Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves

Transcription

1 S S symmetry Article Convergence Analysis on a Second Order Algorithm or Orthogonal Projection onto Curves Xiaowu Li 1, ID, Lin Wang 1,, Zhinan Wu,, Linke Hou 3, *,, Juan Liang 4, *, and Qiaoyang Li 5, *, 1 College o Data Science and Inormation Engineering, Guizhou Minzu University, Guiyang 55005, China; lixiaowu00@16.com (X.L.); wanglin@gzmu.edu.cn (L.W.) School o Mathematics and Computer Science, Yichun University, Yichun , China; zhi_nan_7@163.com 3 Center or Economic Research, Shandong University, Jinan 50100, China 4 Department o Science, Taiyuan Institute o Technology, Taiyuan , China 5 Center or the World Ethnic Studies, Guizhou Minzu University, Guiyang 55005, China * Correspondence: abram75@163.com (L.H.); liangjuan76@16.com (J.L.); liqiaoy@gzmu.edu.cn (Q.L.); Tel.: (L.H.); (J.L.); (Q.L.) These authors contributed equally to this work. Received: 11 August 017; Accepted: 7 September 017; Published: 1 October 017 Abstract: Regarding the point projection and inversion problem, a classical algorithm or orthogonal projection onto curves and suraces has been presented by Hu and Wallner (005). The objective o this paper is to give a convergence analysis o the projection algorithm. On the point projection problem, we give a ormal proo that it is second order convergent and independent o the initial value to project a point onto a planar parameter curve. Meantime, or the point inversion problem, we then give a ormal proo that it is third order convergent and independent o the initial value. Keywords: point projection; planar parametric curve; second order convergence; third order convergence 1. Introduction A vital problem is the projection o a point onto a surace or parametric curve in order to get the oot point and calculate the corresponding parameter values, which is widely applied in the ield o Geometric Modeling [1 7], Computer Graphics and Computer Vision [5 7], Computer Aided Geometric Design [8 1], geometric design [13 15], geometry sculpt [16], scientiic research and engineering applications [17] and computer animation [18 0]. To ind the orthogonal projection o a point onto suraces or parametric curves, Limaien and Trochu [1] constructed an auxiliary unction and obtain all its zeros. Based on the geometric method [4], Li et al. [] use the geometric iterative method with second order approximation properties. The common eatures o [] and [4] are that they both use the curvature inormation, and they are independent o the initial value, respectively. I the algorithm in Hu et al. [4] is applied in the spatial parametric curve, the order o convergence will be one rather than two. In addition, the line segment between the center o circle, test point and osculating circle will not intersect in some cases. Pottmann et al. [3] adopt the idea o the ICP algorithm and provide a new method to geometrically align a point cloud to a surace and to the related registration problems. An alternative notion has been presented in [3], which depends on both instantaneous kinematics and the geometry o the squared distance unction corresponding to a surace. Evidenced by local convergence analysis, as well as by experiment results, the newly proposed ICP method is aster to converge than the classic one. Piegl et al. [5] present a method to project a point onto NURBS suraces with three stpes: decompose a NURBS surace into quadrilaterals, project the test point onto the nearest quadrilateral, and inally get the parameter rom the nearest quadrilateral. Mortenson [1] converted the projection problem Symmetry 017, 9, 10; doi: /sym

2 Symmetry 017, 9, 10 o 13 into the problem o obtaining a polynomial s root, and then adopted the Newton-Raphson approach to get the root. Chen et al. [18] used clipping circle technology and proposed an approach to ind the minimum distance between a point and a NURBS curve. In the same vein, Chen et al. [19] used clipping sphere technology and provided an approach to ind the minimum distance between a clamped B-Spline surace and a point. Similar to [18,19], Young-Taek Oh et al. [] used an eicient culling technique to reduce redundant curves and suraces and proposed an eicient method to project a point to its nearest point on a set o reeorm suraces and curves. Chen et al. [0] proposed a new quadratic clipping approach to calculate a root o a polynomial (t) o degree n within an interval. Distinct rom the classic method in R 1 space, it approximates (t, (t)) and gets three quadratic curves in R space, which results in a higher order o approximation. Among various methods to ind the roots to a polynomial equation in computer graphics and computer aided geometric design, clipping methods with the Bernstein Bézier orm demonstrate great numerical stability. To search the roots o a polynomial (t) in an interval, Chen et al. [11] proposed a rational cubic clipping method, o which the approximation order and the corresponding convergence rate to search a single root are both seven. Based on their previous method in [11], Chen et al. [1] revise a rational cubic clipping approach to get two bounding cubic within O(n) time, which could get a aster convergence rate o ive instead o our in a previous attempt. For Bézier curves, Chen et al. [3] improved the algebraic pruning method to prevent invalid roots ater conversion o the point projection problem into a root inding problem, which is applicable or NURBS curves because, in NURBS curves, is not hard to be changed into Bézier orm. Song et al. [4] presented an algorithm to solve the problem o orthogonal projection o parametric curves onto B-spline suraces. They use a tracing method and then create a polyline approximating the pre-image curve o the projection curve. Bharath Ram Sundar et al. [5] have constructed a measure o distance computation between suraces and curves. To ind point inversion or a parametric surace, Wang et al. [6] proposed a new method to give a simple procedure to get the parameters determined by a given point on a surace. They essentially ormulate ordinary dierential equation systems, which are determined by the intersection curve segment or by the projection curve segment. Li et al. [7] combined a irst order algorithm with Newton s second order algorithm to propose the hybrid second order approach, which goes aster than the current methods and does not depend on the initial value. The adoption o ADMM in [8] contributes to reine the estimate in each iteration because it can incorporate inormation about the direction o estimates gotten in previous steps. In sum, those methods use lots o techniques, or instance, converting the projection problem into a root searching problem or a nonlinear equation system, subdivision approaches, and geometric approaches as well as circular clipping approaches. One widely used, ast and robust method is introduced by the paper [4] and consists o two steps: calculate parameter values by projecting points onto curvature circles in the irst step and then compute parameter increments using Taylor s expansion o the surace or curve in the second step. The paper [4] claims to be quadratic in its convergence rate but does not prove that. In this paper, regarding the point projection problem, we ormally prove that projecting a point onto a planar parameter curve is o second order convergence and independent o the initial value. On the point inversion problem, we ormally prove that it is o third order convergence and independent o the initial value.

3 Symmetry 017, 9, 10 3 o 13. Convergence Analysis In this part, we conduct a convergence analysis o iteration, presented in ormula Equation (5) in paper [4]. We parameterize the curvature circle c such that it shares the same Taylor s polynomial with the curve c. Then this equality holds: q = c(t 0 + t) = c(t 0 + t) + o( t ) = c(t 0 ) + tc (t 0 ) + t c (t 0 ) + o( t ) (1) In R, we deal with the Equation (1) using the method in the paper [4] and get det(q c(t n ), c (t n )) = t det(c (t n ), c (t n )) + o( t ), which yields the iterative ormula t = det(q c(t n), c (t n )) det(c (t n ), c (t n )) = 1 k c 3 det(q c(t n), c (t n )) () On the point projection problem, to veriy that the algorithm deined by Equation () (hereater, the iterative method ()) is quadratically convergent, we irstly derive the expression o the ootpoint q. The corresponding parameter is denoted as α when we orthogonally project a test point p onto a parameter curve c(t), where p = (p 1, p ) and c(t) = ( 1 (t), (t)). By the deinition o the minimum distance between a curve and a point, we will get the relation as ollows (p h) n = 0 (3) where h = ( 1 (α), (α)) and normal vector n = ( (α), 1 (α)). The unit normal vector, the curvature, the relative curvature, and the radius o curvature circle o curve c(t) can be expressed as ollows, respectively, β(t) = ( ( (t) 1, (t) ) (4) 1 (t)) + ( ( (t)) 1 (t)) + ( (t)) k(t) = 1 (t) (t) 1 (t) (t) (( 1 (t)) + ( (t)) ) 3/ (5) k r (t) = 1 (t) (t) 1 (t) (t) (( 1 (t)) + ( (t)) ) 3/ (6) R(t) = 1 κ(t) = (( 1 (t)) + ( (t)) ) 3 1 (t) (t) 1 (t) (t) In this paper, we prove the case where 1 (t) (t) 1 (t) (t) = 1 (t) (t) 1 (t) (t). Additionally, the proo method also holds or the case 1 (t) (t) 1 (t) (t) = ( 1 (t) (t) 1 (t) (t)) only with an opposite sign to the previous case. So the ormula Equation (7) can be written as The center o curvature circle o the planar curve c(t) is R(t) = (( 1 (t)) + ( (t)) ) 3 1 (t) (t) 1 (t) (t) (8) m = (m 1, m ) = c(t) + β(t)/k r (t) (9) (7)

4 Symmetry 017, 9, 10 4 o 13 Now, inserting Equations (4) and (6) into Equation (9), we get m 1 (t) = 1 (t) (t)(( 1 (t)) + ( (t)) ) 1 (t) (t) 1 (t) (t) (10) and m (t) = (t) + 1 (t)(( 1 (t)) + ( (t)) ) 1 (t) (t) 1 (t) (t) (11) The line connecting the test point p and the center o curvature circle m can be parameterized as { x = p 1 + (m 1 p 1 )u y = p + (m p )u (1) where u is a parameter. Also, the equation o the curvature circle is (x m 1 ) + (y m ) = R (13) As the ootpoint q = (q 1, q ) is the intersection o the line Equation (1) and the curvature circle Equation (13), substituting Equation (1) into Equation (13), we get Solving Equation (14), we have (p 1 + (m 1 p 1 ) u m 1 ) + (p + (m p ) u m ) = R (14) u = 1 ± R (m1 p 1 ) + (m p ) (15) Since the ootpoint q lies between the center o curvature circle m and the test point p, the parameter u must belong to (0, 1). By Equation (15), the parameter u will be u = 1 R (m1 p 1 ) + (m p ) (16) From Equations (3), (10), (1) and (16), we get horizontal coordinate q 1 o the intersection q q 1 =p 1 + (m 1 p 1 )u =p 1 + (m 1 p 1 )(1 R (m1 p 1 ) + (m p ) ) (17) From Equations (11), (1) and (16), we obtain vertical coordinate q o the intersection q q =p + (m p )u =p + (m p )(1 R (m1 p 1 ) + (m p ) ) (18) From Equation (17), we have q 1 1 (t) = 1 (t) + p 1 + (m 1 p 1 )u = 1 (t) + p 1 + (m 1 p 1 )(1 R (m1 p 1 ) + (m p ) ) (19)

5 Symmetry 017, 9, 10 5 o 13 From Equation (18), we get q (t) = (t) + p + (m p )u = (t) + p + (m p )(1 R (m1 p 1 ) + (m p ) ) (0) Till now, we provide the detailed illustration o the iterative method (). Next, we will use a method similar to those in the literature [9 31] to prove the theorem. The standard convergence analysis method in numerical analysis in [9 31] are used, namely, e n+1 = C 0 en k + o(en k+1 ), where e n+1 = t n+1 α, e n = t n α, k is a positive integer, C 0 is constant. Theorem 1. Let q(t) c(t) = (q 1 (t) 1 (t), q (t) (t)) be a real parametric curve unction in two dimensions, suppose q(t) c(t) has irst and second derivatives within an interval I. When q(t) c(t) has only a single root α I, and t n stays close suiciently to α, then the iterative method () will be o second order convergence or the point projection problem. ( Proo. Let e n = t n α, a i = (1/i!) (i) ) ( 1 (α), b i = (1/i!) (i) ) (α), i = 0, 1,, 3,...,. Assume that α is a simple root o q c(t) = (q 1 1 (t), q (t)) = 0, i.e., q(α) c(α) = (q 1 (α) 1 (α), q (α) (α)) = 0, q (α) c (α) = (q 1 (α) 1 (α), q (α) (α)) = 0. For 1(t), (t), we take the Taylor s expansion around α, 1 (t n ) = a 0 + a 1 e n + a e n + o(e 3 n) (1) (t n ) = b 0 + b 1 e n + b e n + o(e 3 n) () Furthermore, we have 1 (t n) = a 1 + a e n + o(e n) (3) and (t n) = b 1 + b e n + o(e n) (4) 1 (t n) = a + o(e n ) (5) (t n) = b + o(e n ) (6) For the sake o simplicity, we use the Maple 18 package to calculate the Taylor s expansion in the ollowing way. Combine the derivation o Equations (4) (0) with the orthogonal projection condition or Equation (3) and Equation (1) (6), and then simpliy these equations, we acquire parameterized increment t as ollows: t = e n 6 δ(a 1 + b 1 ) (a 1a p b 1 a 1 b p a + a 1 b b 0 a a 1 a b 0b 1 b 1 b p a 1 + b 1 b p a + b 1 b b 0a 1 b 1 b b 0 a )e n + o(e 3 n) (7) where δ = a a 1b 1 p b a 1 + p a b 1 + b 0 a 1 b b 0 a b 1, p = a 0a 1 a 1 p 1 + b 0 b 1 b 1 and p also satisies the relationship (p 1 1 (α)) 1 (α) + (p (α)) (α) = 0, which is transormed rom the Equation (3). Because t = e n+1 e n (8)

6 Symmetry 017, 9, 10 6 o 13 Substituting Equations (8) into (7), we can get the ollowing error equation similar to that in [3], e n+1 = 6 δ(a 1 + b 1 ) (a 1a p b 1 a 1 b p a + a 1 b b 0 a a 1 a b 0b 1 b 1 b p a 1 + b 1 b p a + b 1 b b 0a 1 b 1 b b 0 a )e n + o(e 3 n) (9) This implies that the algorithm deined by the iterative method () is quadratically convergent or the point projection problem. The proo is completed. Then, we will illustrate that the algorithm expressed by the iterative method () is third order convergent or the point inversion problem. Theorem. Let q(t) c(t) = (q 1 (t) 1 (t), q (t) (t)) be a real parametric curve unction in two dimensions, suppose q(t) c(t) has irst and second derivatives within an interval. When q(t) c(t) has only one single root α I, and t n stays close suiciently to α, so the iterative method () will be o cubic convergence or point the inversion problem. Proo. Given the point inversion problem is a special case o the point projection problem, the proo o this theorem will use the results in the proo o Theorem 1. For the same part, we will not repeat. Since the test point p is on the parametric curve c(t), rom Equation (3), we can know that (p 1, p ) = (a 0, b 0 ). From Equation (9) and (p 1, p ) = (a 0, b 0 ), we acquire the ollowing error equation in [3], e n+1 = (a 1a + b 1 b ) (a 1 + b 1 ) e 3 n + o(e 4 n) (30) This implies that the algorithm expressed by the iterative method () is third order convergent or the point inversion problem. The proo is completed. Theorem 3. The iterative method () is independent o the initial value or point projection and inversion problem. Proo. Firstly, we presents the interpretation or Figure 1. Assume there exists a horizontal axis t and planar parametric curve c(t). There are two points on the planar parametric curve c(t): the irst point c(t n ) with t n on the horizontal axis, and the second point c(α) is the corresponding orthogonal projection point o the test point p onto the planar parametric curve c(t), where α is the corresponding parameter value on the horizontal axis. While, according to the iterative method (), the line segment determined by point p and point c(α) and tangent line o planar parametric curve c(t) at t = α are mutually perpendicular. The tangent line o the planar parametric curve c(t) through point c(t n ) will determine a ootpoint q, which is obtained by orthogonally projecting test point p onto the tangent line o the curve c(t) at t = t n. The corresponding parametric value t n+1 o ootpoint q on the horizontal axis is obviously the next iterative value used by the iterative method (). The corresponding parametric value o the middle point o the line segment determined by the point c(t n ) and the ootpoint q is M. Secondly, we present the proo o Theorem 3. When the iterative method () only has one solution, it is independent o the initial value or point projection and inversion problem.

7 Symmetry 017, 9, 10 7 o 13 c(t ) n t n c(α) t n+1 c(t) Figure 1. Graphic demonstration or convergence analysis. The proo method is similar to the ones in reerences [33 35]. It is not diicult to show that the corresponding parameter o the irst dimensional or the planar parametric curve on the two-dimensional horizontal plane is t. When the iterative method () begins to iterate, as the iterative parameter value satisies the inequality relationship t n < α, by the graphical demonstration, the parameter corresponding to the ootpoint q will be t n+1. The middle point o point (t n+1, 0) and point (t n, 0) is (M, 0), which implies the relationship M = t n + t n+1. Additionally, because o 0 < t = t n+1 t n, there is an inequality relationship t n < M < α. This result suiciently yields the inequality t n α < t n+1 α < α t n = (t n α), namely, there is an iterative error expression e n+1 < e n, where e n = t n α. I t n > α, then the proving method is similar to the case with t n < α. Thus, or the planar parametric curve c(t), there is an iterative error expression e n+1 < e n in the two-dimensional horizontal plane, namely, the iterative method () is independent o the initial iterative parametric value. Alternatively, we will present a descriptive proo that the iterative method () is independent o the initial iterative parametric value. This proo method is also similar to the one in the literature [33 35]. Suppose that the starting point lies on the let o α, then ootpoint q will lie on the right to the starting point, and hence t will be positive. By the iterative method (), the iterative sequence is expressed as ollows, t n = t n 1 + t n 1 (31) where t n 1 = det(q c(t n 1), c (t n 1 )) det(c (t n 1 ), c (t n 1 )) = 1 k c 3 det(q c(t n 1), c (t n 1 )) I t n < α, the sequence t n is strictly monotonically increasing. I t n > α, by at most three iterations, the sequence t n converge to α. Notice that this iterative sequence perorms like an attenuated pendulum and converges to α. Furthermore, when the starting point lies to the right o α, the convergence holds in the same way. To summarize, the ootpoint q depends on the test point p all the time, rather than on the starting iteration point. So the iterative method () is independent o the initial value. The proo is completed. Remark 1. The iterative method () is independent o the initial value, but it only requires that c(t) is C, while the methods in the literature [33 35] need to satisy the suicient conditions or global convergence. In addition, the iterative method () its in with robustness and stabilization in Proessor Les A. Piegl s view [36]. Remark. I k(t) = 0 o the Equation (5), then t = t 0. Hence, we calculate the distance between the point c(t 0 ) on the curve c(t) and the test point p, i.e., d 1 = p c(t 0 ). At the same time, in order to deal with

8 Symmetry 017, 9, 10 8 o 13 the possible ailure o the projection or inversion, we use the parameter perturbation method and increment the parameter t 0 by a small positive constant ε, t 0 = t 0 + ε, so that the iteration can continue. During the iteration process, when the derivative o the curve c(t) at t = t 0 becomes zero, the same perturbation approach is applied. Eventually, d =min{d 1, d, d 3,...} becomes the minimum distance between the curve c(t) and the test point p. 3. Numerical Experiments This section provides numerical evidence to compare the behavior o the Newton iterative algorithm and the iterative method () discussed above. Example 1. We consider the curve c(t) = (t, sin(t)) with test point p = (, ) and the corresponding parameter o projection point α = Table 1 shows the results o Newton iteration and the iterative method (). The results veriy that the iterative method () has better convergence and robustness, while the Newton algorithm is dependent on the initial value but is unstable (See Figure ). Table 1. Step sizes t 1, t in Example 1 or the Newton method and the iterative method (). p = (, ), t 0 = 0.3 iterative methods Step Newton method t 1 NC NC NC NC NC NC The iterative method () t p = (, ), t 0 = 4. iterative methods Step Newton method t 1 NC NC NC NC NC NC The iterative method () t Note: we use NC in Table 1 to represent that the Newton method does not converge, and the same notation applies in Table to Table 6. Figure. Graphic demonstration or Example 1. Example. We use the curve c(t) = (t, cos(t)) with test point p = (, 5) and the corresponding parameter o projection point α = Table shows the results o Newton iteration and the iterative method (). The results also veriy that the iterative method () has better convergence and robustness, while Newton algorithm is dependent on the initial value but is unstable (See Figure 3). Table. Step sizes t 1, t in Example or the Newton method and the iterative method (). p = (, 5), t 0 = 1.7 iterative methods Step Newton method t 1 NC NC NC NC NC NC NC The iterative method () t p = (, 5), t 0 =.4 iterative methods Step Newton method t 1 NC NC NC NC NC NC NC The iterative method () t

9 Symmetry 017, 9, 10 9 o 13 Figure 3. Graphic demonstration or Example. Example 3. We consider the curve c(t) = (t + sin(t), sin(sin(t)) + cos(t)) with test point is p = ( 1, 1) and the corresponding parameter o projection point α = Table 3 shows the number o iterations or the Newton iteration and the iterative method (). The results show the iterative method () has better convergence and robustness, at the same time, the Newton s method sometimes converges slowly and depends on the initial value but is unstable (See Figure 4). Table 3. Comparison o the robustness and eectiveness or the Newton s method and the iterative method (). t The iterative method () Newton s method NC NC Figure 4. Graphic demonstration or Example 3. Example 4. We use the curve c(t) = (t, sin(t) + cos(t)), t [3, 6]. Test point p = (, 6) and the corresponding parameter o projection point is Table 4 shows the number o iterations or Newton iteration and the iterative method (). Results show the iterative method () has better convergence and robustness, at the same time, Newton s method is sometimes dependent on the initial value but is unstable (See Figure 5). Table 4. Comparison o the robustness and eectiveness or the Newton s method and the iterative method (). t The iterative method () Newton method NC NC NC NC NC NC NC NC NC NC t The iterative method () Newton method NC NC NC NC NC NC NC NC NC NC

10 Symmetry 017, 9, o 13 Figure 5. Graphic demonstration or Example 4. Example 5. We consider the curve c(t) = (t, sin(0.5t)), t [ 6, ]. Test point is p = (3, 7), and the corresponding parameter o projection point is Table 5 shows number o iterations or Newton iteration and the iterative method (). Results show the the iterative method () has better convergence and robustness, at the same time, Newton s method is sometimes dependent on the initial value but is unstable (See Figure 6). Table 5. Comparison o the robustness and eectiveness or the Newton s method and the iterative method (). t The iterative method () Newton method NC NC NC NC NC NC NC NC NC NC t The iterative method () Newton method NC NC NC NC NC NC NC NC NC NC Figure 6. Graphic demonstration or Example 5. Example 6. We use the curve c(t) = (t, cos(t)), t [ 1, 8]. The test point is p = ( 1, 5), and the corresponding parameter o projection point is Table 6 shows the number o iterations or Newton iteration and the iterative method (). Results show the iterative method () has better convergence and robustness, at the same time, Newton s method is sometimes dependent on the initial value but is unstable (See Figure 7).

11 Symmetry 017, 9, o 13 Table 6. Comparison o the robustness and eectiveness or the Newton s method and the iterative method (). t The iterative method () Newton method NC NC NC NC NC NC NC NC NC NC Figure 7. Graphic demonstration or Example 6. Remark 3. The iterative method () orthogonally projects a test point onto a parametric curve c(t). For the case with multiple orthogonal points, our approach is changed in the ollowing way: (1) Divide a parameter interval [a, b] o parametric curve c(t) into M subintervals with equal length. () Randomly select an initial iterative parametric value in each subinterval. (3) Using the iterative method () and using each initial iterative parametric value, iterate, respectively. Suppose that the iterative parametric values are α 1,α,...,α M, respectively. (4) Calculate the local minimum distances d 1, d,..., d M,where d i = p c(α i ). (5) Calculate the global minimum distance d = p c(α) = { d 1,d,...d M }. I we attempt to search all the solutions as ast as possible, divide a parameter interval [a, b] into M subintervals with equal length such that M is suiciently large. According to Figures 5 and 7, either blue point is the only orthogonal projection point o the test point, but not the point with minimum distance. For t [ 6, 6] in Figure 5, our parameter values are , , , , respectively. According to Remark 3, it is easy to ind that the orthogonal projection point determined by the parameter value o is the one with minimum distance, but the orthogonal projection points determined by the other parameter values are not. For t [ 10, 3] in Figure 7, eight parameter values are , , , , , , , , respectively. In the same way, only the orthogonal projection point determined by the parameter value o is the one with minimum distance. 4. Conclusions In this paper, regarding the point projection problem, we ormally prove that projection o a point onto a planar parameter curve is quadratically convergent and independent o the initial value. On the point inversion problem, we ormally prove that the inversion problem is cubically convergent and independent o the initial value. In the uture, we would like to present a ormal proo o convergence on projecting a point onto a spatial parametric curve and surace in literature [4]. Acknowledgments: We would like to thank the editor and two anonymous reviewers or their eorts to improve greatly the manuscript. We acknowledge the support by the National Natural Science Foundation o China (Grant No , Grant No ), Scientiic and Technology Foundation Funded Project o Guizhou

12 Symmetry 017, 9, 10 1 o 13 Province (Grant No. [014]09, Grant No. [014]093), Feature Key Laboratory or Regular Institutions o Higher Education o Guizhou Province (Grant No. [016]003), National Bureau o Statistics Foundation Funded Project (Grant No. 014LY011), Key Laboratory o Pattern Recognition and Intelligent System o Construction Project o Guizhou Province (Grant No. [009]400), Inormation Processing and Pattern Recognition or Graduate Education Innovation Base o Guizhou Province. Linke Hou is supported by Shandong Provincial Natural Science Foundation o China (Grant No. ZR016GM4). Juan Liang is supported by Scientiic and Technology Key Foundation o Taiyuan Institute o Technology (Grant No. 016LZ0). Qiaoyang Li is supported by Fund o National Social Science (No. 14XMZ001) and Fund o Chinese Ministry o Education (No. 15JZD034). Author Contributions: The contributions o all o the authors are the same. All o them have worked together to develop the present manuscript. Conlicts o Interest: The authors declare no conlict o interest. Reerences 1. Limaiem, A.; Trochu, F. Geometric algorithms or the intersection o curves and suraces. Comput. Graph. 1995, 19, Li, X.; Wu, Z.; Hou, L.; Wang, L.; Yue, C.; Xin, Q. A geometric orthogonal projection strategy or computing the minimum distance between a point and a spatial parametric curve. Algorithms 016, 9, Pottmann, H.; Leopoldseder, S.; Hoer, M. Registration without ICP. Comput. Vis. Image Underst. 004, 95, Hu, S.M.; Wallner, J. A second order algorithm or orthogonal projection onto curves and suraces. Comput. Aided Geom. Des. 005,, Piegl, L.A.; Tiller, W. Parameterization or surace itting in reverse engineering. Comput. Aided Des. 001, 33, Ma, Y.L.; Hewitt, W.T. Point inversion and projection or NURBS curve and surace: Control polygon approach. Comput. Aided Geom. Des. 000, 0, Oh, Y.T.; Kim, Y.J.; Lee, J.; Kim, M.S.; Elber, G. Continuous point projection to planar reeorm curves using spiral curves. Vis. Comput. 011, 8, Srijuntongsiri, G. An iterative/subdivision hybrid algorithm or curve/curve intersection. Vis. Comput. 011, 7, Chen, X.D.; Ma, W. Geometric point interpolation method in R 3 space with tangent directional constraint. Comput. Aided Des. 01, 44, Chen, X.D.; Yang, C.; Ma, W. Coincidence condition o two Bézier curves o an arbitrary degree. Comput. Graph. 016, 54, Chen, X.D.; Ma, W.; Ye, Y. A rational cubic clipping method or computing real roots o a polynomial. Comput. Aided Geom. Des. 015, 38, Chen, X.D.; Ma, W. Rational cubic clipping with linear complexity or computing roots o polynomials. Appl. Math. Comput. 016, 73, Lin, H.W. Local progressive-iterative approximation ormat or blending curves and patches. Comput. Aided Geom. Des. 010, 7, Lin, H.W. The convergence o the geometric interpolation algorithm. Comput. Aided Des. 010, 4, Lin, H.; Qin, Y.; Liao, H.; Xiong, Y. Aine arithmetic-based B-Spline surace intersection with gpu acceleration. IEEE Trans. Vis. Comput. Graph. 014, 0, Sederberg, T.W.; Lin, H.; Li, X. Curvature o singular Bézier curves and suraces. Comput. Aided Geom. Des. 011, 8, Lin, H.; Zhang, Z. An eicient method or itting large data sets using T-Splines. SIAM J. Sci. Comput. 013, 35, A305 A Chen, X.D.; Yong, J.H.; Wang, G.; Paul, J.C.; Xu, G. Computing the minimum distance between a point and a NURBS curve. Comput. Aided Des. 008, 40, Chen, X.D.; Xu, G.; Yong, J.H.; Wang, G.; Paul, J.C. Computing the minimum distance between a point and a clamped B-spline surace. Graph. Model. 009, 71, Chen, X.D.; Ma, W. A planar quadratic clipping method or computing a root o a polynomial in an interval. Comput. Graph. 015, 46, Mortenson, M.E. Geometric Modeling; Wiley: New York, NY, USA, 1985; pp

13 Symmetry 017, 9, o 13. Oh, Y.T.; Kim, Y.J.; Lee, J.; Kim, M.S.; Elber, G. Eicient point-projection to reeorm curves and suraces. Comput. Aided Geom. Des. 01, 9, Chen, X.; Zhou, Y.; Shu, Z.; Su, H. Improved Algebraic Algorithm On Point Projection For Bézier Curves. In Proceedings o the Second International Multi-Symposium o Computer and Computational Sciences (IMSCCS 007), the University o Iowa, Iowa City, IA, USA, August Song, H.C.; Yong, J.H.; Yang, Y.J.; Liu, X.M. Algorithm or orthogonal projection o parametric curves onto B-spline suraces. Comput. Aided Des. 011, 43, Sundar, B.R.; Chunduru, A.; Tiwari, R.; Gupta, A.; Muthuganapathy, R. Footpoint distance as a measure o distance computation between curves and suraces. Comput. Graph. 014, 38, Wang, X.; Zhang, W.; Huang, X. Computation o point inversion and ray-surace intersection through tracing along the base surace. Vis. Comput. 015, 31, Li, X.; Wang, L.; Wu, Z.; Hou, L.; Liang, J.; Li, Q. Hybrid second-order iterative algorithm or orthogonal projection onto a parametric surace. Symmetry 017, 9, Li, K.; Sundin, M.; Rojas, C.R.; Chatterjee, S.; Jansson, M. Alternating strategies with internal ADMM or low-rank matrix reconstruction. Signal Process. 016, 11, Li, X.; Mu, C.; Ma, J.; Wang, C. Sixteenth-order method or nonlinear equations. Appl. Math. Comput. 010, 15, Li, X.; Mu, C.; Ma, J.; Hou, L. Fith-order iterative method or inding multiple roots o nonlinear equations. Numer. Algoritm. 011, 57, Li, X.; Xin, Q.; Wu, Z.; Zhang, M.; Zhang, Q. A geometric strategy or computing intersections o two spatial parametric curves. Vis. Comput. 013, 9, Traub, J.F. Iterative Methods or Solution o Equations; Prentice-Hall: Englewood Clis, NJ, USA, Melman, A. Geometry and Convergence o Euler s and Halley s Methods. SIAM Rev. 1997, 39, Melman, A. Analysis o third order Methods or Secular Equations. Math. Comput. 1998, 67, Traub, J.F. A Class o Globally Convergent Iteration Functions or the Solution o Polynomial Equations. Math. Comput. 1966, 0, Piegl, L.A. Ten challenges in computer-aided design. Comput. Aided Des. 005, 37, c 017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions o the Creative Commons Attribution (CC BY) license (