Rendering the intersections of implicit surfaces

The need to determine the lines or curves of intersections between two surfaces of similar or different geometric shapes occurs in the fabrication of ducts, boiler mountings, pipe fittings, aircraft and automobile bodies. If two geometric shapes with curved surfaces meet or penetrate each other, the lines of intersection is a curve. The curve of intersection is determined by plotting the projections of points which are common to both surfaces. This is done by choosing section planes which cut the surfaces in lines or circles. The intersections of these lines give points on the required curve.

Topological and differential-equation methods for surface intersections

This paper present an new algorithm for calculating intersections of parametric surface and implicit surface. An elevation function is obtained by inducing the parametric function into implicit function. Using our new proposed isoline extraction algorithm which based on tracing the `tangent vector' of the isoline starting at the initial point, we obtain the zero isolines of elevation function in parameter domain. The intersection of parametric surface and implicit surface is achieved by projecting the zero isolines onto the parametric surface.

The Intersection of Two Ringed Surfaces and Some Related Problems

Implicit representation of parametric curves and surfaces

An algorithm for drawing pictures of three-dimensional objects, with surfaces made up of patches of quadric surfaces, is described.The emphasis of this algorithm is calculating the intersections of quadric surfaces. A parameterization scheme is used. Each quadric surface intersection curve (QSIC) is represented as a set of coefficients and parameter limits. Each value of the parameter represents at most two points, and these may easily be distinguished. This scheme can find the co-ordinates of points of even quartic (fourth-order) intersection curves, using equations of no more than second order.Methods of parameterization for each type of QSIC are discussed, as well as surface bounding and hidden surface removal.

This paper discusses an attempt to devise an efficient (involving minimal computations), accurate (numerically high precision), exhaustive (detecting all possible solutions), and robust (working without failures) method for detecting intersection of two parametric surfaces. The method starts with subdivision to ensure that all solutions are detected. Later it switches over to numerical iterative refinement for efficient and accurate evaluation of the intersection curve. The switching takes place only when the convergence of the refinement method is guaranteed. The necessary theory to arrive at a computable condition leading to this guarantee has been developed using fixed-point and contractionmapping theorems from topology and mathematical analysis. The implementation is discussed elaborating the data structures and the algorithms used for (1) detecting segments of the intersection curve, (2) generating points on these segments using refinement, and (3) tracing a continuous curve by identifying neighboring segments and joining them in order.

The paper integrated genetic algorithm and marching method into a novel algorithm to solve the surface intersection problem. By combining genetic algorithm with local searching method the efficiency of evolution is greatly improved. By fully utilizing the global searching ability and instinct attribute for parallel computation of genetic algorithm and the local rapid convergence of marching method, the algorithm can compute the intersection robustly and generate correct topology of intersection curves. The details of the new algorithm are discussed here. The algorithm have been implemented in a prototype system named TigerSurf based on Windows/NT platform, and a soundly result is gotten from test datum.

A fourth-order topological invariant of magnetic or vortex lines

How to obtain intersection of curves and surfaces is a fundamental problem in many areas such as computer graphics, CAD/CAM, computer animation, and robotics. Especially, how to deal with singular cases, such as tangency or superposition, is a key problem in obtaining intersection results. A method for solving the intersection problem based on the coordinate transformation is presented. With the Lagrange multiplier method, the minimum distance between the center of a circle and a quadric surface is given as well. Experience shows that the coordinate transformation could significantly simplify the method for calculating intersection results. The location of the tangent point is revised according to the tangency condition. It can improve the stability of the intersection of given curves and surfaces in singularity cases. The new algorithm is applied in a three dimensional CAD software (GEMS), produced by Tsinghua University.

Algebraic methods for intersections of quadric surfaces in GMSOLID

There are many space subdivision and space partitioning techniques used in many algorithms to speed up computations. They mostly rely on orthogonal space subdivision, resp. using hierarchical data structures, e.g. BSP trees, quadtrees, octrees, kd-trees, bounding volume hierarchies etc. However in some applications a non-orthogonal space subdivision can offer new ways for actual speed up. In the case of convex polygon in E2 a simple Point-in-Polygon test is of the O(N) complexity and the optimal algorithm is of O(log N) computational complexity. In the E3 case, the complexity is O(N) even for the convex polyhedron as no ordering is defined.New Point-in-Convex Polygon and Point-in-Convex Polyhedron algorithms are presented based on space subdivision in the preprocessing stage resulting to O(1) run-time complexity. The presented approach is simple to implement. Due to the principle of duality, dual problems, e.g. line-convex polygon, line clipping, can be solved in a similarly.

Solving polynomial systems using no-root elimination blending schemes

The authors propose that recursive subdivision techniques be used for some of these tasks and for some special subclasses. Specifically, they advocate the use of interval computation methods that treat the parametric range as an interval. These techniques can be extended to evaluating piecewise continuously differentiable functions over intervals. Also presented is a general schema for designing algorithms using interval evaluation of functions. The power of this methodology is illustrated through the design of algorithms for commonly required processing tasks on geometric objects.

View Video Presentation: https://doi.org/10.2514/6.2023-0911.vid Higher-Order finite-element CFD methods have the potential to reduce the computational cost to achieve a desired solution error. These techniques have been an area of research for many years and are becoming more widely available in popular CFD codes. CFD visualization software is lagging behind the development of higher-order CFD analysis codes. This paper discusses a technique for visualizing isosurfaces and streamlines in higher-order element solutions with reduced memory usage. The technique recursively subdivides higher-order elements into smaller linear sub-elements where the isosurface can be extracted using standard marching-tets techniques. Memory usage is minimized by discarding unneeded sub-elements. In a previous paper this technique was demonstrated with higher-order quadratic hexahedra, tetrahedra, prism, and pyramid elements with Lagrangian polynomial basis functions. In this paper, the technique is extended to cubic elements and streamlines, and performance optimization are discussed. The results are compared to other techniques for visualization of higher-order element isosurfaces.