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lqyygy木蟲 (著名寫手)
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三角網(wǎng)曲面的求交運算被廣泛應用于幾何造型、計算機輔助設計、GIS空間分析和三維地質(zhì)建模中，基于傳統(tǒng)計算幾何的三角網(wǎng)曲面求交運算具有計算方式不清晰、對不同幾何對象的幾何運算和數(shù)學運算不統(tǒng)一，其向高維擴展相對較難等問題，本文引入共形幾何代數(shù)理論，嘗試構(gòu)建不同維度、不同類型地理對象的統(tǒng)一表達與運算框架，進而實現(xiàn)多維統(tǒng)一的曲面求交運算及基于其上的變化檢測算法。最后通過實例驗證算法的有效性、準確性并進行地學解釋探討。 基于共形幾何代數(shù)的多維統(tǒng)一分析框架，根據(jù)地理建模和應用需求，構(gòu)建多維地理對象的拆分與重構(gòu)，實現(xiàn)基于多重向量的多維統(tǒng)一的地理對象統(tǒng)一表達與存儲機制。結(jié)合地理分析需求，利用所選取的代數(shù)空間所定義的基本運算規(guī)則及基本算子構(gòu)造運算空間，構(gòu)建面向多維空間對象計算和檢索的索引機制，實現(xiàn)對多維對象的有效剖分、檢索以及分析。 借鑒CGA多維統(tǒng)一的對象表達和運算結(jié)構(gòu)，面向曲面求交算法的需要，實現(xiàn)三角面片的相交檢測和求交運算，提出多維統(tǒng)一的三角網(wǎng)曲面求交運算的算法框架，構(gòu)建了基于曲面求交的變化檢測算法。上述算法從數(shù)學底層上統(tǒng)一了所涉及的不同維度的對象表達和拓撲關系運算，為地理對象的表達和模擬提供了全新的技術支撐，也為拓展多維統(tǒng)一的空間分析算法提供了思路。 基于南極冰蓋模擬數(shù)據(jù)實現(xiàn)了多維統(tǒng)一的曲面求交運算，并基于曲面求交運算的結(jié)果，對曲面進行局部約束三角重構(gòu)、變化區(qū)域提取和面積、體積等變化指標的計算，檢測出南極冰蓋的的變化趨勢。南極冰蓋案例模擬研究顯示，本文構(gòu)建的曲面變化檢測算法具有較好的檢測效果，顯示基于CGA曲面求交的變化檢測算法具有較好的通用性和可推廣性，為高效的多維融合空間分析算法構(gòu)建提供有效借鑒。 Triangulated surfaces intersection operation has been widely used in geometric modeling, computer-aided design, GIS spatial analysis and 3D geological modeling. The operation based on traditional computational geometry has some shortcomings that the method is not always intuitively clear, geometric operation and the mathematical calculation are inconsistent when deals with different geometrical objects,and it is relatively difficult to expand to the high-dimensional and so on. This thesis introduces the conformal geometric algebra theory, tries to build the framework of a unified expression and operation of different types and different dimensional geometric objects, and then implements multi-dimensional integrated structure of triangulated surfaces intersection algorithm and change detection algorithm based thereon. Finally, the accuracy, validity and geographical interpretation of mentioned algorithms are discussed through the typical case studies. Based on the framework of conformal geometric algebra multidimensionally-unified representation and computation, ways to divide and reconstruct the multidimensional geographic objects are developed, and expressions and storage mechanism based on multivector are implemented, according to the geographical modeling and application requirements. And then, taking into account requirements of geographical spatial analysis, the thesis builds the index of the multi-dimensional space object for computing and retrieval mechanism, achieves effective division, retrieval and analysis of multi- dimensional objects using the basic operational rules defined in the algebraic space and computing space constructed by basic operators. Referring to multidimensional unified objects expression and operation structure of conformal geometric algebra and according to the surface intersection algorithm needs, the thesis achieves the detection of triangular facets intersect and intersection computation, proposes multi-dimensional integrated structure of triangulation surface intersection algorithm framework, and based on this surface intersection algorithm, change detection algorithm is further constructed. Aforementioned algorithms unify the expression and computing of different dimensions objects, basic topology involved in the underlying of the mathematics, which provids new technogical supporting for the expression and simulation of geographical objects. The research also provdes new potential ways to extend the unified multidimensional spatial analysis methods. Multi-dimensional Surface Intersection Algorithm is realized using Antarctic ice sheet's simulation data. And then by the results from intersection operation, we can triangle-reconstruct surface with local constraints,extract the change region,calculate the factors -area and volume,and detect Antarctic ice’s variation tendency.The Antarctic icecap’s simulation case study shows that the surface change-detection algorithm of this paper has better detection results, and that indicates the surface change detection algorithm based CGA'S Surface Intersection has good genenrality and extengding.That provides an effective reference to build a efficient,multi-dimensional integration geographic space analysis algorithm . |
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