Homepage of Xunnian Yang(杨勋年)


Associate Professor, School of mathematical sciences, Zhejiang University

Contact me:

Yang Xunnian

School of mathematical sciences,

Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang Province 310027, China

Email: yxn@zju.edu.cn

Tel: 0571-87951609-8107



Undergraduate: Calculus, Complex analysis, Data structure, Linear algebra, ODE.

Graduate: Computer Aided Geometric Design, Discrete differential geometry



a.     Research fields:

  • Computer aided geometric design

  • Computer graphics

  • Image processing


b.     Current research topics:

  • Spline techniques and nonlinear subdivision

  • Numerical differential geometry and visualization

  • PDE/ODE based image/geometric processing

  • Computational photography and image vectorization


c.     Demos:

  • Matrix weighted NURBS curve fitting and fairing

helix smoothing

  • Dynamic evaluation of free-form curves and surfaces


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), A3401-A3420.

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 pre-computed 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, Computer-Aided Design, 114:112-121.

Pythagorean-hodograph (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 n-1, 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 point-normal pairs or point-normal-curvature 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 Hermite-type data by matrix weighted NURBS curves, Computer-Aided Design, 102:22-32.

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 Hermite-type 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 Hermite-type 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 non-uniformly 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 free-form curves and surfaces, SIAM Journal on Scientific Computing, 39(2):B424-B441.

In this paper we show that free-form 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 free-form curves and surfaces can be computed in a dynamical way. Compared with traditional methods for evaluating free-form 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 free-form curves and surfaces. Second, the evaluation needs only arithmetic operations even when the free-form 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, Computer-Aided Design, 77:86-97.

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, 42:40-53.

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):129-136.

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): 701-711.

This paper considers the problem of G1 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):583-597.

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 C1 and C2 discontinuity measures for surface meshes. A simple algorithm is developed for detection of C1 or C2 discontinuity joint lines on triangular meshes with even highly non-uniform triangulations.  [PDF].


Xunnian Yang and Jianmin Zheng(2013), Curvature tensor computation by piecewise surface interpolation, Computer-Aided Design, 45(12): 1639-1650.

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 C0 continuity across boundary curves of adjacent patches and  G1 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):189-201.

We propose (1) the Gregory normal patch; (2) the side-vertex 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 T-spline surface skinning, Computer-Aided Design, 44(12):1269-1276.

This paper considers the problem of constructing a smooth surface to fit rows of data points. A special class of T-spline 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 T-spline surfaces is proposed, which does not require the knot compatibility of sectional curves. T-spline skinning surface usually has much fewer control points than a lofted B-spline 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): 642-650.

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 point-orientation pairs for shape completion.  [PDF].


Chongyang Deng and Xunnian Yang(2010), A simple method for interpolating meshes of arbitrary topology by Catmull-Clark surfaces, The Visual Computer, 26(2):137-146.

We present an efficient new algorithm for constructing Catmull-Clark surface that interpolates a given mesh, where the control mesh of the interpolating surface is obtained by one Catmull-Clark 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 B-splines, Computer Aided Geometric Design, 25(9):837-849.

We present a local fitting algorithm for converting smooth planar curves to B-splines. A G1 contijuous Bezier spline curve is first obtained by fitting the sampled data and then the Bezier spline is merged into a C2 continuous B-spline curve.  [PDF]

Xunnian Yang and Weiping Yang(2006), Cone spline approximation via fat conic spline fitting, Computer-Aided Design, 38(6): 703-712.

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):243-260.

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 T-spline, The Visual Computer, 21(3):139-151.


Weighted T-spline volumes are a natural generalization of NURBS volumes but permit more flexible control lattices. Thus, w-TFFD 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, Computer-Aided Design, 37(5):497-508.


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]


skeleton metamorphsis
Wujun Che, Xunnian Yang and Guozhao Wang (2004), Skeleton-driven 2D distance metamorphosis using intrinsic shape parameters, Graphical Models ,  66(2):102-126.

A novel algorithm is presented for 2-D 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, Computer-Aided Design,  36(5):461-472.

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):303-317.


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, Computer-Aided Design, 34(13):1037-1046.

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G1 arc splines. We obtain a near optimal fitting arc spline in the end.   [PDF]

Yonggang L, Guozhao Wang, and Xunnian Yang, (2002), Uniform hyperbolic polynomial B-spline curves, Computer Aided Geometric Design, 19(6):379-393.


This paper presents a new kind of uniform splines, called hyperbolic polynomial B-splines, 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, Computer-Aided Design, 33(1):35-43.

We fair and fit planar point set by minimal energy arc splines. A fair point set can be obtained together with a fair G1 arc curve within a given tolerance of the original data. [PDF]


Last modified, October, 2019.