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>Tubular surface <svg height="11.8174pt" id="M213" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 14.924 11.8174" width="14.924pt" 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="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.841,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg> due to Darboux frame.</div>

Advances in Mathematical Physics

fig1

Figure 1

Figure 1: 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 1