How best to organize classes and methods?

Is there a website with articles and link to the article:
web.com/art/artname
Where art - section with articles artname - article url it is searching in the database. In the router it looks like this:

$class = $request->$controller.'.controller';
$obj = new $class;
$act = $obj->showArticle($request->action);

But then you want other functionality, such as authorization and address web.com/auth/login automatically called by auth.controller but then the problem because in the auth.no showArticle controller method, and it is hard-coded in the router.

Options:

1) to Increase a length of the URL for a section in which to indicate the action:

web.com/art/show/artname
web.com/art/edit/artname
web.com/auth/enter/login

Then you can do it clearly:

$class = $request->$controller.'.controller';
$obj = new $class;
$act = $obj->$request->action($request->data);

2. Implement all classes single main method by default, for example main and increase the number of classes without increasing section in the URL, edit the article, then, would be at web.com/artedit/artname

$class = $request->$controller.'.controller';
$obj = new $class;
$act = $obj->main($request->action);

3. Do branching if-statement in the router, a bad option in my opinion, you do not want to think about it ))

if($controller == 'art'){
 $act = $obj->showArticle($request->action);
}
elseif($controller == 'artedit') {
 $act = $obj->edit($request->action);
}