Log-Aesthetic Curves for Shape Completion Problem

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="960302.fig.005a" src="https://static.hindawi.com/articles/jam/volume-2014/960302/figures/960302.fig.005a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Euler spiral algorithm</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="960302.fig.005b" src="https://static.hindawi.com/articles/jam/volume-2014/960302/figures/960302.fig.005b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>LAC algorithm with two tangents setting for s-shaped curve and <svg height="8.76568pt" id="M160" style="vertical-align:-0.1625805pt" version="1.1" viewbox="0 -8.6031 37.6754 8.76568" width="37.6754pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="568" vert-adv-y="568"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,11.482,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,23.219,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,31.172,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg></td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="960302.fig.005c" src="https://static.hindawi.com/articles/jam/volume-2014/960302/figures/960302.fig.005c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>LAC algorithm with two tangents setting for s-shaped curve and <svg height="8.76568pt" id="M161" style="vertical-align:-0.1625805pt" version="1.1" viewbox="0 -8.6031 29.7226 8.76568" width="29.7226pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="568" vert-adv-y="568"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,11.482,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,23.219,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg></td></tr></table></td></tr></table>

<table>The results of fitting Euler spiral and LAC segment for two missing parts of a jug.</table>

Journal of Applied Mathematics

Log-Aesthetic Curves for Shape Completion Problem

Figure 5