Lesson plan
Surface fairing
In mathematics, Surface fairing is an aspect of mesh smoothing. The goal of surface fairing is to compute shapes that are as smooth as possible. On an abstract level, mesh smoothing is concerned with the design and computation of smooth functions f : S \rightarrow \R^d on a triangle mesh. Mesh fairing does not just slightly smooth the function f in order to remove the high frequency noise. It also smooths the function as much as possible in order to obtain, e.g., an as-smooth-as-possible surface patch or an as-smooth-as-possible shape deformation. How to actually measure smoothness or fairness obviously depends on the application, but in general fair surfaces should follow the principle of simplest shape: the surface should be free of any unnecessary details or oscillations. This can be modeled by a suitable energy that penalizes unaesthetic behavior of the surface. A minimization of this fairness energy—subject to user-defined constraints—eventually yields the desired shape. Example applications include the construction of smooth blend surfaces and hole filling by smooth patches.
Article summary
In mathematics, Surface fairing is an aspect of mesh smoothing. The goal of surface fairing is to compute shapes that are as smooth as possible. On an abstract level, mesh smoothing is concerned with the design and computation of smooth functions f : S \rightarrow \R^d on a triangle mesh. Mesh fairing does not just slightly smooth the function f in order to remove the high frequency noise. It also smooths the function as much as possible in order to obtain, e.g., an as-smooth-as-possible surface patch or an as-smooth-as-possible shape deformation. How to actually measure smoothness or fairness obviously depends on the application, but in general fair surfaces should follow the principle of simplest shape: the surface should be free of any unnecessary details or oscillations. This can be modeled by a suitable energy that penalizes unaesthetic behavior of the surface. A minimization of this fairness energy—subject to user-defined constraints—eventually yields the desired shape. Example applications include the construction of smooth blend surfaces and hole filling by smooth patches.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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