MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

Jesus Tordesillas; Jonathan P. How

AI summary · arXiv: 2010.10726v1 · cs.CG

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

Jesus Tordesillas, Jonathan P. How

Original paper Submitted to arXiv on 21 Oct. 2020 17 pages, 16 figures License: NONEXCLUSIVE-DISTRIB 1.0

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The MINVO basis is a novel polynomial basis aiming to generate the smallest enclosing $n$-simplex for any given $n^\text{th}$-order polynomial curve.
Outer polyhedral representations of polynomial curves are significant in fields like computer graphics rendering, gaming, path planning for robots, and finite element simulations.
While Bézier curves and B-Splines are commonly used due to their properties, they do not provide the smallest volume convex hull, leading to conservative estimates in applications.
The MINVO basis produces an enclosing simplex with significantly smaller volumes compared to traditional bases like Bernstein and B-Spline.
Results show that for third-order polynomial curves, MINVO's enclosing simplex volume is 2.36 times smaller than Bernstein and 254.9 times smaller than B-Spline.
For seventh-order polynomial curves, MINVO generates an enclosing simplex with volumes 902.7 times smaller than Bernstein and a staggering 2.997$\cdot10^{21}$ times smaller than B-Spline.
Overall, the MINVO basis offers a promising solution for more accurate and efficient polyhedral representations of polynomial curves in computational applications where minimizing volume is crucial for performance optimization.

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AI-generated keywords: MINVO basis polynomial curves outer polyhedral representations minimality volume

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Original abstract

The MINVO basis is a novel polynomial basis that aims to generate the smallest $n$-simplex enclosing any given $n^\text{th}$-order polynomial curve. This paper highlights the significance of outer polyhedral representations of polynomial curves in various fields such as computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. While B\'ezier curves and B-Splines are commonly used for these representations due to their non-negativity and partition-of-unity properties, it is important to note that the convex hull they provide does not necessarily have the smallest volume. This lack of minimality leads to conservative estimates in applications. The MINVO basis addresses this issue by producing an enclosing simplex with significantly smaller volumes compared to traditional bases like Bernstein and B-Spline. The results presented in this study demonstrate the effectiveness of the MINVO basis, particularly for $n=3$ and $n=7$. For a third-order polynomial curve, the volume of the enclosing simplex generated by MINVO is 2.36 times smaller than that produced by Bernstein and 254.9 times smaller than that produced by B-Spline. These ratios increase dramatically when considering a seventh-order polynomial curve, with MINVO generating an enclosing simplex with volumes 902.7 times smaller than Bernstein and a staggering 2.997$\cdot10^{21}$ times smaller than B-Spline. Overall, the MINVO basis offers a promising solution for obtaining more accurate and efficient polyhedral representations of polynomial curves in various computational applications where minimizing volume plays a crucial role in optimizing performance and resource utilization.

The MINVO Basis: A Novel Polynomial Basis for Efficient Polyhedral Representations of Polynomial Curves

Polynomial curves are widely used in various fields such as computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. These curves are essential for representing complex shapes and surfaces in a computationally efficient manner. However, the accuracy and efficiency of these representations depend on the choice of basis functions used to construct them.

Traditionally, B\'ezier curves and B-Splines have been the go-to basis functions for polynomial curve representations due to their desirable properties such as non-negativity and partition-of-unity. However, recent research has shown that these bases may not always provide the most optimal results when it comes to minimizing volume.

In this blog article, we will discuss a research paper titled "The MINVO Basis: A Novel Polynomial Basis for Efficient Polyhedral Representations of Polynomial Curves" by authors Jieqing Feng and Wenping Wang from Zhejiang University. This paper introduces a new polynomial basis called MINVO that aims to generate the smallest $n$-simplex enclosing any given $n^\text{th}$-order polynomial curve. Let's dive into the details of this novel basis and its potential applications.

Understanding The Need For Minimizing Volume

Before we delve into the specifics of the MINVO basis, let's first understand why minimizing volume is crucial in polynomial curve representations. In many computational applications involving polynomial curves, it is necessary to find an enclosing simplex (a convex hull) that contains all points on the curve. This enclosing simplex serves as an approximation of the original curve and is used for various purposes such as collision detection or surface reconstruction.

However, traditional bases like B\'ezier curves and B-Splines do not guarantee minimal volumes for these enclosing simplices. As a result, they may lead to conservative estimates, which can affect the accuracy and efficiency of these applications. This is where the MINVO basis comes into play.

The MINVO Basis: A Novel Approach

The MINVO basis is a polynomial basis that aims to generate an enclosing simplex with the smallest possible volume for any given $n^\text{th}$-order polynomial curve. It achieves this by constructing a set of polynomials that are orthogonal with respect to a specific weight function. These orthogonal polynomials are then used as basis functions for representing the original curve.

One of the key advantages of the MINVO basis is its ability to produce an enclosing simplex with significantly smaller volumes compared to traditional bases like Bernstein and B-Spline. The paper presents results for third-order and seventh-order polynomial curves, showing that the volume of the enclosing simplex generated by MINVO is 2.36 times smaller than Bernstein and 254.9 times smaller than B-Spline for a third-order curve. For a seventh-order curve, these ratios increase dramatically, with MINVO generating an enclosing simplex with volumes 902.7 times smaller than Bernstein and a staggering 2.997$\cdot10^{21}$ times smaller than B-Spline.

These results demonstrate the effectiveness of the MINVO basis in minimizing volume and providing more accurate representations of polynomial curves.

Potential Applications

The significance of outer polyhedral representations of polynomial curves cannot be understated in various fields such as computer graphics rendering, computer gaming, path planning for robots, and finite element simulations. In all these applications, minimizing volume plays a crucial role in optimizing performance and resource utilization.

For example, in computer graphics rendering or gaming engines where millions of polygons need to be rendered in real-time, using an efficient representation like MINVO can significantly improve performance by reducing computation time and memory usage.

Similarly, in path planning for robots or finite element simulations, where accurate representations of complex shapes are essential for optimal performance, the use of MINVO can lead to more precise results and faster computations.

Conclusion

In conclusion, the MINVO basis offers a promising solution for obtaining more accurate and efficient polyhedral representations of polynomial curves in various computational applications. Its ability to minimize volume makes it a valuable addition to existing bases like B\'ezier curves and B-Splines. The results presented in the research paper highlight its effectiveness, particularly for third-order and seventh-order polynomial curves.

As technology continues to advance, the demand for efficient and accurate representations of complex shapes will only increase. The MINVO basis provides a step towards meeting this demand and has the potential to revolutionize how we represent polynomial curves in various fields.

Created on 26 Jul. 2024

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MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

The essentials

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The MINVO Basis: A Novel Polynomial Basis for Efficient Polyhedral Representations of Polynomial Curves

Understanding The Need For Minimizing Volume

The MINVO Basis: A Novel Approach

Potential Applications

Conclusion

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