Conformal Geometric Algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this work, we demonstrate the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA. In such a framework, and for the first time, one may perform real-time cuts and tears as well as drill holes on a rigged 3D model. These operations can be performed before and/or after model animation, while maintaining deformation topology. Moreover, our framework permits generation of intermediate keyframes on-the-fly based on user input, apart from the frames provided in the model data. We are motivated to use CGA as it is the lowest-dimension extension of dual-quaternion algebra that amends the shortcomings of the majority of existing animation and deformation techniques. Specifically, we no longer need to maintain objects of multiple algebras and constantly transmute between them, such as matrices, quaternions and dual-quaternions, and we can effortlessly apply dilations. Using such an all-in-one geometric framework allows for better maintenance and optimization and enables easier interpolation and application of all native deformations. Furthermore, we present these three novel algorithms in a single CGA representation which enables cutting, tearing and drilling of the input rigged model, where the output model can be further re-deformed in interactive frame rates. These close to real-time cut,tear and drill algorithms can enable a new suite of applications, especially under the scope of a medical VR simulation.

中文翻译：

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切割，撕裂，钻孔变形模型的多合一几何算法

保形几何代数（CGA）是一个框架，它允许表示对象（例如点，平面和球体）以及变形（例如平移，旋转和膨胀）作为统一矢量（称为多矢量）来表示。在这项工作中，我们通过基于CGA的新颖，集成的操纵角色模拟框架展示了多向量用法的优点。在这样的框架中，这是第一次，可以在装配的3D模型上执行实时剪切和撕​​裂以及钻孔。这些操作可以在模型动画之前和/或之后执行，同时保持变形拓扑。此外，除了模型数据中提供的框架外，我们的框架还允许根据用户输入即时生成中间关键框架。我们有兴趣使用CGA，因为它是双四元数代数的最小维度扩展，它弥补了大多数现有动画和变形技术的不足。具体来说，我们不再需要维护多个代数的对象并在它们之间不断变换，例如矩阵，四元数和对偶四元数，并且我们可以毫不费力地应用膨胀。使用这种多合一的几何框架可以更好地进行维护和优化，并且可以更轻松地内插和应用所有原始变形。此外，我们在单个CGA表示中展示了这三种新颖的算法，可对输入装配模型进行切割，撕裂和钻孔，其中输出模型可在交互帧速率下进一步变形。这些接近实时切割，