On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.0608pt" id="M1" height="20.3615pt" version="1.1" viewBox="-0.0657574 -15.3007 17.3744 20.3615" width="17.3744pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-16" d="M567 0V44H236V319H535V363H236V618H567V662H70V0H567ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.012,0,0,-0.012,10.674,-7.578)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g><g transform="matrix(.012,0,0,-0.012,10.674,4.995)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg>

<div>Bishop spherical Darboux image has a cusp as <span class="nowrap"><svg height="8.55521pt" id="M362" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 26.1796 8.55521" width="26.1796pt" 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="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.507,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.77,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg>.</span></div>

Abstract and Applied Analysis

fig4

Figure 4

Figure 4: On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.0608pt" id="M1" height="20.3615pt" version="1.1" viewBox="-0.0657574 -15.3007 17.3744 20.3615" width="17.3744pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-16" d="M567 0V44H236V319H535V363H236V618H567V662H70V0H567ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.012,0,0,-0.012,10.674,-7.578)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g><g transform="matrix(.012,0,0,-0.012,10.674,4.995)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg> 