Exploring New Geometry by Touching, Seeing, and Feeling

日常生活中对几何形状的理解结合了视觉印象、触觉感受，以及它们在看到某物并伸手触摸它以确定其物理性质的过程中的统一，这可能会传达新的信息。 交互式计算机图形系统可以与3D触摸响应（触觉）界面相结合，不仅允许模拟我们熟悉的3D世界，本研究的重点是利用交互式图形学与触觉学相结合的方法，开发研究几何结构的新方法，理解力超出了人类的正常能力。 结果将以教学动画的形式以及用于创建这些动画的软件环境提供。典型的问题包括直接操纵和研究当3D结从它所处的空间的固体块中移除时产生的可变形体积，留下结的补充，以及对复杂碰撞的探索自由变形的表面在4D空间是必不可少的研究性质的两个复杂的变量。预期的结果，追求这样的方法将提高人类的能力和直觉有关的具体挑战几何可视化问题，提供方法，使科学问题的发现有关的复杂几何结构，否则可能不会被问到，并最终推进整个概念，使人类的理解使用计算机为基础的技术。

The understanding of geometric shapes in everyday life combines visual impressions, tactile sensation, and their unification in the process of seeing something and reaching out to touch it to determine its physical properties, which may convey new information. Interactive computer graphics systems can be combined with 3D-touch-responsive (haptic) interfaces to permit not only the simulation of our familiar 3D world, but also the extension of our available sensations to additional dimensions and unfamiliar geometric objects by using the abstract power of computer modeling.This research focuses on exploiting interactive graphics combined with haptics to develop new methods for investigating geometric structures whose understanding is beyond the normal capabilities of the unaided human. Results will be made available in the form of pedagogical animations as well as the software environments used to create them. Typical problems include the direct manipulation and study of deformable volumes resulting when a 3D knot is removed from the solid block of space it lives in, leaving the knot complement, and the exploration of the intricate collision-free deformations of surfaces in 4D space that are essential for studying the properties of functions of two complex variables.The anticipated outcome of pursuing such approaches will be to enhance human capabilities and intuition pertaining to specific challenging geometric visualization problems, to provide methods enabling the discovery of scientific questions related to complex geometric structures that might not otherwise have been asked, and ultimately to advance the entire concept of empowering human understanding using computer-based technology.