- Author:
- Jai Prakash Menon
Free-form ('sculptured') surfaces are traditionally represented as unions of parametric patches of high implicit degree. Recently low-degree algebraic patches have been introduced for representing free-form surfaces. This thesis draws on the new line of work on algebraic patches, and introduces, as a first result, a new representation for free-form surfaces called a Constructive Shell Representation (CSR). A CSR is a union of truncated tetrahedra, called trunctets, forming a 'thick shell' that contains the free-form surface. One bounding face of each trunctet is an algebraic patch which is a subset of the free-form surface; the other faces are planar.
CSRs for surfaces that are boundaries of (free-form) solids may be very useful for solid modeling, and this class of CSRs is the main focus of this thesis. Traditionally, Boundary representations (Breps) and Constructive Solid Geometry (CSG), and secondarily cell decompositions and enumeration (e.g. octrees), have been the popular schemes for representing solids unambiguously. CSRs combine useful properties of both Brep and CSG schemes, and provide, as a second result, a complete new hybrid Brep/CSG representation scheme for free-form solids.
A third result is methods for representing free-form solids in CSG, using Boolean combinations of CSR-subsets and other naturally induced flat-faced solids. (Free-form solids have been difficult to represent in CSG, mainly because the popularly used parametric patches do not associate naturally with the low-degree algebraic halfspaces on which CSG technology is based. CSRs finesse the problem because they are built using patches that are subsets of low-degree algebraic halfspaces.)
Other results in the thesis include CSR-based simplifications for line/solid classification, construction of ray representations for free-form solids, domain extensions for Brep $\to$ CSG conversion, and new algorithms for accelerating Brep $\to$ CSG conversion. As a consequence, CSRs help bring symmetric dual-rep $\langle$CSG, Brep$\rangle$ solid modeling systems to fruition.
One chapter is devoted to potential applications of CSRs for free-form surfaces that are not boundaries of solids; there are many open issues in this area. We show that well known, robust CSG algorithms may be used with 'general' surfaces represented by CSRs.
An experimental system, called Q scPATCH has been implemented to support CSRs of surfaces and CSR/CSG representations of solids. Q scPATCH exploits the RayCasting Engine (a highly parallel computer) to support modeling applications using ray representations.
Index Terms
- Constructive shell representations for free-form surfaces and solids
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