Homepage of Xunnian Yang(杨勋年)
Associate Professor, School of mathematical sciences, Zhejiang University 
Contact me: Yang Xunnian School of mathematical sciences, Email: yxn@zju.edu.cn Tel: 0571879516098107
Teaching: Undergraduate: Calculus, Complex analysis, Data structure, Linear algebra, ODE. Graduate: Computer Aided Geometric Design, Discrete differential geometry
Research: a. Research fields:
b. Current research topics:
c. Demos:

Selected from Full publications: 

Xunnian
Yang and Jialin Hong (2019), Dynamic evaluation of exponential polynomial curves and surfaces via basis transformation , SIAM Journal on Scientific Computing,
41(5), A3401A3420.
In this paper we present a robust and efficient algorithm for dynamic evaluation of exponential polynomial curves and surfaces. Based on properties that spaces spanned by general exponential polynomials are translation invariant and polynomial spaces are invariant with respect to a linear transformation of the parameter, the transformation matrices between bases with or without translated or linearly transformed parameters are explicitly computed. Points on curves or surfaces with equal or changing parameter steps can then be evaluated dynamically from a start point using a precomputed matrix. This method needs only arithmetic operations for evaluating exponential polynomial curves and surfaces, and it can give robust and accurate evaluation results for any chosen parameter steps. Dynamic evaluation of polynomial curves with changing parameter steps can reduce time costs significantly than computing each point individually by classical methods. [PDF]. 
Xunnian
Yang (2019), Geometric interpolation by PH curves with quadratic or quartic rational normals, ComputerAided Design, 114:112121.
Pythagoreanhodograph (PH) curves have nice properties and have found important applications in geometric modeling and CNC machining. While the unit normals of PH curves of degree n are generally rational curves of degree n1, this paper investigates PH curves of arbitrary degrees but with only quadratic rational unit normals when the curves are convex or with quartic rational unit normals when the curves have single inflection points. PH curves with quadratic or quartic rational normals have simple Gauss maps and hodographs of the curves are given by low degree tangent vector fields together with simple real scaling functions. Practical algorithms for interpolation of pointnormal pairs or pointnormalcurvature pairs together with unit normals at selected parameter coordinates or at inflection points by the investigated PH curves without or with the constraint of arc lengths have been given. [PDF]. 

Xunnian
Yang (2018), Fitting and fairing Hermitetype data by matrix weighted NURBS curves, ComputerAided Design,
102:2232.
This paper proposes techniques to fit and fair sequences of points together with normals or tangents at the points by matrix weighted NURBS curves. Given a set of Hermitetype data, a matrix weighted NURBS curve is constructed by choosing the input points as control points and computing the weight matrices using the normals or tangents. Matrix weighted NURBS curves constructed from Hermitetype data can be fair and fit the input points closely when the original data were regularly sampled from curves with smoothly varying tangents and curvatures. If the original data are nonuniformly spaced or noisy, fair fitting curves can still be obtained by repeatedly sampling points from previously constructed curves and constructing new matrix weighted NURBS curves using the resampled data. [PDF]. 

Xunnian
Yang and Jialin Hong (2017), Dynamic evaluation of freeform curves and surfaces, SIAM Journal on Scientific Computing,
39(2):B424B441.
In this paper we show that freeform curves with properly defined basis functions are the solutions of linear differential systems. By employing typical numerical methods for solving the differential systems, points and derivatives of freeform curves and surfaces can be computed in a dynamical way. Compared with traditional methods for evaluating freeform curves and surfaces there are two advantages of the proposed technique. First, the proposed method is universal and efficient for evaluating a large class of freeform curves and surfaces. Second, the evaluation needs only arithmetic operations even when the freeform curves and surfaces are defined using some transcendental functions. [PDF]. 

Weidong Wu
and Xunnian
Yang (2016), Geometric Hermite interpolation by a family of intrinsically defined planar curves, ComputerAided Design,
77:8697.
This paper proposes techniques of interpolation of intrinsically defined planar curves to Hermite data. In particular, a family of planar curves corresponding to which the curvature radius functions are polynomials in terms of the tangent angle are used for the purpose. The Cartesian coordinates, the arc lengths and the offsets of this type of curves can be explicitly obtained provided that the curvature functions are known. For given G1 or G2 boundary data with or without prescribed arc lengths the free parameters within the curvature functions can be obtained just by solving a linear system. By choosing low order polynomials for representing the curvature radius functions, the interpolating curves can be spirals that have monotone curvatures or fair curves with small numbers of curvature extremes. [PDF]. 

Xunnian
Yang (2016),
Matrix
weighted rational
curves and surfaces, Computer Aided Geometric Design, 42:4053.
Rational curves and surfaces are powerful tools for shape representation and geometric modeling. However, the real weights are generally difficult to choose except for a few special cases such as representing conics. This paper presents an extension of rational curves and surfaces by replacing the real weights with matrices. The matrix weighted rational curves and surfaces have the same structures as the traditional rational curves and surfaces but the matrix weights can be defined in geometric ways. Particularly, the shapes of the extended rational Bezier, NURBS or the generalized subdivision curves and surfaces can be controlled using control points together with control normals. It is also shown that matrix weighted NURBS curves and surfaces can pass through their control points, thus curve or surface reconstruction by the extended NURBS model becomes simple and direct. [PDF]. 

Weidong Wu
and Xunnian
Yang (2015),
Variational surface design under normal field guidance, Journal of
Computational Design and Engineering,
2(3):129136.
This paper proposes a novel method for shape design of a Bezier surface with given boundary curves. The surface is defined as the minimizer of an extended membrane functional or an extended thin plate functional under the guidance of a specified normal field together with an initial prescribed surface. For given boundary curves and the guiding normal field, the free coefficients of a Bezier surface are obtained by solving a linear system. Unlike previous PDE based surface modeling techniques which construct surfaces just from boundaries, our proposed method can be used to generate smooth and fair surfaces that even follow a specified normal field. [PDF]. 


Xunnian
Yang (2014),
Geometric Hermite
interpolation by logarithmic arc splines, Computer Aided Geometric Design,
31(9): 701711.
This paper considers the problem of G^{1} curve interpolation using a special type of discrete logarithmic spirals. A "logarithmic arc spline" is defined as a set of smoothly connected circular arcs. The arcs of a logarithmic arc spline have equal angles and the curvatures of the arcs form a geometric sequence. Given two points together with two unit tangents at the points, a practical algorithm is developed for computing the interpolating logarithmic arc splines. Compared to known methods for logarithmic spiral interpolation, the proposed method has the advantages of unbounded winding angles, simple offsets and NURBS representation. [PDF]. 
Xunnian
Yang, Jianmin Zheng and Desheng Wang(2014),
A Computational approach to joint line detection on triangular meshes, Engineering with Computers,
30(4):583597. We present formulae for evaluating differential quantities at vertices of triangular meshes that may approximate potential piecewise smooth surfaces with discontinuous normals or discontinuous curvatures at the joint lines. We also define the C^{1} and C^{2} discontinuity measures for surface meshes. A simple algorithm is developed for detection of C^{1} or C^{2} discontinuity joint lines on triangular meshes with even highly nonuniform triangulations. [PDF]. 


Xunnian
Yang and Jianmin Zheng(2013),
Curvature tensor computation by piecewise surface interpolation, ComputerAided Design,
45(12): 16391650. This paper presents a new method for curvature tensor estimation on a triangular mesh by replacing flat triangles with triangular parametric patches. Piecewise parametric surfaces that have C^{0} continuity across boundary curves of adjacent patches and G^{1} continuity at the joint vertices are obtained by a local interpolation scheme. A closed form expression of Taubin integral is derived based on the piecewise parametric surfaces. Principal curvatures and principal directions are then computed from the Taubin integral. [PDF]. 
Xunnian
Yang and Jianmin Zheng(2013),
Shape aware normal interpolation for curved surface shading from polyhedral
approximation, The Visual Computer, 29(3):189201. We propose (1) the Gregory normal patch; (2) the sidevertex interpolating normal scheme, for normal interpolation along with interpolation of cubic Bezier triangle for rendering curved surfaces from rough triangular meshes. More realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes. [PDF]. 

Xunnian
Yang and Jianmin Zheng(2012),
Approximate Tspline surface skinning, ComputerAided Design,
44(12):12691276. This paper considers the problem of constructing a smooth surface to fit rows of data points. A special class of Tspline surfaces is examined, which is characterized to have a global knot vector in one parameter direction and individual knot vectors from row to row in the other parameter direction. A skinning algorithm using these Tspline surfaces is proposed, which does not require the knot compatibility of sectional curves. Tspline skinning surface usually has much fewer control points than a lofted Bspline surface that fits the data points with the same error bound. [PDF]. 


Hailing
Zhou, Jianmin Zheng and Xunnian
Yang(2012),
Euler arc splines for curve completion, Computers and Graphics, 36(6):
642650. We propose a special kind arc spline called Euler arc spline which is an extension of Euler curve. Euler arc splines have several nice properties desired by aesthetics of curves and they can be represented by NURBS curves. We also propose an algorithm to construct an Euler arc spline to interpolate two given pointorientation pairs for shape completion. [PDF]. 

Chongyang Deng and
Xunnian Yang(2010),
A simple method for interpolating meshes of arbitrary topology by CatmullClark
surfaces,
The Visual Computer, 26(2):137146.
We present an efficient new algorithm for constructing CatmullClark surface that interpolates a given mesh, where the control mesh of the interpolating surface is obtained by one CatmullClark subdivision of the given mesh with modified geometric rule. [PDF] . 

Chongyang Deng and
Xunnian Yang(2008),
A local fitting algorithm for converting planar curves to Bsplines,
Computer Aided Geometric Design, 25(9):837849. We present a local fitting algorithm for converting smooth planar curves to Bsplines. A G^{1} contijuous Bezier spline curve is first obtained by fitting the sampled data and then the Bezier spline is merged into a C^{2} continuous Bspline curve. [PDF] 

Xunnian Yang and
Weiping Yang(2006),
Cone spline approximation via fat conic spline fitting, ComputerAided Design,
38(6): 703712. Fat conic section and fat conic spline are defined. The problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. [PDF]


Xunnian Yang(2006),
Normal based subdivision scheme
for curve design, Computer Aided Geometric Design, 23(3):243260. We present a new kind of nonlinear and geometric driven subdivision scheme for curve interpolation. Displacement vector for every new vertex is given by normal vectors at old vertices. A shape preserving subdivision scheme is given. [PDF]

Wenhao Song and Xunnian Yang (2005), Freeform deformation with weighted Tspline, The Visual Computer, 21(3):139151.
Weighted Tspline volumes are a natural generalization of NURBS volumes but permit more flexible control lattices. Thus, wTFFD holds virtues of traditional FFDs and is more adaptive to objects with arbitrary topology or complex shapes. [PDF] 

Xunnian Yang (2005), Surface interpolation of meshes by geometric subdivision, ComputerAided Design, 37(5):497508.
Two new nonlinear subdivision schemes are introduced for surface interpolation of triangular meshes. The final interpolating surface inherits the shape of the initial control mesh more fairly and naturally. [PDF] [ppt]


Wujun Che, Xunnian Yang and Guozhao Wang (2004), Skeletondriven 2D distance metamorphosis using intrinsic shape parameters, Graphical Models , 66(2):102126.
A novel algorithm is presented for 2D shape interpolation. The skeletons of two given shapes are computed and the smooth transformation of distance fields is driven by metamorphosis of skeletons. [PDF] 


Xunnian Yang(2004),
Curve fitting and fairing using conic splines,
ComputerAided Design, 36(5):461472.
We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. A G2 (except inflection) and fair curve is obtained which fits given point data or curve. [PDF] 

Xunnian Yang(2003), High accuracy approximation of helices by quintic curves, Computer Aided Geometric Design, 20(6):303317.
In this paper we present methods for approximating a helix segment by quintic Bezier curves or quintic rational Bezier curves based on the geometric Hermite interpolation technique in space. [PDF] 

Xunnian Yang (2002), Efficient circular arc interpolation based on active tolerance control, ComputerAided Design, 34(13):10371046. In this paper, we present an efficient suboptimal algorithm for fitting smooth planar parametric curves by G^{1} arc splines. We obtain a near optimal fitting arc spline in the end. [PDF] 

Yonggang Lü, Guozhao Wang, and Xunnian Yang, (2002), Uniform hyperbolic polynomial Bspline curves, Computer Aided Geometric Design, 19(6):379393.
This paper presents a new kind of uniform splines, called hyperbolic polynomial Bsplines, generated over the space Ω =span{sinh t, cosh t, t^(k−3), t^(k−4), . . . , t, 1} in which k is an arbitrary integer larger than or equal to 3. [PDF]


Xunnian Yang, and Guozhao Wang, (2001), Planar point set fairing and fitting by arc splines, ComputerAided Design, 33(1):3543. We fair and fit planar point set by minimal energy arc splines. A fair point set can be obtained together with a fair G^{1 }arc curve within a given tolerance of the original data. [PDF] 
Last modified, October, 2019.