Harmonic Evolute Surface of Tubular Surfaces via <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06570053pt" id="M1" height="11.534pt" version="1.1" viewBox="-0.0657574 -11.4683 11.7752 11.534" width="11.7752pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-13" d="M70 662V0H388C518 0 626 42 626 191C626 259 594 320 530 347C583 374 608 427 608 484C608 628 469 662 352 662H70ZM491 597C539 573 564 537 564 482C564 423 542 393 491 370V597ZM447 369C425 367 403 366 381 366H236V618H388C407 618 428 613 447 606V369ZM491 318C551 303 582 249 582 189C582 120 558 76 491 52V318ZM447 52C427 47 408 44 389 44H236V322H381C403 322 425 321 447 319V52ZM192 44H114V618H192V44Z"/></g></svg>-</span>Darboux Frame in Euclidean 3-Space

<div>Harmonic surface <svg height="11.8174pt" id="M229" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 10.5526 11.8174" width="10.5526pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.486,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg> via <span class="nowrap"><svg height="8.73791pt" id="M230" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.68809 8.92059 8.73791" width="8.92059pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M70 662V0H388C518 0 626 42 626 191C626 259 594 320 530 347C583 374 608 427 608 484C608 628 469 662 352 662H70ZM491 597C539 573 564 537 564 482C564 423 542 393 491 370V597ZM447 369C425 367 403 366 381 366H236V618H388C407 618 428 613 447 606V369ZM491 318C551 303 582 249 582 189C582 120 558 76 491 52V318ZM447 52C427 47 408 44 389 44H236V322H381C403 322 425 321 447 319V52ZM192 44H114V618H192V44Z"></path></g></svg>-</span>Darboux frame.</div>

Advances in Mathematical Physics

fig4

Figure 4

Figure 4: Harmonic Evolute Surface of Tubular Surfaces via <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06570053pt" id="M1" height="11.534pt" version="1.1" viewBox="-0.0657574 -11.4683 11.7752 11.534" width="11.7752pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-13" d="M70 662V0H388C518 0 626 42 626 191C626 259 594 320 530 347C583 374 608 427 608 484C608 628 469 662 352 662H70ZM491 597C539 573 564 537 564 482C564 423 542 393 491 370V597ZM447 369C425 367 403 366 381 366H236V618H388C407 618 428 613 447 606V369ZM491 318C551 303 582 249 582 189C582 120 558 76 491 52V318ZM447 52C427 47 408 44 389 44H236V322H381C403 322 425 321 447 319V52ZM192 44H114V618H192V44Z"/></g></svg>-</span>Darboux Frame in Euclidean 3-Space 

Advances in Mathematical Physics

Harmonic Evolute Surface of Tubular Surfaces via -Darboux Frame in Euclidean 3-Space

Figure 4