Freeform Surface Research Papers - Academia.edu (original) (raw)

Minkowski sums have numerous applications in robot path planning, collision detection, assembly planning and more. While there are algorithms and implementations for computing Minkowski sums of polygons, implementing such algorithms for freeform geometry (e.g., B-spline curves) presents new and difficult challenges.
In this work we have implemented algorithms for the computation of Minkowksi
sums of freeform curves. We present our symbolic representation of the boundary of the Minkowski sum as an implicit bivariate polynomial equation in the parameter space.
The solution is a univariate curve in the parameter space that is mapped to a curve in the Euclidean plane. This curve is further processed to remove unwanted branches. The computations required for this processing are performed by defining conditions on the curves, and applying a constraint solver to compute the result based on these conditions.
We extend our results to the computation of Minkowski sums of freeform surfaces in R3. We represent the boundary of the Minkowski sum as a system of implicit multivariate polynomial equations in the parameter space. The solution is a bivariate manifold in the parameter space, which is sampled and mapped to the Euclidean space.
Finally, we show how to compute a surface Minkowski sum that is constrained to a plane. Unlike the full surface Minkowski sum, the solution to this problem is a univariate curve and therefore easier to process. The result can be applied, for example, in path planning of a freeform robot moving on a floor while avoiding three-dimensional freeform obstacles.
Keywords: Minkowksi sums, freeform geometry

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- Robotics (Computer Science), Computational Geometry, Robot Path Planning, Freeform Surface

We have developed a deconvolution freeform lens array design approach to generate high-contrast structured
light illumination patterns. This method constructs the freeform lens array according to the point
response obtained by deconvoluting the prescribed illumination pattern with the blur response of the
extended light source. This design method is more effective in designing a freeform lens array to achieve
accurate structured light patterns. For a sinusoidal fringe pattern, the contrast ratio can be as high as
97%, compared to 62% achieved by the conventional ray mapping method

- by Donglin Ma
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- LED Lighting, Deconvolution, Geometrical Optics, Non-Imaging Optics

This paper focuses on the manufacture of architectural free-form GRC panels with 3D printed internal grid. Since double-curvature formworks and molds could account for up to 40 to 80% of the production cost of the pieces, especially in small series product, avoiding these casting requirements is a highly attractive prospect economically. In this paper a feasible, moldless, 3D printing method combined with a Glass Reinforced Concrete alternative is considered as a solution to this drawback. Furthermore, it explores current methods involved in free-form GRC panel production, as well as the approach to additive manufacturing as an alternative for eluding formworks. Thus, GRC and 3D printing processes are described in detail and then evaluated as complementary techniques in this hybrid technology. Resumen Este artículo se centra principalmente en la fabricación de paneles arquitectónicos de forma libre de GRC, con una malla interna impresa en 3D. Debido a que los moldes y los encofrados de doble curvatura representan entre el 40 al 80% del precio de producción de los paneles, la opción de evitar el uso del molde, especialmente en pequeñas series, es una idea muy atractiva económicamente. Por lo tanto, para resolver dicho inconveniente, se presenta un método fiable, sin necesidad de molde, con impresión 3D, y combinando con hormigón reforzado con fibra de vidrio. Además, se exploran los métodos actuales que están involucrados en la producción de paneles de GRC, así como también la aproximación de la manufactura aditiva, como una alternativa para evitar los encofrados. Los procesos de GRC y de impresión 3D son descritos en detalle, y posteriormente evaluados como técnicas complementarias en esta tecnología híbrida.

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be ap- plied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing CGA-specific algorithms, these methods participate in the advancement of geometric thinking, leveraging an intuitive spatial logic that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions.

- by Pablo Colapinto
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- Robotics, Applied Mathematics, Computer Science, Computer Graphics

The purpose of this thesis is to research on the viability of industriali- sing opaque free-form architectural envelopes by using no-mould GRC panels. Because the automation, digitalisation and computer numerically controlled processes are changing the way we design, plan and build buildings, we are getting increasingly closer to solving the current demands for envelopes with sinuous shapes affordably and universally.

We have developed a new method to design freeform illumination optics by introducing a double-pole
coordinate system in ray mapping. This method establishes a much more accurate ray mapping by moving
the two poles of the spherical coordinate system to the southernmost point of the sphere and overlapping
them together. It can reduce surface error and improve illumination uniformity significantly. The
residual surface error (RSE) of the freeform lens designed in the double-pole coordinate system is one
magnitude smaller than that of the lens designed in the θ; φ coordinate system and is only 1/3 of that of
the freeform surface designed in the u; v coordinate system.

- by Donglin Ma
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- LED Lighting, Illumination, Geometrical Optics, Non-Imaging Optics

The research aims to investigate the architectural uses of free-form panels made of natural stone, reinforced through the use of carbon fiber (or glass fiber) sheets on the back. The combination of these two materials allows reducing the thickness of the stone components up to few millimetres and the achievement of a great static performance. The project is a collaboration between the authors (from Polytechnic University of Bari) and the Ticino-based company Generelli SA.

This article examines, from the energy viewpoint, a new lightweight, slim, high energy efficient, light-transmitting envelope system, providing for seamless, free-form designs for use in architectural projects. The research was based on envelope components already existing on the market, especially components implemented with granular silica aerogel insulation, as this is the most effective translucent thermal insulation there is today. The tests run on these materials revealed that there is not one that has all the features required of the new envelope model, although some do have properties that could be exploited to generate this envelope, namely, the vacuum chamber of vacuum insulated panels (VIP), the monolithic aerogel used as insulation in some prototypes, and reinforced polyester barriers. By combining these three design components — the high-performance thermal insulation of the vacuum chamber combined with monolithic silica aerogel insulation, the free-form design potential provided by materials like reinforced polyester and epoxy resins—, we have been able to define and test a new, variable geometry, energy-saving envelope system.

- by Luis Alberto Alonso Pastor
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- Energy, Energy Efficiency Buildings, Aerogels, New Materials in design
- by Abir Acharjee
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- Geometry And Topology, Architecture, Architectural Geometry, Freeform Surface