curve shortening flow mathematics curve shortening flow process smooth curve plane moving points curve speed proportional curve shortening flow example geometric flow dimensional case mean flow names process include shortening flow geometric heat flow arc length evolution points smooth simple closed curve way curve remains simple smooth loses area constant rate perimeter decreases quickly possible continuous curve evolution curve non convex total absolute decreases convex convex ratio curve decreases curve circular shape collapsing single point simple smooth closed curves evolve remain collapses point circle simple closed curve maintains shape curve shortening flow curves cross infinite length shape including grim curve infinite curve translates upwards rotate remaining size shape curve shortening flow numerically curve finite difference method motion alternative methods include computing resulting curve repeatedly applying median filter digital image black white represent inside outside curve curve shortening flow originally studied model metal sheets later applied image analysis multi scale representation shapes model reaction systems behavior cellular pure mathematics curve shortening flow find closed model behavior higher dimensional flows definitions flow process points mathematical space continuously change locations properties time specifically dimensional geometric flow curve shortening flow points undergoing flow belong curve changes shape curve plane determined locations points curve shortening flow point curve moves direction normal curve rate proportional evolving curve represented function arc length curve time evolution curve curve shortening flow described partial differential equation formula form heat equation unit normal ingredients equation arc length time unaffected translations plane follows flow defined equation translations precisely plane scaled constant factor flow remains essentially unchanged slowed sped factor non smooth curves order flow defined given curve sufficiently smooth continuous flow starts curve remains reaching blows smooth curve crossings possible happens curve collapses point immersed curves types cases care possible continue flow past curve single point simple closed curve extension flow non smooth curves based level set method possibilities curves zero measure including smooth curves instantly evolve smooth curves evolve smooth curve curves measure instead immediately evolve area smooth boundaries curve example instantly smooth despite locally connected examples reverse evolution curve shortening flow behaved curves complicated finite time non surfaces curve shortening flow results curve shortening flow plane dimensional order avoid additional types important convex infinity defined mean compact set compact convex hull defined curve shortening flow cause curve depart convex hull condition prevents parts curve reaching boundary space curves curve shortening flow studied curves dimensional space normal case defined plane derivative respect arc length normalized unit components frame defined points zero product normal remains defined points allowing curve shortening flow defined curves space cross according flow flow lead curves plane curve shortening flow space curves way define flow past plane curves curves possible extend definition flow general curves instance level set method extended definitions allow parts curves sets area commonly studied variation problem involves networks interior smooth curves junctions curves meet junctions exactly curves meeting angles conditions seen optimal tree dimensional foam soap bubbles flow defined short term eventually reach singular state curves meeting junction way continue flow past behavior avoidance principle radius stretch factor smooth simple closed curves undergo curve shortening flow simultaneously remain flow progresses reason smooth curves way creates crossing time crossing curves necessarily crossing situation curves point necessarily pull apart pushing crossing reason single simple closed curve evolve cross avoidance principle implies smooth curve eventually reaches point infinite collapses point given smooth curve surrounded circle remain collapses reaches circle flow remaining circular collapses avoidance principle remain contained reasoning radius smallest circle decrease rate fast decrease radius circle undergoing flow avoidance principle single curve terms ratio arc length shorter arcs distance pairs points called stretch factor shows stretch factor strictly decreasing local case ends diameter circle case stretch factor constant property implies avoidance principle curve touch stretch factor infinite touching points length curve undergoes curve shortening flow length decreases rate given formula formula interval taken curve arc length curve non negative smooth closed curve exist arcs strictly positive length decreases generally evolution curves normal speed rate change length formula interpreted inner product given evolution curve shortening flow curve shortening flow described gradient flow length flow locally decreases length curve quickly possible relative norm flow property gives curve shortening flow area simple closed curve area enclosed curve constant rate units area unit time independent curve total time curve shrink point proportional area regardless initial shape area curve reduced constant rate inequality circle greatest possible area simple closed curves given length follows circles slowest curves collapse point curve shortening flow curves time collapse circle length constant rate area reduction conservation law satisfied curve shortening flow implies possible express point curve eventually collapses integral curve function points derivatives expression lead forbidden second conservation law combining constant rate area loss avoidance principle possible prove point lies circle concentric minimum circle area difference areas circle given curve total absolute total absolute smooth curve integral absolute value arc length curve formula expressed sum angles normal consecutive pairs points convex curves larger non convex curves serving measure non curve new points created curve shortening flow angles representation total absolute sum decreases consecutive points reach angle position eliminated total absolute increase curve convex curves constant non convex curves decreases hamilton smooth simple closed curve undergoes curve shortening flow remains smoothly embedded self intersections eventually convex remain convex time points curve shape curve converge circle curve single point behavior summarized saying simple closed curve round point result proved convergence circle convex curves contract point specifically showed ratio ratio squared curve length area number circle larger convex curve decreases quickly proved smooth convex curves eventually contract point forming proved non convex curve eventually convex provide simpler proof result based stretch factor similar results extended closed curves curves satisfying local condition curves sides curve infinite area evolved curve remains smooth free time curve finite area curve finite total absolute evolution reaches time proportional area finite area curve near curves sufficiently behaved functions ray direction solution shape unique shape rays networks formed rays line smooth curves connecting rays hamilton holds region curves convex shape self crossing curves curves self crossings reach contracting point instance smooth immersed curve single crossing resembling figure infinity symbol unequal areas eventually smaller lobe collapse point equal areas remain equal evolution curve ratio curve collapses locally convex self crossing curve approaches loops self similar way approaches grim curve described loop collapses total absolute lost exactly smooth simple closed curve remain smooth simple case collapse point finite time remain smooth simple forever case curve necessarily closed surface immersed curves self crossings self set times lose crossing consequence number self crossing points non decreasing formula according formula evolving curve time reversed heat non increasing result analyze evolution specific curves curves self similar evolution simple closed curve circle circle simple closed curve keeps shape curve shortening flow examples curves non simple include self crossings non closed extend infinity shape particular line stays unchanged curve shortening flow lines curves unaffected curve shortening flow exist complex stable networks curves plane grim curve log moves upwards changing shape way curve similar grim translated curve shortening flow shifted direction symmetry axis curve changing shape orientation grim curve property called model physics literature family self crossing closed curves derived projections knots shrink remain self similar curve shortening flow come known curves work mentioned earlier rediscovered independently curves locally convex described support functions suitably scaled versions support functions obey differential equation formula positive periodic solutions corresponding curves self similar evolution period strictly formula curves including infinite remain self similar complicated motions including rotation combinations rotation expansion translation networks smooth curves meeting junctions angles self similar solutions include double bubble surrounding equal areas lens shape bounded arcs circles rays having corners lens fish shaped network bounded line segment rays convex curve self similar networks involve larger number curves family networks grows remains self similar tree like networks curves meeting angles triple junctions fan rays meet common ray case shapes smooth curve rays evolution shapes defined variants curve shortening flow given fan rays different solution type solutions provide unique definition curve shortening flow starting fan rays ancient solutions ancient solution flow problem curve evolution backwards time self similar solutions shrink stay size growing ancient solutions sense backwards reversing self similarity transformation undergo forwards curve shortening flow instance circle grim curves ancient solutions closed curves circle curves form ancient solutions class curves called work curves function angle formula formula limiting shape reverse evolution pair grim curves approaching opposite directions coordinate system given implicit curve equation formula physics literature shapes known model general classes curves functions diverse collection ancient solutions known numerical order curve shortening flow efficiently continuous curve continuous evolution curve need replaced tracking tracking methods long fluid dynamics model track motion boundaries different materials steep material properties weather fronts shock waves single material methods involve motion boundary directly simulate motion boundary underlying fluid treating boundary property fluid methods simulate curve shortening flow curve undergoing flow boundary shock tracking methods curve shortening curve undergoing evolution finite difference method derive approximate normal values determine time step cases normal motion points combined motion spread uniformly curve write methods fast accurate complicated extend versions curve shortening flow apply complicated simple closed curves necessary deal changes methods warns conditions stability determined easily time step chosen hoc finite method formula adding small term based operator modification called help prove existence flows numerical simulation method lions proven converge numerical method listed equipped bounds convergence rate comparison forward backward accurate finite difference methods suggest numerical method computing curve shortening flow maintains curve steps current curve placing new sample points uniform measured normalized arc length locations points function small standard effect replacing point location weighted average locations nearby points curve weights standard chosen small step sample points nearly uniform method curve shortening distribution limit number sample points grows normalized arc length radius median describe scheme operating square grid effectively array curve evolved represented assigning value black exterior curve white interior curve giving indicator function interior curve representation updated alternating steps image heat simulate evolution heat equation short time step result image transform indicator function radius proportional square root time step set numerical value numerical value greater image original values new positions order scheme accurate time step large cause curve points low small cause radius minimum radius size min max small allow suitable intermediate time step chosen method evolution networks curves meeting junctions dividing plane regions applying method simultaneously region instead method alternatively described applying median filter weights possible use heat grid high resolution near curve waste time memory far curve contribute outcome instead values image version method uses image values represent signed distance curve achieve accuracy require lower resolution applications metal sheets early reference curve shortening flow model physical process heat treatment causes boundaries grains metal shift unlike soap films forced differences air pressure surfaces constant mean grain boundaries subject local effects cause according mean flow dimensional case flow curve shortening flow corresponds sheets metal thin grains effectively dimensional boundaries dimensional shape analysis image processing computer vision suggest applying curve shortening flow outline shape derived digital image order remove noise shape provide scale space provides simplified description shape different levels resolution method involves computing curve shortening flow tracking points curve progress flow drawing plots positions points curve time points typically removed curve pairs curve convex according hamilton lifetime pair points corresponds feature shape method describe computing numerical curve shortening flow method scale space observe scale space given shape assert uniquely determines shape robust small variations shape compare related alternative definitions scale space shapes find scale space intensive robust noise strongly influenced small scale shape differences reaction reaction systems modeled allen equation limiting behavior fast reaction slow local energy energy level system settle regions different local fronts boundaries regions evolving according curve shortening flow cellular cellular cell infinite grid cells finite set states cells update states simultaneously based configuration small set neighboring cells life like cellular rule grid infinite square exactly cell states set neighbors cell neighbors moore neighborhood update rule depends number neighbors states complicated function states particular life like rule introduced called twisted majority rule rule update rule sets new value cell majority cells given neighbors cells split state state case new value cell minority majority detailed dynamics rule complicated including existence small stable structures aggregate started cells random states tends form large regions cells state boundaries regions evolving according curve shortening flow construction closed curve shortening flow prove inequality surfaces non increasing function distance origin surface smooth compact set given area minimum perimeter area necessarily circle centered origin proof applies curve shortening flow curves metric circle boundary compact set compares change perimeter curves reduced point flow curve shortening flow prove smooth equivalent sphere form simple closed curves related flows geometric flows related curve shortening flow include following ones behavior crystals materials important variants curve shortening flow speed flow depends orientation curve way define energy curve integral smooth function normal form gradient flow energy according normal speed curve flows proportional analog flow simulated curve numerical experiments initial curves appear converge shape point alternatively let curve flow speed usual smooth functions orientation flow point reflection resulting flow shown obey avoidance principle analog hamilton curve shortening flow investigated flow normal speed curve proportional cube root resulting flow corresponding time scaling plane larger symmetry group similarity curve shortening flow flow hamilton applies simple closed curve eventually convex collapses point transforming curve equal normal speeds points called transform curves evolved way general develop sharp corners trace forms axis curve closely related curve evolution moves straight segments curve equal speeds allows concave corners quickly unit speed instead forms different type skeleton given curve straight skeleton surfaces higher dimensions definition including dependent measures mean measures ways defining geometric flows based including mean flow normal speed embedded surface mean flow flow metric space based flow gradient flow energy functional combining mean curve shortening flow special case mean flow dimensional curves inspired curve shortening flow smooth curves researchers studied methods flowing stay applications including pattern formation distributed systems robots length preserving flows solve carpenter rule problem computer vision active model edge detection image based curve shortening curves based combination features image
Curve-shortening flow