13.2.3 Approximations In general, an offset curve is functionally more complex than its progenitor curve because of the square root involved in the expression of the unit normal vector. Li[13 for example has shown that offset of a parabola is a rational curve and its singular point at infinity was studied by Farouki and Sederberg[8. However, this result has not been generalized to higher order curves Farouki and Neff [6 have shown that the two-sided offsets of planar rational polynomial curves are high-degree implicit algebraic curves fo(a, y)=0 of potentially complex shape. These equations can not typically be separated into two equations describing interior and exterior offsets individually. The degree of this implicit offset curve is no= 4n-2-2m where n is the degree of polynomial generator curve r=[r(t),y(t) and m is the degree of o(t)= GCD('(t),y(t)). For example the degree of the two-sided offset curve of a parabola r(t)=(t, t2)is 6 and of a general polynomial cubic curve is 10 with o(t) constant If the progenitor surface is a NUrBS curve, then its offset is usually not a NURBS curve except for straight lines and circles Because of the wide application of offset surfaces and the difficulty in directly incorpo- rating such entities in geometric modeling systems, due to their potential analytic and algebraic complexity, a number of researchers have developed approximation algorithms for these types of geometries in terms of piecewise polynomial or rational polynomial functions 27, 10 Summary of an Approximation Algorithm 27, see also Figure 13.9 Input is a NUrBs curve 2. Offset each leg of polygon by d 3. Intersect consecutive legs of polygon to find new vertices 4. Check deviation of the approximate offset with the true offset using as weights(for rational function) the weights of the progenitor curve 5. If the deviation is larger than the given tolerance subdivide the curve into two and go back to step 1. If the deviation is smaller than the given tolerance stop Approximated offset Curve Progen tor curve Figure 13.9: Offset curve approximation13.2.3 Approximations • In general, an offset curve is functionally more complex than its progenitor curve because of the square root involved in the expression of the unit normal vector. Lu¨ [13] for example has shown that offset of a parabola is a rational curve and its singular point at infinity was studied by Farouki and Sederberg [8]. However, this result has not been generalized to higher order curves. Farouki and Neff [6] have shown that the two-sided offsets of planar rational polynomial curves are high-degree implicit algebraic curves fo(x, y) = 0 of potentially complex shape. These equations can not typically be separated into two equations describing interior and exterior offsets individually. The degree of this implicit offset curve is no = 4n − 2 − 2m, where n is the degree of polynomial generator curve r = [x(t), y(t)] and m is the degree of φ(t) = GCD(x 0 (t), y 0 (t)). For example the degree of the two-sided offset curve of a parabola r(t) = (t,t 2 ) is 6 and of a general polynomial cubic curve is 10 with φ(t) a constant. • If the progenitor surface is a NURBS curve, then its offset is usually not a NURBS curve, except for straight lines and circles. • Because of the wide application of offset surfaces and the difficulty in directly incorpo￾rating such entities in geometric modeling systems, due to their potential analytic and algebraic complexity, a number of researchers have developed approximation algorithms for these types of geometries in terms of piecewise polynomial or rational polynomial functions [27, 10]. • Summary of an Approximation Algorithm [27], see also Figure 13.9: 1. Input is a NURBS curve. 2. Offset each leg of polygon by d. 3. Intersect consecutive legs of polygon to find new vertices. 4. Check deviation of the approximate offset with the true offset using as weights (for rational function) the weights of the progenitor curve. 5. If the deviation is larger than the given tolerance subdivide the curve into two and go back to step 1. If the deviation is smaller than the given tolerance stop. d d d Approximated Offset Curve Progenitor Curve Figure 13.9: Offset curve approximation. 9