Polyarc bounded complex interval arithmetic

TL;DR

This work introduces the polyarcular interval, an extension of complex interval arithmetic that exactly represents a broad class of primitive intervals (rectangular, polar, circular) and closely matches polygonal representations in precision with a favorable computational profile. By grounding the theory in Minkowski algebra and algebro-geometric boundary analysis, the authors develop three complementary envelope-evaluation methods (implicit, parametric, and mixed) and a Gauss-map framework to identify boundary contributions, enabling polyarcular arithmetic to remain closed under many operations. They provide rigorous bounds on boundary behavior, a backtracking mechanism to diagnose representation tightness, and a comprehensive computational treatment including type casting and trimming. A case study on antenna tolerance analysis demonstrates that polyarcular intervals can yield perfect tightness with modestly increased cost, highlighting practical impact for engineering applications. The work culminates with a discussion of the trade-offs between polyarcular and polygonal approaches and suggests directions for further code release and exploration of complex interval arithmetic.

Abstract

Complex interval arithmetic is a powerful tool for the analysis of computational errors. The naturally arising rectangular, polar, and circular (together called primitive) interval types are not closed under simple arithmetic operations, and their use yields overly relaxed bounds. The later introduced polygonal type, on the other hand, allows for arbitrarily precise representation of the above operations for a higher computational cost. We propose the polyarcular interval type as an effective extension of the previous types. The polyarcular interval can represent all primitive intervals and most of their arithmetic combinations precisely and has an approximation capability competing with that of the polygonal interval. In particular, in antenna tolerance analysis it can achieve perfect accuracy for lower computational cost then the polygonal type, which we show in a relevant case study. In this paper, we present a rigorous analysis of the arithmetic properties of all five interval types, involving a new algebro-geometric method of boundary analysis.

Figures (8)

• Figure 1: Sum, product, and reciprocal of primitive intervals. Dark and light blue lines indicate the boundary of operand $\boldsymbol{A}$ and its translated and rotate-and-scaled copies ($\boldsymbol{A} \oplus\boldsymbol{B}$,$\boldsymbol{A}\otimes \boldsymbol{B}$), while red lines indicate the boundary of operand $\boldsymbol{B}$ and its reciprocal ($\boldsymbol{B}^{-1}$). Red dots identify internal points of the interval $\boldsymbol{B}$ used to translate and rotate-and-scale $\boldsymbol{A}$. Solid black line identifies the polyarcular operation results, while dotted line idenfies the convex polygonal operation results. The dashed black line identifies the boundary of the primitive wrapper of the results.
• Figure 2: Example of a complex interval represented by various complex interval types. The gray area identifies the complex interval; solid black lines identify the polygonal and polyarcular boundary curves. The grey dash-dotted lines indicate the boundaries of inclusive rectangular, polar and circular interval type objects.
• Figure 3: Graphical summary of complex interval subspaces, types and operations. We figure illustrates a number of complex intervals from the following subspaces: rectangular ($\mathcal{R}$), polar ($\mathcal{P}$), circular ($\mathcal{C}$), polygonal ($\mathcal{G}$), polyarcular ($\mathcal{A}$). In the equation the interval symbol with the type designation is followed by the amount and type of stored parameters, and finally the notation of the narrowest subspace to which it belongs. The $\oplus$ and $\otimes$ indicate the binary addition and multiplication operations; $\ominus$ and $\oslash$ indicate the unary negative and reciprocal operations; $\cup$ and $\cap$ indicate the binary union and intersection operations. Arrows around operators indicate whether the result of an operation is in the same subspace or is outside of it. Hollow arrows indicate significant special cases (trivial exceptions, such as division by zero and union of non-intersecting intervals are not indicated).
• Figure 4: Two polar intervals and their sum combined with their Gauss maps on the same plot. The three concentric curves in the unit circle shows their Gauss maps, where the argument and the color indicates the curve normal angle, and the radius indicates the position on the curve. The arrows show the curve normal vectors.
• Figure 5: Evaluation of the product of two arcs. The plot in the top left shows the sampled result and the implicit boundary curves besides the operands on the complex plane. The plot on the right shows the sampled result on its implicit surface constrained by planes over the complex plane. The plot in the bottom left shows the parametric condition of the boundary crossing the envelope. Similar figures for all edge and arc arithmetic combinations can be found in the Supplementary Material.

Theorems & Definitions (76)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.1
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Lemma 2.1