All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
The line that is drawn in the area surrounding the Ka'bah to indicate the beginning point of the Tawaaf (circumambulation around the Ka'bah) is for the people to know the starting point and end of the Tawaaf because many of them do not know exactly where to start. Indeed some of them start the Tawaaf at the door of the Ka'bah or a short distance away from the black stone. This, of course, invalidates the Tawaaf and it is not counted as one round according to the preponderant opinion of scholars. This is the view of the Shaafi'ee and Hanbali schools, and according to one narration of Abu Haneefah. So if one does not make up the one round that is not valid consecutively after doing the other rounds, then all the Tawaaf is void according to some scholars as they set a condition of continuity between one round and the other for the Tawaaf to be valid.
To conclude, the line is drawn to clarify the starting and ending point of the Tawaaf and this is not contradicting the Sunnah of the Prophet to an extent to say that one should disregard the line and follow the Sunnah. The most important thing is that one should start the Tawaaf whilst parallel to the black stone, so if a person does so without searching the line to start from it, his Tawaaf is considered correct and acceptable.
Allaah knows best.