a unit vector parallel to the shadow:

    $$\hat{w} = \frac{w}{L} = \frac{u}{|u|} = \frac{u}{\sin(\Xi)\sin(\psi)}$$

    See [Normalization](../nb/shadow_angle.md#normalization)

    Parameters:
        dialface: Unit bivector encoding the face of the sundial see [analemma.algebra.frame.dialface][])
        shadow_bivector: The bivector $S = s\wedge g$ for sun ray $s$ and gnomon $g$

    Returns:
        Unit vector parallel to the shadow, $\hat{w}$.
    """
    u = dialface_shadowbivector_intersection(dialface, shadow_bivector)
    return util.update_coeffs(u / sin(Xi) / sin(Psi), sp.factor)