{"schema_version":"1.0","canonical_url":"https://patentable.app/patents/US-8521486","patent":{"patent_number":"US-8521486","title":"Methods for searching for arrangements of shapes subject to boundary distance constraints among those shapes","assignee":null,"inventors":[],"filing_date":"2010-07-27T00:00:00.000Z","publication_date":"2013-08-27T00:00:00.000Z","cpc_codes":["G06F","G06F","G06F","G06F"],"num_claims":6,"abstract":"This disclosure describes methods for searching for arrangements of shapes subject to boundary distance constraints among those shapes. The first embodiment includes constructing 110 functions that are exact or approximate analytical representations of those constraints. Next is constructing 120 an analytical optimization problem which incorporates the constraint functions of 110. Further is selecting 130 the initial values of one or more optimization variables. And then solving 140 the optimization problem constructed in 120 using one or more analytical optimization methods and one or more of the initial values of 130. Further included is a sequencing structure 150. The second embodiment includes constructing 210 functions that are exact or approximate analytical representations of those constraints. Next is constructing 215 one or more constraint functions. And next is constructing 220 an analytical optimization problem that incorporates the constraint functions of 210 and 215. Further is selecting 230 the initial values of one or more optimization variables. And then solving 240 the optimization problem constructed in 220 using one or more analytical optimization methods and one or more of the initial values of 230. Further included is a sequencing structure 250. "},"analysis":{"summary":null,"layman_explanation":null,"technical_analysis":null,"business_analysis":null,"faqs":null,"topics":[],"tech_cluster":null},"seo":{"title":"Methods for searching for arrangements of shapes subject to boundary distance constraints among those shapes","description":"This disclosure describes methods for searching for arrangements of shapes subject to boundary distance constraints among those shapes. The first embodiment includes constructing 110 functions that ar","keywords":[]},"attribution":{"source":"Patentable","source_url":"https://patentable.app","canonical_url":"https://patentable.app/patents/US-8521486","license":"CC-BY-4.0-like","license_terms":"AI-generated analysis on this page (summary, layman_explanation, technical_analysis, business_analysis, faqs) may be reused with attribution and a visible link back to the canonical URL above. Patent abstracts, claims, and bibliographic data are USPTO public domain.","required_link":"https://patentable.app/patents/US-8521486","citation_suggestion":"Patentable. \"Methods for searching for arrangements of shapes subject to boundary distance constraints among those shapes\" (US-8521486). https://patentable.app/patents/US-8521486","copyright_holder":"Nomic Interactive Technology LLC"},"links":{"html":"https://patentable.app/patents/US-8521486","json":"https://patentable.app/api/llm-context/US-8521486","site":"https://patentable.app","llms_txt":"https://patentable.app/llms.txt"},"generated_at":"2026-05-30T16:23:56.918Z"}