Intersection Curves Of Surfaces Research Articles

Singular Points of Curves

In this paper, we consider the key problem of geometric modeling, connected with the construction of the intersection curves of surfaces. Methods for constructing the intersection curves in complex cases are found: by touching and passing through singular points of surfaces. In the first part of the paper, the problem of determining the tangent line of two surfaces given in parametric form is considered. Several approaches to the solution of the problem are analyzed. The advantages and disadvantages of these approaches are revealed. The iterative algorithms for finding a point on the line of tangency are described. The second part of the paper is devoted to methods for overcoming the difficulties encountered in solving a problem for singular points of intersection curves, in which a regular iterative process is violated. Depending on the type of problem, the author dwells on two methods. The first of them suggests finding singular points of curves without using iterative methods, which reduces the running time of the algorithm of plotting the intersection curve. The second method, considered in the final part of the article, is a numerical method. In this part, the author introduces a function that achieves a global minimum only at singular points of the intersection curves and solves the problem of minimizing this function. The application of this method is very effective in some particular cases, which impose restrictions on the surfaces and their arrangement. In conclusion, this method is considered in the case when the function has such a relief, that in the neighborhood of the minimum point the level surfaces are strongly elongated ellipsoids. All the images given in this article are the result of the work of algorithms on methods proposed by the author. Images are built in the author’s software environment.

Reduced curvature formulae for surfaces, offset surfaces, curves on a surface and surface intersections

We introduce the concept of reduced curvature formulae for 3-D space entities (surfaces, curves). A reduced formula entails only derivatives of the functions involved in the entity's representation and admits no further algebraic simplifications. Although not always the most compact, reduced curvature formulae entail only basic arithmetic operators and are more efficient computationally compared to alternative unreduced formulae. Reduced formulae are presented for the normal, mean and Gaussian curvatures of a surface and the curvature of curves on a surface, where each surface or curve on a surface may be defined parametrically or implicitly. Reduced formulae are also presented for the curvature of surface intersection curves, where each of the intersecting surfaces may be a given surface or an offset of a given surface and each given surface may be defined parametrically or implicitly. Known formulae are cited, without derivation, to form a collection, in one place, of new and of known results scattered in the literature. Each curve curvature formula is presented together with a formula for the respective binormal vector, from which formulae for the Frenet frame and torsion of the curve can be derived.

Построение линии пересечения некоторых сложных поверхностей

The paper considers various ways to construct the
 intersection curves of surfaces, available in the scientific literature.
 The analysis shows that the exact build intersection surfaces finds
 extensive practical application when designing constructions from
 sheet material, which must be built. In this case, the accuracy of
 manufacturing elements construction depends largely on the accuracy
 of the projections of the intersection of surfaces on an integrated
 drawing. In this work the examples build the intersection
 of several complex surfaces 2nd order among themselves and with
 the curve line using 2D and 3D technologies. In this case, the accuracy
 of manufacturing elements construction depends largely on
 the accuracy of the projections of the intersection of surfaces on
 an integrated drawing. In this work the examples build the intersection
 of several complex surfaces 2nd order among themselves and
 with the curve line using 2D and 3D technologies. Provides an
 algorithm of building solid model of certain surfaces, which significantly
 reduces the time to solving problems at the intersection
 of the surfaces, making visibility into the solution, create a realistic
 model of the object. In General, this work demonstrates that
 one of the stages of further reform in teaching descriptive geometry
 is the evolutionary path of development, outlining the many issues
 (e.g., intersection of surfaces) using electronic kuhlmann. In this
 case, students will receive a decision not by pushing buttons, and
 applying the methods of descriptive geometry, simultaneously developing
 spatial imagination and computer skills are already in the
 first year. And the final stage of teaching descriptive geometry will
 transition to 3D modelling, when the above tasks will instantly and
 clearly resolved by simply pressing buttons.

An Approach for Extracting Neighborhood Points of Surface Intersection Curves

This paper comes up with an approach for surface reconstruction from arbitrary, unorganized and intersecting curves. First of all, camera calibration determines the interior and exterior camera parameters. Based on laser triangulation measurement, the point in world coordinate system can be computed. The extracted curves are processed in a volumetric in which a vector field is built and updated to compute the neighbor-hood points. Then the surface reconstruction can be obtained with MC algorithm. This approach makes surface reconstruction and curves registration a unified procedure. This algorithm solves the problem of linear complexity respect to input curves, and makes it suitable for interactive modeling. The experiment verifies its available, reduces the effect of registration errors and has a good result.

On the curvature of curves and surfaces defined by normalforms

The normalform h=0 of a curve (surface) is a generalization of the Hesse normalform of a line in R 2 (plane in R 3 ). It was introduced and applied to curve and surface design in recent papers. For determining the curvature of a curve (surface) defined via normalforms it is necessary to have formulas for the second derivatives of the normalform function h depending on the unit normal and the normal curvatures of three tangential directions of the surface. These are derived and applied to visualization of the curvature of bisectors and blending curves, isophotes, curvature lines, feature lines and intersection curves of surfaces. The idea of the normalform is an appropriate tool for proving theoretical statements, too. As an example a simple proof of the Linkage Curve Theorem is given.

Intersection curves of surfaces

Deals with the development of algorithms to calculate the penetration curves of intersecting bodies. Although starting from ideas of a finite element environment where such procedures can be very helpful when generating complicated meshes, the algorithms also cover requirements of the describing geometry or computer graphics.

Marching along surface⧸surface intersection curves with an adaptive step length

A method for generating points on the intersection of two C2 smooth parametric surfaces is presented. In each generated point the tangent and the curvature of the intersection curve are obtained from the surface positions, first and second derivatives. Initial approximation of the next point lies on a parabola approximating the intersection curve in a vicinity of the last point found. The length of the parabola between the two points is evaluated so that its maximal deviation from the chord joining the points is not greater than a given deviation tolerance. The new point is relaxed to the intersection curve.

特異点近傍における曲面間交線の追跡

Robust differential equation solvers which can be applied in tracing intersection curves of surfaces are discussed and recommended for use in regions very near to singular points. And also behaviours of intersection curves around the singular points are analyzed, because the differential equations do not hold at the singular points where normal vectors of the two surfaces become parallel. With these two approaches tracing of intersection curves can be confidently performed even in critical cases and the tracing time is automatically adjusted in accordance with difficulty of the tracing. Numerical examples of critical cases are shown.

Geometric hermite approximation of surface patch intersection curves

This paper introduces two new tools for attacking the problem of approximating the intersection curve of two parametric surface patches: the Bézier clipping algorithm for curve⧸surface intersection, and geometric Hermite approximation of surface⧸surface intersection curves. The curve⧸surface intersection algorithm is used to compute a set of endpoint pairs for all components of the intersection curve. Thereafter, a G k piecewise parametric approximation of the intersection curve is directly computed with no further subdivision or marching. An error bound is determined directly from the approximation. If the error is too large, each unsatisfactory approximating curve is split in half and new approximations are made directly. This procedure is O( h 2 k+2 ) convergent, which means that each time the members of a G k sequence of approximating curves are split in half, the new error is 2 −(2 k+2) of the previous error (in the limit). Thus, doubling the number of approximating curves in a G 3 sequence reduces the error typically by two orders of magnitude (1⧸256).

Geometric approaches to nonplanar quadric surface intersection curves

Quadric surfaces occur frequently in the design of discrete piece parts in mechanical CAD/CAM. Solid modeling systems based on quadric surfaces must be able to represent intersection curves parametrically and in a fashion that allows the underlying surfaces to be partitioned. An algebraic approach originally developed by Levin meets these needs but is numerically sensitive and based on solutions to fourth-degree polynomial equations. In this paper we develop geometric approaches that are robust and efficient, and do not require solutions to polynomials of degree higher than 2.

Boundary evaluation of non-convex primitives to produce parametric trimmed surfaces

To integrate a CSG-based solid modeler into an existing wireframe/surface modeling system, new boundary evaluation technology has been developed. This scheme uses exact representations for the simple quadric surfaces and both exact and approximate representations of higher-order curved surfaces. It supports parametric primitives (box, wedge, sphere, cylinder, cone, torus), procedural primitives (extrusion, revolution, tube) and a sculptured surface primitive. The output includes curves, parametric trimmed surfaces, and a data structure of adjacency information.An existing boundary evaluator (PADL-2's) has been enhanced to allow a general non-convex faceted primitive with planar and quadric facets. This new hybrid evaluator combines two techniques for curve/primitive classification. PADL-2's existing halfspace-based classification is reserved for the simple convex primitives, and a new ray firing based classification is applied to the non-convex primitives. After evaluation, approximate intersection curves (from intersections involving higher order surfaces) are refined to a specified tolerance by exploiting an exact parametric representation of the surfaces of the primitives. The refined curves and the quadric surface intersection curves are used to create a parametric trimmed surface representation of the solid. This combination of techniques and representations offers advantages in accuracy, robustness and efficiency suitable to a production environment.