Twitter deprecated Twitter API v.1. So this info is also deprecated now. You can access the same api by using the URL https://api.twitter.com/1.1/search/tweets.json. Though this endpoint does not seem to be identical to the previous endpoint outlined before.
As far as I know Twitter Search API is the best thing we've got as of now. And Twitter Search API supports searching timeline of any user. In fact "Search" text box of Twitter (at the top of Twitter page) uses the search API, so you can experiment using that search box.
For example searching
twitter from:yourusername
shows your tweets with word twitter .
You can use the same query for searching from your timeline, or from any users timeline. An API call identical to the example query would be
Best Answer
5mi
= 5 mile. Defines the radius to search around New York.