Parametric triangular Bézier surface interpolation with approximate continuity

Liu, Yingbin; Mann, Stephen

doi:10.1145/1364901.1364956

Cited by 13 publications

(5 citation statements)

References 9 publications

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“…This approach can result in mismatches even of position. Our approach differs from datum plane interpolation, as well as from the notion of approximate smoothness explored in [LM08], in that we leverage an underlying guide surface. Without the guide the degrees of freedom from dropping exact smoothness constraints are very difficult to harness towards class A quality.…”

Section: Introductionmentioning

confidence: 99%

Can bi‐cubic surfaces be class A?

Karčiauskas

Peters

2015

Computer Graphics Forum

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a) 2-beam join (b) Catmull-Clark [CC78] (c) C 1 /G 1 [GZ94] (d) rational [LSNC09] (e) this paper Figure 1: Class A surfaces avoid oscillating highlight lines. (a) Joining two crossing beams, modeled by bi-3 B-spline surfaces, via a multi-sided cap (red). Of the four displayed bi-3 constructions only construction (e) qualifies as class A.Abstract 'Class A surface' is a term in the automotive design industry, describing spline surfaces with aesthetic, nonoscillating highlight lines. Tensor-product B-splines of degree bi-3 (bicubic) are routinely used to generate smooth design surfaces and are often the de facto standard for downstream processing. To bridge the gap, this paper explores and gives a concrete suggestion, how to achieve good highlight line distributions for irregular bi-3 tensor-product patch layout by allowing, along some seams, a slight mismatch of normals below the industryaccepted tolerance of one tenth of a degree. Near the irregularities, the solution can be viewed as transforming a higher-degree, high-quality formally smooth surface into a bi-3 spline surface with few pieces, sacrificing formal smoothness but qualitatively retaining the shape.

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Section: Introductionmentioning

confidence: 99%

Can bi‐cubic surfaces be class A?

Karčiauskas

Peters

2015

Computer Graphics Forum

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“…Liu and Mann [25] introduced a piecewise interpolation polynomial by producing a quintic triangular Bézier patch for each data point using a Bézier control point based on the virtual mesh of the provided dataset. The resulting surface, however, only obtained G 1 .…”

Section: Quintic Triangular Bézier Patch Using Dataset's Virtual Meshmentioning

confidence: 99%

“…u, v, w ðÞ are barycentric coordinates, and the control points are P i,j,k . Liu and Mann [25] constructed a piecewise triangular surface that interpolates data vertices onto a specified triangular mesh M. The number of incident edges for each data vertex V is referred to as the valence of V. Because they assume M has arbitrary topology and is triangulated without singularities, V 0 s valence is always greater than 2. The mesh M is likewise considered to be closed in their study.…”

Section: Quintic Triangular Bézier Patch Using Dataset's Virtual Meshmentioning

confidence: 99%

“…When the valences of the three vertices of a data triangle are identical, the patch formed by Loop's method is quintic; if the three valences are all six, the patch degree decreases to quartic [26]. The boundary curves in Liu and Mann's [25] method are constructed similarly to the first step of Loop's scheme, but with the center control point set differently. The twist terms are solved in their scheme in the same way as Loop's scheme is solved in the second stage.…”

Section: Quintic Triangular Bézier Patch Using Dataset's Virtual Meshmentioning

confidence: 99%

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Fitting Parametric Polynomials based on Bézier Control Points

Fadhel¹

2023

Recent Research in Polynomials [Working Title]

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This chapter provides an overview at fitting parametric polynomials with control points coefficients. These polynomials have several properties, including flexibility and stability. Bézier, B-spline, Nurb, and Bézier trigonometric polynomials are the most significant of these kinds. These fitting polynomials are offered in two dimensions (2D) and three dimensions (3D). This type of polynomial is useful for enhancing mathematical methods and models in a variety of domains, the most significant of which being interpolation and approximation. The utilization of parametric polynomials minimizes the number of steps in the solution, particularly in programming, as well as the fact that polynomials are dependent on control points. This implies having more choices when dealing with the generated curves and surfaces in order to produce the most accurate results in terms of errors. Furthermore, in practical applications such as the manufacture of automobile exterior constructions and the design of surfaces in various types of buildings, this kind of polynomial has absolute preference.

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“…Together with the schemes that we are going to present in the following subsections, we would also like to cite [6,7,8,9,10] as interesting schemes related to the interpolation problem we are considering. However, we are not going to include them in our discussion because [6,7] do not fit into the class of analytically representable curved patches, [8,9] use neighboring triangles to construct cubic patches with approximate-G 1 continuity and, although in [10] Walton and Meek propose an interesting method leading to a G 1 surface, they make use of a blending type approach that results in a patch that is not purely polynomial.…”

Section: Continuous Surface Schemesmentioning

confidence: 99%

A comparison of local parametric C0 Bézier interpolants for triangular meshes

Boschiroli¹,

Fünfzig

Romani

et al. 2011

Computers & Graphics

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Parametric curved shape surface schemes interpolating vertices and normals of a given triangular mesh with arbitrary topology are widely used in computer graphics for gaming and real-time rendering due to their ability to effectively represent any surface of arbitrary genus. In this context, continuous curved shape surface schemes using only the information related to the triangle corresponding to the patch under construction, emerged as attractive solutions responding to the requirements of resource-limited hardware environments. In this paper we provide a unifying comparison of the local parametric C 0 curved shape schemes we are aware of, based on a reformulation of their original constructions in terms of polynomial Bézier triangles. With this reformulation we find a geometric interpretation of all the schemes that allows us to analyse their strengths and shortcomings from a geometrical point of view. Further, we compare the four schemes with respect to their computational costs, their reproduction capabilities of analytic surfaces and their response to different surface interrogation methods on arbitrary triangle meshes with a low triangle count that actually occur in their real-world use.

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Parametric triangular Bézier surface interpolation with approximate continuity

Cited by 13 publications

References 9 publications

Can bi‐cubic surfaces be class A?

Can bi‐cubic surfaces be class A?

Fitting Parametric Polynomials based on Bézier Control Points

A comparison of local parametric C0 Bézier interpolants for triangular meshes

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