# 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.

### 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
...