Bezier curve is one of the most fundamental primitives for the modeling of fonts and two-dimensional (2D) computer graphics objects. How to efficiently render the Bezier curve becomes an important task for many embedded applications. This thesis first proposed a novel adaptive curve-rendering algorithm which can determine the coordinates of all the crossing points of the curve and scan-lines with the required accuracy for the graphics fill operation. Next, for the rendering of stroked Bezier curves, this thesis proposed several possible rendering circuit architectures. The performance and gate count of these architectures have been estimated, and compared in this thesis. It has been found that the design based on the table-lookup normal vector calculator can lead to the fastest circuit, while the design based on the Cordic operator represents the most economic design. A basic Bezier curve rendering circuit has been implemented in this thesis, and used to accelerate a prototype OpenVG embedded systems.

[pt] Neste trabalho apresentamos uma proposta para auxiliar o estudo de funções polinomiais no ensino médio por intermédio das Curvas de Bézier. Para isto introduzimos as curvas de Bézier utilizando o algoritmo de De Casteljau e discutimos suas propriedades. Em seguida aplicamos a formulação das curvas de Bézier não paramétricas para representar funções polinomiais e discutimos alguns resultados observados em sala de aula. / [en] In this work we present a proposal to assist the study of polynomial functions in high school through the Bezier curves. For this we introduce the Bezier curves using the De Casteljau algorithm and discuss its properties. Then apply the formulation of Bezier curves non-parametric to represent polynomial functions and discuss some results observed in the classroom.

Toward support of the use of geometry in advanced simulation, a freeorm deformation (FFD) tool was designed, developed, and tested using object oriented (OO) techniques. The motivation for creating this FFD tool in-house was to provide engineers and researchers a cost efficient, quick, and easy way to computationally manipulate models without having to start from scratch while readily seeing the resulting geometry. The FFD tool was built using the OO scripting language, Python, the OO GUI toolkit, Qt, and the graphics toolkit, OpenGL. The tool produced robust and intuitive results for two-dimensional shapes especially when multiple point manipulation was utilized. The use of ?grouping? control points also provided the user the ability to maintain certain desired shape features such as straight lines and corners. This in-house FFD tool could be useful to engineers due to the ability to customize source code.

When interpolating a data mesh using triangular Bézier patches, the requirement of C¹ or G¹ continuity imposes strict constraints on the control points of adjacent patches. However, fulfillment of these continuity constraints cannot guarantee that the resulting surfaces have good shape. This thesis presents an approach to constructing surfaces with approximate C¹/G¹ continuity, where a small amount of discontinuity is allowed between surface normals of adjacent patches. For all the schemes presented in this thesis, although the resulting surface has C¹/G¹ continuity at the data vertices, I only require approximate C¹/G¹ continuity along data triangle boundaries so as to lower the patch degree. For functional data, a cubic interpolating scheme with approximate C¹ continuity is presented. In this scheme, one cubic patch will be constructed for each data triangle and upper bounds are provided for the normal discontinuity across patch boundaries. For a triangular mesh of arbitrary topology, two interpolating parametric schemes are devised. For each data triangle, the first scheme performs a domain split and constructs three cubic micro-patches; the second scheme constructs one quintic patch for each data triangle. To reduce the normal discontinuity, neighboring patches across data triangle boundaries are adjusted to have identical normals at the middle point of the common boundary. The upper bounds for the normal discontinuity between two parametric patches are also derived for the resulting approximate G¹ surface. In most cases, the resulting surfaces with approximate continuity have the same level of visual smoothness and in some cases better shape quality.

When interpolating a data mesh using triangular Bézier patches, the requirement of C¹ or G¹ continuity imposes strict constraints on the control points of adjacent patches. However, fulfillment of these continuity constraints cannot guarantee that the resulting surfaces have good shape. This thesis presents an approach to constructing surfaces with approximate C¹/G¹ continuity, where a small amount of discontinuity is allowed between surface normals of adjacent patches. For all the schemes presented in this thesis, although the resulting surface has C¹/G¹ continuity at the data vertices, I only require approximate C¹/G¹ continuity along data triangle boundaries so as to lower the patch degree. For functional data, a cubic interpolating scheme with approximate C¹ continuity is presented. In this scheme, one cubic patch will be constructed for each data triangle and upper bounds are provided for the normal discontinuity across patch boundaries. For a triangular mesh of arbitrary topology, two interpolating parametric schemes are devised. For each data triangle, the first scheme performs a domain split and constructs three cubic micro-patches; the second scheme constructs one quintic patch for each data triangle. To reduce the normal discontinuity, neighboring patches across data triangle boundaries are adjusted to have identical normals at the middle point of the common boundary. The upper bounds for the normal discontinuity between two parametric patches are also derived for the resulting approximate G¹ surface. In most cases, the resulting surfaces with approximate continuity have the same level of visual smoothness and in some cases better shape quality.

In this thesis, we present a method for computing an artistic representation of ocean waves using Bezier curves. Wave forms are loosely based on procedural wave models and are designed to emulate those found in both art and nature. The wave forms are generated using a slice method which is user defined by structured input, thus providing the artist with full control over crest shape and placement. Wave propagation is obtained by interpolating between defined crest shapes and positions. We also present a method for computing a stylized representation of breaking crests in shallow water. Artists may use our model to create many interesting wave forms, including basic sinusoidal waves and waves with breaking crests that have a rotation that is cyclical in time. The major drawbacks to our solution are that data entry can be tedious and it can be difficult to produce waves that animate with a natural appearance.

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Yes / Given a prescribed boundary of a Bezier surface we compare the Bezier surfaces generated by two different methods, i.e. the Bezier surface minimising the Biharmonic functional and the unique Bezier surface solution of the Biharmonic equation with prescribed boundary. Although often the two types of surfaces look visually the same, we show that they are indeed different. In this paper we provide a theoretical argument showing why the two types of surfaces are not always the same.

Splines are a common mixed-initiative technique for road generation. A designer draws the shape of the curve but the mesh can be procedurally generated along the spline. This relationship improves the workflow of building roads in virtual environments and video games without taking away all of the control of the designer. Whilst this technique is useful when building single roads such as race tracks, it unfortunately struggles when dealing with more complex road networks that feature intersections. These intersections struggle with overlapping meshes and flickering textures without a straightforward solution. This problem significantly limits the usefulness of spline tools when generating roads. This paper aims to solve part of this problem by suggesting a method for detecting intersections in splines that support procedural mesh generation.