Perturbation functions and operations in geometric modeling

Abstract

The free forms based on the analytical perturbation functions have an advantage of spline representation of surfaces, that is, a high degree of smoothness, and an advantage of arbitrary form for a small number of perturbation functions. We have investigated geometric operations on functionally defined objects on the basis of the perturbation functions. We have analyzed the collision detection algorithm by means of recursive object space subdivision. The free-form representation created by mean of the analytical perturbation functions have the following advantages: fewer surfaces for mapping curvilinear objects, short database description, fewer operation for geometric transformations and data transfer, simple animation and deformation of objects and surfaces. For shape creating we propose a set of algorithms and software based on function-defined surfaces that perform an interactive rate and enable intuitive operations. Interactive modification of the function model with fast visualization allows us to provide both the interactivity and any required level of detail leading to a photo-realistic appearance of the resulting shapes. Our investigation in the volume-oriented visualization technology has made it possible to reveal some advantages in both the scene representation technique and the rendering algorithm. The main merits of our approach are the following: reduced number of surfaces for describing curvilinear objects (representation of objects by free-form surfaces reduces 100 times and more the database description compared with their representation by polygons); efficiency of the masking rendering technique combining simple computationwith fast search and discard of spaces out of the scene objects; the possibility to process voxel arrays bounded by freeformsurfaces; reduction of the load on the geometry processor and decrease of data flow from it to the raster subsystem; simple animation and morphing of scenes. The proposed visualization algorithm along with the possibility to visualize arbitrary surfaces of free-forms and inhomogeneous volume spaces offers a wide scope of application. The free-form representation has a wide spectrum of applications (interactive graphics systems for visualizing functionally defined objects, CAD 3-D simulation systems, 3-D web visualization, prototyping, etc.). More effective tools for designing, manipulating, and deforming free-form3-D shapes are needed in CAD, animation, and virtual reality applications.

Вільні форми, засновані на аналітичних функціях збурень, мають перевагу сплайн-представлення поверхонь, тобто високий ступінь гладкості і перевага довільної форми для невеликої кількості функцій збурень. Ми дослідили геометричні операції на функціонально визначених об'єктах на основі функцій збурень. Ми проаналізували алгоритм виявлення зіткнень за допомогою рекурсивного підрозділу космічного об'єкта.