URL discrepancy between localhost and live

I use a PHP IDE (Eclipse) for quick renderings of my site.

My local files are stored at https://localhost/<strong>problemsite</strong>

My live files are stored at https://<strong>problemsite</strong>.domainname.com

The live site is set up as a sub-domain, but can also be typed https://www.domain.com/<strong>problemsite</strong>

The problem is that when I define images, for example background in the imported CSS files, I can successfully load them using /wp-content/themes/thematic/library/imgs/file.gif

…however, on my local computer, the files will not resolve unless I absolutely link it as <strong>https://localhost/problemsite</strong>/wp-content/themes/thematic/library/imgs/file.gif

I have tried any number of periods before the file and have scoured many sites before coming here. I even tried adding this snippet:

define('WP_HOME', 'https://'.$_SERVER['HTTP_HOST'].'/problemsite');
define('WP_SITEURL', 'https://'.$_SERVER['HTTP_HOST'].'/problemsite');

to the wp-config.php file, but to no avail.

This is the first problem I have encountered that I have not been able to find a resolution for.