id: asdf://asdf-format.org/transform/manifests/transform-1.2.0
extension_uri: asdf://asdf-format.org/transform/extensions/transform-1.2.0
title: Transform extension 1.2.0
description: |-
  A set of tags for serializing data transforms.
tags:
- tag_uri: tag:stsci.edu:asdf/transform/add-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/add-1.1.0
  title: Perform a list of subtransforms in parallel and then add their results together.
  description: |-
    Each of the subtransforms must have the same number of inputs and
    outputs.
- tag_uri: tag:stsci.edu:asdf/transform/affine-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/affine-1.2.0
  title: An affine transform.
  description: |-
    Invertibility: All ASDF tools are required to be able to compute the
    analytic inverse of this transform.
- tag_uri: tag:stsci.edu:asdf/transform/airy-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/airy-1.2.0
  title: The Airy projection.
  description: |-
    Corresponds to the `AIR` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.2.0)
    for the definition of the full transformation.
- tag_uri: tag:stsci.edu:asdf/transform/bonne_equal_area-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/bonne_equal_area-1.2.0
  title: Bonne's equal area pseudoconic projection.
  description: |-
    Corresponds to the `BON` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \frac{\pi}{180^\circ} A_\phi R_\theta / \cos \theta \\
      \theta &= Y_0 - R_\theta$$

    where:

    $$R_\theta &= \mathrm{sign} \theta_1 \sqrt{x^2 + (Y_0 - y)^2} \\
      A_\phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= R_\theta \sin A_\phi \\
      y &= -R_\theta \cos A_\phi + Y_0$$

    where:

    $$A_\phi &= \frac{180^\circ}{\pi R_\theta} \phi \cos \theta \\
      R_\theta &= Y_0 - \theta \\
      Y_0 &= \frac{180^\circ}{\pi} \cot \theta_1 + \theta_1$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/cobe_quad_spherical_cube-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/cobe_quad_spherical_cube-1.1.0
  title: COBE quadrilateralized spherical cube projection.
  description: |-
    Corresponds to the `CSC` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/compose-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/compose-1.1.0
  title: Perform a list of subtransforms in series.
  description: |-
    The output of each subtransform is fed into the input of the next
    subtransform.

    The number of output dimensions of each subtransform must be equal
    to the number of input dimensions of the next subtransform in list.
    To reorder or add/drop axes, insert `remap_axes` transforms in the
    subtransform list.

    Invertibility: All ASDF tools are required to be able to compute the
    analytic inverse of this transform, by reversing the list of
    transforms and applying the inverse of each.
- tag_uri: tag:stsci.edu:asdf/transform/concatenate-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/concatenate-1.1.0
  title: Send axes to different subtransforms.
  description: |-
    Transforms a set of separable inputs by splitting the axes apart,
    sending them through the given subtransforms in parallel, and
    finally concatenating the subtransform output axes back together.

    The input axes are assigned to each subtransform in order.  If the
    number of input axes is unequal to the sum of the number of input
    axes of all of the subtransforms, that is considered an error case.

    The output axes from each subtransform are appended together to make
    up the resulting output axes.

    For example, given 5 input axes, and 3 subtransforms with the
    following orders:

    1. transform A: 2 in -> 2 out
    1. transform B: 1 in -> 2 out
    1. transform C: 2 in -> 1 out

    The transform is performed as follows:

    ```
      :    i0    i1       i2       i3    i4
      :    |     |        |        |     |
      :  +---------+ +---------+ +----------+
      :  |    A    | |    B    | |    C     |
      :  +---------+ +---------+ +----------+
      :    |     |     |     |        |
      :    o0    o1    o2    o3       o4
    ```

    If reordering of the input or output axes is required, use in series
    with the `remap_axes` transform.

    Invertibility: All ASDF tools are required to be able to compute the
    analytic inverse of this transform.
- tag_uri: tag:stsci.edu:asdf/transform/conic_equal_area-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/conic_equal_area-1.2.0
  title: Alber's conic equal area projection.
  description: |-
    Corresponds to the `COE` projection in the FITS WCS standard.

    See
    [conic](ref:transform/conic-1.2.0)
    for the definition of the full transformation.

    The transformation is defined as:

    $$C &= \gamma / 2 \\
      R_\theta &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin \theta} \\
      Y_0 &= \frac{180^\circ}{\pi} \frac{2}{\gamma} \sqrt{1 + \sin \theta_1 \sin \theta_2 - \gamma \sin((\theta_1 + \theta_2)/2)}$$

    where:

    $$\gamma = \sin \theta_1 + \sin \theta_2$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/conic_equidistant-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/conic_equidistant-1.2.0
  title: Conic equidistant projection.
  description: |-
    Corresponds to the `COD` projection in the FITS WCS standard.

    See
    [conic](ref:transform/conic-1.2.0)
    for the definition of the full transformation.

    The transformation is defined as:

    $$C &= \frac{180^\circ}{\pi} \frac{\sin\theta_a\sin\eta}{\eta} \\
      R_\theta &= \theta_a - \theta + \eta\cot\eta\cot\theta_a \\
      Y_0 = \eta\cot\eta\cot\theta_a$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/conic_orthomorphic-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/conic_orthomorphic-1.2.0
  title: Conic orthomorphic projection.
  description: |-
    Corresponds to the `COO` projection in the FITS WCS standard.

    See
    [conic](ref:transform/conic-1.2.0)
    for the definition of the full transformation.

    The transformation is defined as:

    $$C &= \frac{\ln \left( \frac{\cos\theta_2}{\cos\theta_1} \right)}
                {\ln \left[ \frac{\tan\left(\frac{90^\circ-\theta_2}{2}\right)}
                                 {\tan\left(\frac{90^\circ-\theta_1}{2}\right)} \right] } \\
      R_\theta &= \psi \left[ \tan \left( \frac{90^\circ - \theta}{2} \right) \right]^C \\
      Y_0 &= \psi \left[ \tan \left( \frac{90^\circ - \theta_a}{2} \right) \right]^C$$

    where:

    $$\psi = \frac{180^\circ}{\pi} \frac{\cos \theta}
             {C\left[\tan\left(\frac{90^\circ-\theta}{2}\right)\right]^C}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/conic_perspective-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/conic_perspective-1.2.0
  title: Colles' conic perspecitve projection.
  description: |-
    Corresponds to the `COP` projection in the FITS WCS standard.

    See
    [conic](ref:transform/conic-1.2.0)
    for the definition of the full transformation.

    The transformation is defined as:

    $$C &= \sin \theta_a \\
      R_\theta &= \frac{180^\circ}{\pi} \cos \eta [ \cot \theta_a - \tan(\theta - \theta_a)] \\
      Y_0 &= \frac{180^\circ}{\pi} \cos \eta \cot \theta_a$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/constant-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/constant-1.2.0
  title: A transform that takes no inputs and always outputs a constant value.
  description: |-
    Invertibility: All ASDF tools are required to be able to compute the
    analytic inverse of this transform, which always outputs zero values.
- tag_uri: tag:stsci.edu:asdf/transform/cylindrical_equal_area-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/cylindrical_equal_area-1.2.0
  title: The cylindrical equal area projection.
  description: |-
    Corresponds to the `CEA` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= x \\
    \theta &= \sin^{-1}\left(\frac{\pi}{180^{\circ}}\lambda y\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \phi \\
    y &= \frac{180^{\circ}}{\pi}\frac{\sin \theta}{\lambda}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/cylindrical_perspective-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/cylindrical_perspective-1.2.0
  title: The cylindrical perspective projection.
  description: |-
    Corresponds to the `CYP` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \frac{x}{\lambda} \\
    \theta &= \arg(1, \eta) + \sin{-1}\left(\frac{\eta \mu}{\sqrt{\eta^2 + 1}}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \lambda \phi \\
    y &= \frac{180^{\circ}}{\pi}\left(\frac{\mu + \lambda}{\mu + \cos \theta}\right)\sin \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/divide-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/divide-1.1.0
  title: Perform a list of subtransforms in parallel and then divide their results.
  description: |-
    Each of the subtransforms must have the same number of inputs and
    outputs.

    Invertibility: This transform is not automatically invertible.
- tag_uri: tag:stsci.edu:asdf/transform/gnomonic-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/gnomonic-1.1.0
  title: The gnomonic projection.
  description: |-
    Corresponds to the `TAN` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.1.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = \tan^{-1}\left(\frac{180^{\circ}}{\pi R_\theta}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta = \frac{180^{\circ}}{\pi}\cot \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/hammer_aitoff-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/hammer_aitoff-1.1.0
  title: Hammer-Aitoff projection.
  description: |-
    Corresponds to the `AIT` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= 2 \arg \left(2Z^2 - 1, \frac{\pi}{180^\circ} \frac{Z}{2}x\right) \\
      \theta &= \sin^{-1}\left(\frac{\pi}{180^\circ}yZ\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= 2 \gamma \cos \theta \sin \frac{\phi}{2} \\
      y &= \gamma \sin \theta$$

    where:

    $$\gamma = \frac{180^\circ}{\pi} \sqrt{\frac{2}{1 + \cos \theta \cos(\phi / 2)}}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/healpix-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/healpix-1.1.0
  title: HEALPix projection.
  description: |-
    Corresponds to the `HPX` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/healpix_polar-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/healpix_polar-1.1.0
  title: HEALPix polar, aka "butterfly", projection.
  description: |-
    Corresponds to the `XPH` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/identity-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/identity-1.1.0
  title: The identity transform.
  description: |-
    Invertibility: The inverse of this transform is also the identity transform.
- tag_uri: tag:stsci.edu:asdf/transform/label_mapper-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/label_mapper-1.1.0
  title: Represents a mapping from a coordinate value to a label.
  description: |-
    A label mapper instance maps inputs to a label.  It is used together
    with
    [regions_selector](ref:transform/regions_selector-1.1.0). The
    [label_mapper](ref:transform/label_mapper-1.1.0)
    returns the label corresponding to given inputs. The
    [regions_selector](ref:transform/regions_selector-1.1.0)
    returns the transform corresponding to this label. This maps inputs
    (e.g. pixels on a detector) to transforms uniquely.
- tag_uri: tag:stsci.edu:asdf/transform/mercator-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/mercator-1.1.0
  title: The Mercator projection.
  description: |-
    Corresponds to the `MER` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= x \\
    \theta &= 2 \tan^{-1}\left(e^{y \pi / 180^{\circ}}\right)-90^{\circ}$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \phi \\
    y &= \frac{180^{\circ}}{\pi}\ln \tan \left(\frac{90^{\circ} + \theta}{2}\right)$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/molleweide-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/molleweide-1.1.0
  title: Molleweide's projection.
  description: |-
    Corresponds to the `MOL` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \frac{\pi x}{2 \sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2}} \\
      \theta &= \sin^{-1}\left(\frac{1}{90^\circ}\sin^{-1}\left(\frac{\pi}{180^\circ}\frac{y}{\sqrt{2}}\right) + \frac{y}{180^\circ}\sqrt{2 - \left(\frac{\pi}{180^\circ}y\right)^2}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \frac{2 \sqrt{2}}{\pi} \phi \cos \gamma \\
      y &= \sqrt{2} \frac{180^\circ}{\pi} \sin \gamma$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/multiply-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/multiply-1.1.0
  title: Perform a list of subtransforms in parallel and then multiply their results.
  description: |-
    Each of the subtransforms must have the same number of inputs and
    outputs.

    Invertibility: This transform is not automatically invertible.
- tag_uri: tag:stsci.edu:asdf/transform/parabolic-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/parabolic-1.1.0
  title: Parabolic projection.
  description: |-
    Corresponds to the `PAR` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \frac{180^\circ}{\pi} \frac{x}{1 - 4(y / 180^\circ)^2} \\
      \theta &= 3 \sin^{-1}\left(\frac{y}{180^\circ}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \phi \left(2\cos\frac{2\theta}{3} - 1\right) \\
      y &= 180^\circ \sin \frac{\theta}{3}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/plate_carree-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/plate_carree-1.1.0
  title: "The plate carr\xE9e projection."
  description: |-
    Corresponds to the `CAR` projection in the FITS WCS standard.

    The main virtue of this transformation is its simplicity.

    The pixel-to-sky transformation is defined as:

    $$\phi &= x \\
    \theta &= y$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \phi \\
    y &= \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/polyconic-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/polyconic-1.1.0
  title: Polyconic projection.
  description: |-
    Corresponds to the `PCO` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/polynomial-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/polynomial-1.2.0
  title: A Polynomial model.
  description: |-
    A polynomial model represented by its coefficients stored in
    an ndarray of shape $(n+1)$ for univariate polynomials or $(n+1, n+1)$
    for polynomials with 2 variables, where $n$ is the highest total degree
    of the polynomial.

    $$P = \sum_{i, j=0}^{i+j=n}c_{ij} * x^{i} * y^{j}$$

    Invertibility: This transform is not automatically invertible.
- tag_uri: tag:stsci.edu:asdf/transform/power-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/power-1.1.0
  title: Perform a list of subtransforms in parallel and then raise each result to
    the power of the next.
  description: |-
    Each of the subtransforms must have the same number of inputs and
    outputs.

    Invertibility: This transform is not automatically invertible.
- tag_uri: tag:stsci.edu:asdf/transform/quad_spherical_cube-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/quad_spherical_cube-1.1.0
  title: Quadrilateralized spherical cube projection.
  description: |-
    Corresponds to the `QSC` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/regions_selector-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/regions_selector-1.1.0
  title: Represents a discontinuous transform.
  description: |-
    Maps regions to transgorms and evaluates the transforms with the corresponding inputs.
- tag_uri: tag:stsci.edu:asdf/transform/remap_axes-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/remap_axes-1.1.0
  title: Reorder, add and drop axes.
  description: |-
    This transform allows the order of the input axes to be shuffled and
    returned as the output axes.

    It is a list made up of integers or "constant markers".  Each item
    in the list corresponds to an output axis.  For each item:

    - If an integer, it is the index of the input axis to send to the
      output axis.

    - If a constant, it must be a single item which is a constant value
      to send to the output axis.

    If only a list is provided, the number of input axes is
    automatically determined from the maximum index in the list.  If an
    object with `mapping` and `n_inputs` properties is provided, the
    number of input axes is explicitly set by the `n_inputs` value.

    Invertibility: TBD
- tag_uri: tag:stsci.edu:asdf/transform/rotate2d-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/rotate2d-1.2.0
  title: A 2D rotation.
  description: |-
    A 2D rotation around the origin, in degrees.
    Invertibility: All ASDF tools are required to be able to compute the analytic inverse of this transform.
- tag_uri: tag:stsci.edu:asdf/transform/rotate3d-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/rotate3d-1.2.0
  title: Rotation in 3D space.
  description: |-
    Euler angle rotation around 3 axes.

    Invertibility: All ASDF tools are required to be able to compute the
    analytic inverse of this transform.
- tag_uri: tag:stsci.edu:asdf/transform/sanson_flamsteed-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/sanson_flamsteed-1.1.0
  title: The Sanson-Flamsteed projection.
  description: |-
    Corresponds to the `SFL` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \frac{x}{\cos y} \\
      \theta &= y$$

    And the sky-to-pixel transformation is defined as:

    $$x &= \phi \cos \theta \\
      y &= \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/scale-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/scale-1.2.0
  title: A Scale model.
  description: |-
    Scale the input by a dimensionless factor.
- tag_uri: tag:stsci.edu:asdf/transform/shift-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/shift-1.2.0
  title: A Shift opeartion.
  description: |-
    Apply an offset in one direction.
- tag_uri: tag:stsci.edu:asdf/transform/slant_orthographic-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/slant_orthographic-1.1.0
  title: The slant orthographic projection.
  description: |-
    Corresponds to the `SIN` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.1.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = \cos^{-1}\left(\frac{\pi}{180^{\circ}}R_\theta\right)$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta = \frac{180^{\circ}}{\pi}\cos \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/slant_zenithal_perspective-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/slant_zenithal_perspective-1.2.0
  title: The slant zenithal perspective projection.
  description: |-
    Corresponds to the `SZP` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.2.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = \tan^{-1}\left(\frac{180^{\circ}}{\pi R_\theta}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta = \frac{180^{\circ}}{\pi}\cot \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/stereographic-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/stereographic-1.1.0
  title: The stereographic projection.
  description: |-
    Corresponds to the `STG` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.1.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = 90^{\circ} - 2 \tan^{-1}\left(\frac{\pi R_\theta}{360^{\circ}}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta = \frac{180^{\circ}}{\pi}\frac{2 \cos \theta}{1 + \sin \theta}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/subtract-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/subtract-1.1.0
  title: Perform a list of subtransforms in parallel and then subtract their results.
  description: |-
    Each of the subtransforms must have the same number of inputs and
    outputs.

    Invertibility: This transform is not automatically invertible.
- tag_uri: tag:stsci.edu:asdf/transform/tabular-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/tabular-1.2.0
  title: A Tabular model.
  description: |-
    Tabular represents a lookup table with values corresponding to
    some grid points.
    It computes the interpolated values corresponding to the given
    inputs. Three methods of interpolation are supported -
    "linear", "nearest" and "splinef2d". It supports extrapolation.
- tag_uri: tag:stsci.edu:asdf/transform/tangential_spherical_cube-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/tangential_spherical_cube-1.1.0
  title: Tangential spherical cube projection.
  description: |-
    Corresponds to the `TSC` projection in the FITS WCS standard.

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/zenithal_equal_area-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/zenithal_equal_area-1.1.0
  title: The zenithal equal area projection.
  description: |-
    Corresponds to the `ZEA` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.1.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = 90^\circ - 2 \sin^{-1} \left(\frac{\pi R_\theta}{360^\circ}\right)$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta &= \frac{180^\circ}{\pi} \sqrt{2(1 - \sin\theta)} \\
               &= \frac{360^\circ}{\pi} \sin\left(\frac{90^\circ - \theta}{2}\right)$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/zenithal_equidistant-1.1.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/zenithal_equidistant-1.1.0
  title: The zenithal equidistant projection.
  description: |-
    Corresponds to the `ARC` projection in the FITS WCS standard.

    See
    [zenithal](ref:transform/zenithal-1.1.0)
    for the definition of the full transformation.

    The pixel-to-sky transformation is defined as:

    $$\theta = 90^\circ - R_\theta$$

    And the sky-to-pixel transformation is defined as:

    $$R_\theta = 90^\circ - \theta$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
- tag_uri: tag:stsci.edu:asdf/transform/zenithal_perspective-1.2.0
  schema_uri: http://stsci.edu/schemas/asdf/transform/zenithal_perspective-1.2.0
  title: The zenithal perspective projection.
  description: |-
    Corresponds to the `AZP` projection in the FITS WCS standard.

    The pixel-to-sky transformation is defined as:

    $$\phi &= \arg(-y \cos \gamma, x) \\
    \theta &= \left\{\genfrac{}{}{0pt}{}{\psi - \omega}{\psi + \omega + 180^{\circ}}\right.$$

    where:

    $$\psi &= \arg(\rho, 1) \\
    \omega &= \sin^{-1}\left(\frac{\rho \mu}{\sqrt{\rho^2 + 1}}\right) \\
    \rho &= \frac{R}{\frac{180^{\circ}}{\pi}(\mu + 1) + y \sin \gamma} \\
    R &= \sqrt{x^2 + y^2 \cos^2 \gamma}$$

    And the sky-to-pixel transformation is defined as:

    $$x &= R \sin \phi \\
    y &= -R \sec \gamma \cos \theta$$

    where:

    $$R = \frac{180^{\circ}}{\pi} \frac{(\mu + 1) \cos \theta}{(\mu + \sin \theta) + \cos \theta \cos \phi \tan \gamma}$$

    Invertibility: All ASDF tools are required to provide the inverse of
    this transform.
