Computational geometry: an introduction
Computational geometry: an introduction
Computer Aided Geometric Design - Special issue: Topics in CAGD
Loop detection in surface patch intersections
Computer Aided Geometric Design
Computer Aided Geometric Design
Improved test for closed loops in surface intersections
Computer-Aided Design
Solid shape
Solutions of tangential surface and curve intersections
Computer-Aided Design
Analytic properties of plane offset curves
Computer Aided Geometric Design
A marching method for parametric surface/surface intersection
Computer Aided Geometric Design
A surface intersection algorithm based on loop detection
SMA '91 Proceedings of the first ACM symposium on Solid modeling foundations and CAD/CAM applications
Visualization and modeling contours of trivariate functions
Visualization and modeling contours of trivariate functions
Computing the differential characteristics of isointensity surface
Computer Vision and Image Understanding
Differential geometry of intersection curves of two surfaces
Computer Aided Geometric Design
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Optimal slicing of free-form surfaces
Computer Aided Geometric Design
Comparing Offset Curve Approximation Methods
IEEE Computer Graphics and Applications
GMP '04 Proceedings of the Geometric Modeling and Processing 2004
Determining the Geometry of Boundaries of Objects from Medial Data
International Journal of Computer Vision
Computation of the solutions of nonlinear polynomial systems
Computer Aided Geometric Design
Computation of singularities and intersections of offsets of planar curves
Computer Aided Geometric Design
Curvature formulas for implicit curves and surfaces
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Sliding windows algorithm for B-spline multiplication
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Swept regions and surfaces: Modeling and volumetric properties
Theoretical Computer Science
| Hi-index | 0.00 |
This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for continuously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. This paper presents the mathematical framework, and develops algorithms accordingly, to continuously and robustly track the intersection curves of two deforming parametric surfaces, with the deformation represented as generalized offset vector fields. The set of intersection curves of 2 deforming surfaces over all time is formulated as an implicit 2-manifold $\mathcal{I}$ in the augmented (by time domain) parametric space $\mathbb R^5$. Hyper-planes corresponding to some fixed time instants may touch$\mathcal{I}$ at some isolated transition points, which delineate transition events, i.e., the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of 5 constraints in 5 variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper-tangent bounding cones. The actual transition events are computed by contouring the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the euclidean space in which the surfaces are embedded.