Discrete differential geometry driven methods for architectural geometry

• Main Author: Yao, Sidan
• Other Authors: He Ying
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2022
• Subjects: Engineering::Computer science and engineering::Computing methodologies::Computer graphics
• Online Access: https://hdl.handle.net/10356/155036
• Institution: Nanyang Technological University

With the rapid growth of free form architectures, the demand for architectural geometry technologies increases dramatically in recent years. Architectural geometry contains knowledge highly relevant to computer graphics and geometry especially discrete differential geometry. In addition, architectural geometry provides new context for some well-established concepts and brings new challenges and new objectives. In this dissertation, we tackle four challenges in architectural geometry. These four proposed solutions cover various processes including architecture maintenance, architecture construction, architecture texture mapping and architecture decoration. In Chapter 4, we present a data model synchronization method that preserves semantic information across editing operations relying only on geometry, UV mappings, and materials. This enables easy integration of existing and future 3D editing techniques with rich data models. The method links the original data model to the edited geometry using point set registration, recovering the existing information based on spatial and UV search methods, and automatically labels the newly created geometry. The implementation synchronized changes in the 3D geometry with a CityGML data model. In Chapter 5, we present a simple yet effective method for constructing 3D self-supporting surfaces with planar quadrilateral (PQ) elements. Starting with a self-supporting surface in triangle mesh, we first compute the principal curvatures and directions of each triangular face using a new approach, yielding more accurate results than existing methods. Then, we smooth the principal direction field to reduce the number of singularities and partition all faces into two groups in terms of principal curvature difference. For each face with small curvature difference, we compute a stretch matrix that turns the principal directions into a pair of conjugate directions. Finally, applying a mixed-integer programming solver to the mixed principal and conjugate direction field, we obtain a planar quadrilateral mesh. Experimental results show that our method is computationally efficient and can yield high-quality PQ meshes that well approximate the geometry of the input surfaces and maintain their self-supporting properties. In Chapter 6, we present a simple and robust algorithm to compute quad layout. We first propose an interpolation strategy to find tracing directions for singularities: given a singularity with index k (which is a multiple of 1/4), there are exactly 4-4k searching directions produced. We then trace integral curves with a rounding strategy to encourage them to go through mesh vertices, which can effectively reduce the number of short segments. Finally, we partition the triangle mesh along the integral curves to extract the quadrilateral patches. Our method does not require any numerical solver and is easy to implement and computationally efficient. Computational results show that our results have consistently fewer quad patches than those of the existing methods. For models with rich geometric details, our method can save up to 50\% patches in the quad layout. In Chapter 7, we present research in contrast-enhanced high-relief modeling. We present a simple and effective method to generate contrast-enhanced high-reliefs. Our key idea is a depth compression function with only two variables for 3D models. In this function, normals and mean curvatures are utilized to enhance the contrast of the resulting high-reliefs. To calculate the two variables in depth compression function, we construct an optimization framework with two objectives, volume minimization and contrast maximization. The variables are obtained when the two terms reach a balance point. Our method narrows down the solution space to only two variables and is easy to implement and produces good-quality high-reliefs. We show results on a range of real 3D models including real-world and synthetic models. In summary, we present four algorithms in architectural geometry. These solutions cover many applications including semantic information update, free form surface construction, surface parameterization and architecture surface decoration. We demonstrate the effectiveness of the proposed methods through extensive evaluation and comparison with the state-of-the-art methods.