Architecture of application with public RESTful API + local access to API

I'm planning a small web application project which will consist of both a website (using PHP) and in the future a mobile application. I want to implement a RESTful API (using PHP) to communicate with the mobile app. But since the API and the website will both be written in PHP and hosted on the same server, it seems a bit odd to make HTTP calls from the website to the public API (or isn't it)?

Anyway, I am considering putting a layer between the API and the business logic, basically just consisting of an object that exposes the same API as the public RESTful API, but as a PHP object that can be accessed directly from the website.

1. Is this a good or bad idea? Why?
2. Is this a well known pattern? If so, what is it called?

I found some sites proposing a similar structure and calling it "API Gateway Object", but I'm not sure if that is an actual well-known pattern or just something they came up with tho.

Here's a sketch of what I have in mind: