Given two distinct points S and E on a closed parametric curve forming the boundary of a simply-connected domain (without holes), this paper provides an algorithm to find the shortest interior path (SIP) between the two points in the domain. The SIP consists of portions of curves along with straight line segments that are tangential to the curve. The algorithm initially computes point-curve tangents and bitangents using their respective constraints. They are then analyzed further to identify potential tangents. A region check is performed to determine the tangent that will form part of the SIP. Portions of the curve that belong to the SIP are also identified during the process. The SIP is identified without explicitly computing the length of the curves/tangents. The curve is represented using non-uniform rational B-splines (NURBS). Results of the implementation are provided.