My PhD thesis was on implicit surfaces. Looking back 1990's, I realize now that I was always interested in creating high genus shapes. During 1990's, we only had implicit surfaces and solid modeling to provide such high genus models. I believed that implicit surface design must be like parametric surface design with control meshes and guaranteed interactivity.
During my PhD, I developed a piecewise implicit framework to provide control meshes. Although, the framework was mathematically interesting, for all practical/engineering purposes, it was too complicated. I have improved and simplified the underlying mathematics and I have eventually achieved interactivity for a certain type of shapes and showed that it is possible to guarantee interactivity with general implicit shapes.
Theoretical Framework: After writing these two papers, I stopped working on Implicit Surfaces. One reason was that these two papers were a good closure. They introduced the theoretical framework that is needed. They showed that what is needed to guarantee interactivity for modeling any general shape with implicit techniques. In 1999, all we needed faster computers to handle reconstruction time. Moreover, with rising popularity of subdivision surfaces, It was hard to motivate any student to work on implicit surfaces. I eventually moved to topological mesh modeling, which can work in sync with subdivision schemes.
| Yildiz, Tolga Talha, Vinayak Krishnamurthy, and Ergun Akleman. "Hölder–Minkowski transcendental operators for associative averaging in implicit modeling." Computer-Aided Design (2026): 104159. Website
Description: Minkowski-type constructions are widely used in implicit modeling as smooth approximations of minimum and maximum operators. Existing smooth implicit operators can be algebraically continuous. Still, they often fail to produce a bounded geometric path between the union and the intersection, and they can fail with repeated use of identical primitives. This paper introduces Hölder–Minkowski transcendental operators, a log-homogeneous reformulation that lifts signed implicit functions to a positive homogeneous space, performs associative aggregation, and projects the result back to the real domain. The resulting operators provide continuous, bounded interpolation between union- and intersection-like behavior while remaining compatible with signed implicit functions. The construction is associative for hierarchical composition and idempotent at both the function and shape levels, preventing repeated primitives from causing uncontrolled growth, shrinkage, or disappearance. Together, these properties support stable exploration of continuous families of implicit shapes while preserving the practical min/max limits of classical set operations. See Website to play several applications to learn how it works and compare with our earlier implicit methods.
|
| E. Akleman, J. Chen, "Constant Time Updateable
Operations for Implicit Shape Modeling",
Proceedings of Implicit Surfaces'99, Bordeaux, France,
pp. 73-80, September 1999. (Extended Abstract)
(Paper)
Description: This paper shows additional capabilities of our constant time updateable operations. If you construct functions using these operations, the resulting functions provide implicit surfaces that can be rendered in computationally efficient way. Moreover, it is possible to create shaders still using similar functions. See various software applications to understand how it works.
|
| E. Akleman, J. Chen, "Coupled Modeling of
Implicit Shapes and Solid Textures", Proceedings
of Implicit Surfaces'99, Bordeaux, France,
pp. 89-95, September 1999. (Extended Abstract)
(Paper)
Description:In this paper, Jianer and I showed that to guarantee interactivity the operations must be associative. We have introduced associative operators that can be an alternative for Ricci and Rvachev operators. The importance of constant time updateability has recently been realized again. These operators will eventually be popular. |
Interactive Implicits: These are papers about interactive software I have developed.
| E. Akleman, "Interactive Construction of
Ray-Quadrics", Proceedings of Implicit
Surfaces'98, Seattle, 105-114, June 1998. (Extended Abstract)
(Paper)
Description: This paper seems to be natural extension of 96 Graphics Interface and 98 Implicit Surfaces papers. However, there was a need for several new innovations in terms of polygonization. The software is completed in 1996. I wrote the paper one year later. |
| E. Akleman, "Interactive Construction of Smoothly
Blended Star Solids", Proceedings of Graphics
Interface'96, Toronto, Canada, pp. 159-167, June 1996.
(Extended Abstract)
(Paper)
Description: This paper was my first real breakthrough. I was able to create faces interactively with implicit surfaces. The paper introduces many innovations. I wrote the functions such a way that I was able to convert an implicit surface to a polygonal surface and render it in real time with GL in 1995. I have also introduced an interface concept similar to control vertices of parametric surfaces. |
Other Supporting Papers:
| E. Akleman, J. Chen, "Generalized Distance
Functions", Proceedings of Shape Modeling'99,
Aizu, Japan, pp. 72-79, March 1999.
(Extended Abstract)
(Paper)
Description: This paper basically says that all convex shapes can be used to define distance functions. We did not really explained that way. We basically generalized Minkovski's equations.
|
| E. Akleman, "Ray-Quadrics", Proceedings of Implicit
Surfaces'96, Eindhoven, Netherland, pp. 89-98, October 1996.
(Extended Abstract)
(Paper)
Description: This was the first paper I was able to create holes. The thumbnail animation cannot be created with my interactive method, but it is a ray-quadric shape. It is computationally efficient to convert such implicit surfaces to polygonal meshes. |
Early Implicit Papers: I have hardcopies of these papers, but, original latex files are lost with my apple computer. If I can find originals or convert the hardcopes to pdf's, I will include these papers.
| E. Akleman, "Generalized Morphing Operations over
Implicitly Represented Shapes", Proceedings of
First Bilkent Computer Graphics Conference
on Advanced Techniques in Animation, Rendering,
and Visualization, pp. 143-149, August 1993.
Description: This was a morphing method between several ray-linear implicit shapes. |
|
| E. Akleman, "Construction of Convex and Star Shapes By Using Yontsal Functions", Proceedings of ISCIS'93, 10th International Symposium on Computer and Information Sciences, pp. 181-190 November 1993.
Description: This paper is an early ray-linears. |
|
