Search NASA⌕ Search

SEARCH · Search NASA

Results for “Bezier curves”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Adapting Shape Parameters for Cubic Bezier Curves

Bezier curves are an established tool in Computer Aided Geometric Design. One of the drawbacks of the Bezier method is that the curves often bear little resemblance to their control polygons. As a result, it becomes increasingly difficult to obtain anything but a rough outline of the desired shape. One possible solution is tomanipulate the curve itself instead of the control polygon. The standard cubic Bezier curve form has introduced into it two shape parameters, gamma 1 and 2. These parameters give the user the ability to manipulate the curve while the control polygon retains its original form, thereby providing a more intuitive feel for the necessary changes to the curve in order to achieve the desired shape.

Isacoff, D.↗
NASA NTRS1990

Curvature continuity of cubic Bezier curves in the solid modeling aerospace research tools design software

Preliminary results are presented of an investigation into the development of a procedure to provide curvature continuity between biparametric cubic Bezier surface patches in the computer-aided design package known as SMART (Solid Modeling Aerospace Research Tools). This initial effort was aimed at providing the designer with the ability to locally impose curvature continuity at the intersection of two Bezier curves without disrupting either the curvature or slope continuity that may exist at the ends of these curves. Such a method was found if the original Bezier control points are all coplanar. If they are not, then it is possible to find a minimum deviation from exact curvature continuity. In cases where this is not sufficient, then an entire piecewise curve must be made curvature continuous simultaneously. A method was developed based on cubic splines which is very fast. The procedure returns new Bezier control points which have both slope and curvature continuity.

Roach, Robert L.↗
DOE OSTI2025

Working with Bezier Curves as bases for Functional Expansion Tallies

Functional expansion tallies (FETs) are a powerful tool for getting more information per history from Monte Carlo simulations, but in the past they have been constrained to orthogonal bases. Bezier curves, from computer aided design (CAD), could be well suited for FET due to their ability to assume many arbitrary shapes, but are non-orthogonal. Recent development has made non-orthongal FET possible. The convergence of B ´ezier curve FETs in both polynomial order and number of samples is explored. These bases are well suited for representing normal distributions, and opens the door to possible other CAD derived FET bases.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2016

A Safe Cooperative Framework for Atmospheric Science Missions with Multiple Heterogeneous UAS using Piecewise Bezier Curves

Autonomous operation of UAS holds promise for greater productivity of atmospheric science missions. However, several challenges need to be overcome before such missions can be made autonomous. This paper presents a framework for safe autonomous operations of multiple vehicles, particularly suited for atmospheric science missions. The framework revolves around the use of piecewise Bezier curves for trajectory representation, which in conjunction with path-following and time-coordination algorithms, allows for safe coordinated operations of multiple vehicles.

Mehdi, S. Bilal↗
NASA NTRS1985

When Is a Bernstein-bezier Curve the Graph of a Function?

The question of determining when a Bernstein-Bezier cubic curve in the plane can be represented as the graph of function in some fixed orthogonal coordinate system is considered. The notion of a curve being monotone in a given direction is introduced to aid in the analysis.

Mclaughlin, H. W.↗
NASA NTRS2016

Spline Trajectory Algorithm Development: Bezier Curve Control Point Generation for UAVs

A greater need for sophisticated autonomous piloting systems has risen in direct correlation with the ubiquity of Unmanned Aerial Vehicle (UAV) technology. Whether surveying unknown or unexplored areas of the world, collecting scientific data from regions in which humans are typically incapable of entering, locating lost or wanted persons, or delivering emergency supplies, an unmanned vehicle moving in close proximity to people and other vehicles, should fly smoothly and predictably. The mathematical application of spline interpolation can play an important role in autopilots' on-board trajectory planning. Spline interpolation allows for the connection of Three-Dimensional Euclidean Space coordinates through a continuous set of smooth curves. This paper explores the motivation, application, and methodology used to compute the spline control points, which shape the curves in such a way that the autopilot trajectory is able to meet vehicle-dynamics limitations. The spline algorithms developed used to generate these curves supply autopilots with the information necessary to compute vehicle paths through a set of coordinate waypoints.

Howell, Lauren R.↗
NASA NTRS2018

Silhouette-Informed Trajectory Generation Through a Wire Maze for Small UAS

Current rapidly-exploring random tree (RRT) algorithms rely on proximity query packages that often include collision checkers, tolerance verification, and distance computation algorithms for the generation of safe paths. In this paper, we broaden the information available to the path-planning algorithm by incorporating silhouette information of nearby obstacles in conflict. A silhouette-informed tree (SIT) is generated through the flight-safe region of a wire maze for a single unmanned aerial system (UAS). The silhouette is used to extract local geometric information of nearby obstacles and provide path alternatives around these obstacles. Thus, focusing the search for the generation of new tree branches near these obstacles, and decreasing the number of samples required to explore the narrow corridors within the wire maze. The SIT is then processed to extract a path that connects the initial location of the UAS with the goal, reduce the number of line segments in this path if possible, and smooth the resulting path using Pythagorean Hodograph Bezier curves. To ensure that the smoothed path remains in the flight-safe region of the configuration space, a tolerance verification algorithm for Bezier curves and convex polytopes in three dimensions is proposed. Lastly, temporal specifications are imposed on the smoothed path in the shape of an arbitrary speed profile.

Puig-Navarro, Javier↗
NASA NTRS1985

A Uniform Subdivision Method for Triangular Bezier Patches

A subdivision method for Bezier triangles is presented that parallels the development of Lane-Riesenfeld for Bezier Curves. The method is developed from the mathematical description of the patch, and the resulting algorithm does not require evaluation of the blending functions. This method subdivides the triangle uniformly so that the lengths of the sides of the subtriangles are uniformly reduced. This method is a member of the class of methods first presented by Goldman, which contain subdivision methods that arise from degenerate subdivision of Bezier tetrahedra. It is shown that a certain method is a natural extension of the Lane-Riesenfeld results for Bezier curves.

Joy, K. I.↗
NASA NTRS1985

A Comparison of Three Curve Intersection Algorithms

An empirical comparison is made between three algorithms for computing the points of intersection of two planar Bezier curves. The algorithms compared are: the well known Bezier subdivision algorithm, which is discussed in Lane 80; a subdivision algorithm based on interval analysis due to Koparkar and Mudur; and an algorithm due to Sederberg, Anderson and Goldman which reduces the problem to one of finding the roots of a univariate polynomial. The details of these three algorithms are presented in their respective references.

Sederberg, T. W.↗
NASA NTRS1984

An urnful of blinding functions

There is a fundamental connection between the mathematical theory of discrete probability distributions and the parametric curves and surfaces of computer-aided geometric design. It is no accident that the blending functions of Bezier curves and surfaces have an obvious probabilistic interpretation, nor is it a coincidence that the normalized uniform B-spline basis functions also model a simple stochastic process. The link between probability and geometry, and how to exploit simple probabilistic arguments to derive many of the classical geometric properties of the parametric curves and surfaces currently in vogue in computer-aided geometric design is discussed. This probabilistic approach is also used to introduce many new types of curves and surfaces, and it is demonstrated how probability theory can be used to simplify, unify, and generalize many well-known results.

Goldman, R. N.↗
NASA NTRS1992

Surface generation and editing operations applied to structural support of aerospace vehicle fuselages

The Solid Modeling Aerospace Research Tool (SMART) is a computer aided design tool used in aerospace vehicle design. Modeling of structural components using SMART includes the representation of the transverse or cross-wise elements of a vehicle's fuselage, ringframes, and bulkheads. Ringframes are placed along a vehicle's fuselage to provide structural support and maintain the shape of the fuselage. Bulkheads are also used to maintain shape, but are placed at locations where substantial structural support is required. Given a Bezier curve representation of a cross sectional cut through a vehicle's fuselage and/or an internal tank, this project produces a first-guess Bezier patch representation of a ringframe or bulkhead at the cross-sectional position. The grid produced is later used in the structural analysis of the vehicle. The graphical display of the generated patches allows the user to edit patch control points in real time. Constraints considered in the patch generation include maintaining 'square-like' patches and placement of longitudinal, or lengthwise along the fuselage, structural elements called longerons.

Schwartz, Susan K.↗
NASA NTRS1989

An interactive grid generation procedure for axial and radial flow turbomachinery

A combination algebraic/elliptic technique is presented for the generation of three dimensional grids about turbo-machinery blade rows for both axial and radial flow machinery. The technique is built around use of an advanced engineering workstation to construct several two dimensional grids interactively on predetermined blade-to-blade surfaces. A three dimensional grid is generated by interpolating these surface grids onto an axisymmetric grid. On each blade-to-blade surface, a grid is created using algebraic techniques near the blade to control orthogonality within the boundary layer region and elliptic techniques in the mid-passage to achieve smoothness. The interactive definition of bezier curves as internal boundaries is the key to simple construction. This procedure lends itself well to zonal grid construction, an important example being the tip clearance region. Calculations done to date include a space shuttle main engine turbopump blade, a radial inflow turbine blade, and the first stator of the United Technologies Research Center large scale rotating rig. A finite Navier-Stokes solver was used in each case.

Beach, Timothy A.↗
NASA NTRS1990

An interactive grid generation procedure for axial and radial flow turbomachinery

A combination algebraic/elliptic technique is presented for the generation of three dimensional grids about turbo-machinery blade rows for both axial and radial flow machinery. The technique is built around use of an advanced engineering workstation to construct several two dimensional grids interactively on predetermined blade-to-blade surfaces. A three dimensional grid is generated by interpolating these surface grids onto an axisymmetric grid. On each blade-to-blade surface, a grid is created using algebraic techniques near the blade to control orthogonality within the boundary layer region and elliptic techniques in the mid-passage to achieve smoothness. The interactive definition of bezier curves as internal boundaries is the key to simple construction. This procedure lends itself well to zonal grid construction, an important example being the tip clearance region. Calculations done to date include a space shuttle main engine turbopump blade, a radial inflow turbine blade, and the first stator of the United Technologies Research Center large scale rotating rig. A finite Navier-Stokes solver was used in each case.

Beach, T. A.↗
NASA NTRS1991

A PC-based inverse design method for radial and mixed flow turbomachinery

An Inverse Design Method suitable for radial and mixed flow turbomachinery is presented. The codes are based on the streamline curvature concept; therefore, it is applicable for current personal computers from the 286/287 range. In addition to the imposed aerodynamic constraints, mechanical constraints are imposed during the design process to ensure that the resulting geometry satisfies production consideration and that structural considerations are taken into account. By the use of Bezier Curves in the geometric modeling, the same subroutine is used to prepare input for both aero and structural files since it is important to ensure that the geometric data is identical to both structural analysis and production. To illustrate the method, a mixed flow turbine design is shown.

Skoe, Ivar Helge↗
NASA NTRS1992

An interactive grid generation technique for turbomachinery

A combination algebraic/elliptic technique is presented for the generation of 3-D grids about turbomachinery blade rows for both axial and radial flow machinery. The technique is build around use of an advanced engineering workstation to construct several 2-D grids interactively on predetermined blade-to-blade surfaces. A 3-D grid is generated by interpolating these surface grids onto an axisymmetric grid. On each blade to blade surface, a grid is created using algebraic techniques near the blade to control orthogonality within the boundary layer region and elliptic techniques in the mid-passage to achieve smoothness. The interactive definition of bezier curves as internal boundaries is the key to simple construction. The approach is adapted for use with the average passage solution technique, although this is not a limitation for most other uses. A variety of examples are presented.

Beach, Tim↗
NASA NTRS1992

Grid generation for 3D turbomachinery configurations

A numerical grid generation algorithm associated with the flow field about turbomachinery geometries is presented. Graphical user interface is developed with the FORMS Library to create an interactive, user-friendly working environment. This customized algorithm reduces the man-hours required to generate a grid associated with turbomachinery geometry, as compared to those required by general-purpose grid generation softwares. Bezier curves are utilized both interactively and automatically to accomplish grid line smoothness and orthogonality. Graphical user interactions are provided in the algorithm and allow the user to design and manipulate the grid lines with a mouse.

Shih, Ming-Hsin↗
NASA NTRS1993

Bgridi - Interactive three-dimensional turbomachinery grid generation system with applications

An interactive, graphics-based grid generation system for 3D turbomachinery geometries has been developed. The system consists of separate modules for geometry modeling and grid generation. The grid generation method generates a series of 2D grids in the blade-to-blade passage to build up the 3D grid. A Poisson equation with forcing functions selected to control orthogonality and spacing on all boundaries is solved to generate the grid. Interactive definition of Bezier curves and surfaces as internal boundaries is used to improve control of grid quality. A multi-block data structure simpifies the creation of structured H-grids about complex turbomachinery geometries. The multi-block strategy facilitates the creation of a grid in the tip flow region, which is an important contributor to losses within the passage. The grid generation system is applied to several complex geometries including a simple radial impeller, a conventional turbine rotor, the SSME LOX turbine rotor with and without a tip flow grid, a multi-passage inducer-impeller, and a tip-flow cascade. In addition, a calculation of the tip-flow cascade flow field is shown.

Shoemaker, J. M.↗
NASA NTRS1992

TIGER: A graphically interactive grid system for turbomachinery applications

Numerical grid generation algorithm associated with the flow field about turbomachinery geometries is presented. Graphical user interface is developed with FORMS Library to create an interactive, user-friendly working environment. This customized algorithm reduces the man-hours required to generate a grid associated with turbomachinery geometry, as compared to the use of general-purpose grid generation softwares. Bezier curves are utilized both interactively and automatically to accomplish grid line smoothness and orthogonality. Graphical User Interactions are provided in the algorithm, allowing the user to design and manipulate the grid lines with a mouse.

Shih, Ming-Hsin↗