Geodesics on Tensor Product Surfaces: Modeling and Calculation
Explore the concept of geodesics on tensor product surfaces, including the calculation methods and applications. Learn about the challenges involved in determining geodesics and the strategies to approximate them efficiently. This article discusses techniques such as path discretization, optimization, and reparameterization for computing geodesics on surfaces. Discover how machine learning can enhance the speed of geodesic calculations and compare various approaches in geometric modeling.
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Presentation Transcript
GEODESICS ON TENSOR PRODUCT SURFACES BETHANY WITEMEYER GEOMETRIC MODELING DECEMBER 15, 2021
WHAT IS A GEODESIC? A shortest path on a surface Typically found using a differential equation on continuous surfaces Public Domain, https://commons.wikimedia.org/w/index.php?curid=647033
THE GOAL Geodesics have many applications, but are hard to calculate Calculate geodesics on tensor product surfaces ? ?=0 ? ?=0 ??,???(?)??(?) Calculate approximate geodesics by discretizing the path and using the straight-line distance Minimize the length of the path by moving the (u,v) coordinates of the intermediate points
THE SOLUTION Subdivide the path segments Don t add the new points to the optimization problem Forces the path to be closer to the actual surface distance
CONCLUSIONS We can calculate approximate geodesics on a tensor product surface! Reparameterize the path using arc length Use machine learning to speed up the calculation Compare other ways of calculating geodesics
