Abstract: The parameter u varies normally in the range [0,1]. It is, however, easy to reparametrize the curve such that its parameter varies in an arbitrary range [a,b], where a and b are real and a < b. If an interval curve P^I_[0,1] (u) is defined by interval control points [pi^-,pi⁺] in the range [0,1] and if we select an interval [a,b], how can we calculate the interval control points [βi^-,βi⁺] such that the reparametrized interval curve Q_([a,b])^I (u) based on them will go from P^I_[0,1] (a) to P^I_[0,1] (b) [i.e., Q^I_[a,b] (0)=P^I_[0,1] (a) and Q^I_[a,b] (1)=P^I_[0,1] (b)] and will be identical to P^I_[0,1] (u) in that interval. In this paper this concept has been discussed to form a new curve over rectangular domain such that its parameter varies in an arbitrary range [a,b] instead of standard parameter [0,1]. We also want that curve gets generated within the given error tolerance limit. The four fixed Kharitonov's polynomials (four fixed curves) associated with the original interval curve are obtained. A new parameterization is applied to the four fixed Kharitonov's polynomials (four fixed curves). Finally, the required interval control points are obtained from the fixed control points of the four fixed Kharitonov's polynomials. Using matrix representation, it has been shown how to determine the control polygon that covers an arbitrary interval [a,b] of the given interval curve. A numerical example is included in order to demonstrate the effectiveness of the proposed method.