Phi-functions

Analytic non-overlap modeling that encodes the interaction of two pieces.

Also called: Phi-functions · Φ-functions · phi-function · quasi-phi-functions

Another primitive for the overlap side of irregular placement (classified as a method in v1). Where the No-Fit Polygon describes the locus of overlap-free relative positions combinatorially, a phi-function gives a single signed analytic value for the relationship between two pieces (positive = apart, zero = touching, negative = overlapping). Its smoothness lets placement be cast as nonlinear programming (NLP), including continuous rotation. See Chernov, Stoyan, Romanova & Pankratov (2012) for the standard definition.

Claims & evidence

Every relationship is a claim with an evidence grade; applicable problem-to-problem relationships also carry an equivalence level. See the evidence policy.

A–X describes evidence reliability; E0–E4 describes mathematical similarity between problems. The two axes are independent.

Relationships from this entry 1

shares method with
Phi-functions to No-Fit Polygon
Phi-functions and the No-Fit Polygon both encode piece-to-piece overlap — the NFP as a combinatorial locus, phi-functions as a continuous analytic expression suited to nonlinear optimization and continuous rotation.
EvidenceA · Peer-reviewed / official
Equiv.E3 · structural analogy
Sources
- Source grade: A — Peer-reviewed / officialPhi-Functions for 2D Objects Formed by Line Segments and Circular Arcs
  Citation details
  Chernov, N., Stoyan, Yu., Romanova, T., Pankratov, A. (2012). Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs. Advances in Operations Research, vol. 2012. Phi-functions encode pairwise interaction analytically (positive = apart, zero = touching, negative = overlapping), enabling nonlinear-programming packing models with continuous rotation.

Relationships from this entry
Relationship	Claim	Evidence	Equiv.	Sources
shares method with Phi-functions to No-Fit Polygon	Phi-functions and the No-Fit Polygon both encode piece-to-piece overlap — the NFP as a combinatorial locus, phi-functions as a continuous analytic expression suited to nonlinear optimization and continuous rotation.	A · Peer-reviewed / official	E3 · structural analogy	Source grade: A — Peer-reviewed / officialPhi-Functions for 2D Objects Formed by Line Segments and Circular Arcs Citation details Chernov, N., Stoyan, Yu., Romanova, T., Pankratov, A. (2012). Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs. Advances in Operations Research, vol. 2012. Phi-functions encode pairwise interaction analytically (positive = apart, zero = touching, negative = overlapping), enabling nonlinear-programming packing models with continuous rotation.

Relationships to this entry 1

uses method
2D Irregular Nesting to Phi-functions
Overlap in irregular placement can also be encoded with phi-functions, an alternative to the No-Fit Polygon that enables nonlinear-programming formulations with continuous rotation.
EvidenceA · Peer-reviewed / official
Sources
- Source grade: A — Peer-reviewed / officialPhi-Functions for 2D Objects Formed by Line Segments and Circular Arcs
  Citation details
  Chernov, N., Stoyan, Yu., Romanova, T., Pankratov, A. (2012). Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs. Advances in Operations Research, vol. 2012. Phi-functions encode pairwise interaction analytically (positive = apart, zero = touching, negative = overlapping), enabling nonlinear-programming packing models with continuous rotation.

Relationships to this entry
Relationship	Claim	Evidence	Equiv.	Sources
uses method 2D Irregular Nesting to Phi-functions	Overlap in irregular placement can also be encoded with phi-functions, an alternative to the No-Fit Polygon that enables nonlinear-programming formulations with continuous rotation.	A · Peer-reviewed / official	—	Source grade: A — Peer-reviewed / officialPhi-Functions for 2D Objects Formed by Line Segments and Circular Arcs Citation details Chernov, N., Stoyan, Yu., Romanova, T., Pankratov, A. (2012). Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs. Advances in Operations Research, vol. 2012. Phi-functions encode pairwise interaction analytically (positive = apart, zero = touching, negative = overlapping), enabling nonlinear-programming packing models with continuous rotation.