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The Research On Nonlinear Projection Problems In Computer Graphics

Posted on:2013-02-03Degree:DoctorType:Dissertation
Country:ChinaCandidate:X B FangFull Text:PDF
GTID:1118330371480976Subject:Computer application technology
Abstract/Summary:PDF Full Text Request
Projection and transformation are two important tools in classical Computer Graphics. As they can be represented by simple matrix multiplication, they are linear in nature. In the past two decades, many nonlinear projections, such as orthogonal projection, minimum distance projection, parallel and central projection, directed projection, least-squares projection, pseudo-perspective projection, multi-perspective projection, panoramic projection, fisheye projection, stereographic projection and circle inversion, attracted much close attention in the fields of Computer Graphics, Computer-Aided Design, Painting art, Photography, Cartology, and etc. All these nonlinear projections greatly enrich the content of traditional Computer Graphics. They produce more artistically appealing visual effects than linear projections and provide effective methods and tools for engineering problems in related domains.Several important nonlinear projection problems in the fields of Computer Graphics and Computer Aided Geometric Design are researched in the thesis, including orthogonal projection of point/curve on curve/surface, skinning of circles on unit sphere via stereographic projection, parallel projection and central projection of curve on surface. Additionally, three common nonlinear projections, the pseudo-perspective projection, fisheye projection and circle inversion, are introduced.A second-order iterative method is proposed for orthogonal projection of parametric curve on implicit surface. By viewing the desired orthogonal projection curve on surface as parametric curve, its first-and second-order differential quantities are firstly analyzed, and then a second-order approach based on Taylor Approximation is formulated and a first-order technique for error adjustment is finally given. Dominant performances of the computational accuracy and efficiency of the proposed method compared with existing algorithms are confirmed by several experiments.A second-order iterative method is presented for orthogonal projection of curve on parametric surface. In terms of the definition of orthogonal projection, the first- and second-order differential quantities of the orthogonal projection curve are firstly analyzed in the parameter domain of the surface, and then a second-order approach based on Taylor Approximation are formulated to tracing the projection point in the parameter domain and a first-order technique for error adjustment is also given. Experimental results show that the proposed method has good performance as compared with existing algorithms in both computational accuracy and efficiency.A geometric interpolation algorithm for skinning of circles on unit sphere is put forward. The problem of skinning of circles on unit sphere is transformed to that of skinning of circles on plane by stereographic projection. For a planar circle sequence, two touching points on each circle are defined by Apollonius circle and then the corresponding unit tangent vectors at those two points are specified. After that, a scheme for construction of skin of each two neighboring circles is proposed by using directed circular arc to interpolate the two-point Hermite data consisted of the two points of contact and the two unit tangent vectors at those two points. The scheme covers C-shaped and S-shaped interpolation algorithms, circle-point interpolation and point-circle interpolation algorithms and circle-circle interpolation algorithm. Skin curves generated by the presented method have G continuity and can avoid the local intersection problem which encountered for existing algorithms. Finally, the skin curve on unit sphere is attained by transformed the skin of planar circle sequence back onto sphere via the inverse stereographic projection.Two algorithms for parallel projection and central projection of parametric curve on parametric surface are presented, respectively. Second-order iteration formulations based on Taylor Approximation are established to match the projection curves in the parameter domain directly and corresponding error adjustment techniques are provided to reduce the deviations caused by the omitted high-order terms. Experimental results show that the second-order algorithms for parallel projection and central projection have superior performance than existing methods in both accuracy and efficiency.Results of the proposed algorithms for orthogonal projection, parallel and central projection are a series of projection points on surfaces. It is desirable to design interpolation curve located on surface and represented with node vector and control points. The proposed C-shaped and S-shaped interpolation algorithms for two-point Hermite data can be further applied for curve approximation with circular arcs. Furthermore, the circle-point, point-circle and circle-circle interpolation algorithms can be used for path design of problem of obstacle avoidance.
Keywords/Search Tags:orthogonal projection, skinning, stereographic projection, parallel projection, central projection
PDF Full Text Request
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