transform/zenithal_perspective-1.3.0

The zenithal perspective projection.

Description

Corresponds to the AZP projection in the FITS WCS standard.

The pixel-to-sky transformation is defined as:

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

where:

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

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

\[\begin{split}x &= R \sin \phi \\ y &= -R \sec \gamma \cos \theta\end{split}\]

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.

Outline

Schema Definitions

This node must validate against all of the following:

  • This type is an object with the following properties:
    • mu
      object
      Distance from point of projection to center of sphere in spherical radii.

      This node must validate against any of the following:

    • gamma
      object
      Look angle, in degrees.

      This node must validate against any of the following:

Original Schema

%YAML 1.1
---
$schema: "http://stsci.edu/schemas/yaml-schema/draft-01"
id: "http://stsci.edu/schemas/asdf/transform/zenithal_perspective-1.3.0"
tag: "tag:stsci.edu:asdf/transform/zenithal_perspective-1.3.0"
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.

allOf:
  - $ref: "zenithal-1.2.0"
  - type: object
    properties:
      mu:
        anyOf:
          - $ref: "../unit/quantity-1.1.0"
          - type: number
        description: |
          Distance from point of projection to center of sphere in
          spherical radii.
        default: 0

      gamma:
        anyOf:
          - $ref: "../unit/quantity-1.1.0"
          - type: number
        description: |
          Look angle, in degrees.
        default: 0
...