Robust approximation of the Medial Axis Transform of LiDAR point clouds as a tool for visualisation

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(1)Delft University of Technology. Robust approximation of the Medial Axis Transform of LiDAR point clouds as a tool for visualisation Peters, Ravi; Ledoux, Hugo DOI 10.1016/j.cageo.2016.02.019 Publication date 2016 Document Version Accepted author manuscript Published in Computers & Geosciences: an international journal. Citation (APA) Peters, R., & Ledoux, H. (2016). Robust approximation of the Medial Axis Transform of LiDAR point clouds as a tool for visualisation. Computers & Geosciences: an international journal, 90, 123-133. https://doi.org/10.1016/j.cageo.2016.02.019 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above.. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10..

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(10) OPOIPNPHFOFPVT HFPNFUSZBXBSF QPJOU TJN QMJĕDBUJPO ✏f (p) f (p). S. *NQMFNFOUBUJPO FYQFSJNFOUT BOE EJTDVTTJPO. p. 8F IBWF JNQMFNFOUFE UIF BMHPSJUINT EFTDSJCFE JO 4FDUJPOT BOE BOE UFTUFE UIFN VTJOH UXP BFSJBM -J%"3 EBUBTFUT ćF ĕSTU JT B QVCMJDMZ BWBJMBCMF EBUBTFU PG B NPVOUBJO SBOHF JO $BMJGPS OJB 64" /BUJPOBM 4DJFODF 'PVOEBUJPO 'SPN UIJT ANPVOUBJO EBUBTFU XF UPPL BO BSFB PG × NFUFST DPOUBJOJOH QPJOUT ćF SFQPSUFE QPJOU EFOTJUZ JT QPJOUT QFS TRVBSF NFUFS ćF TFDPOE EBUBTFU JT GSPN UIF NVOJDJQBMJUZ PG 3PUUFSEBN ćF /FUIFSMBOET BOE IBT B SFQPSUFE QPJOU EFOTJUZ PG QPJOUT QFS TRVBSF NFUFS 'SPN UIJT AVSCBO EBUBTFU XF TF MFDUFE BO BSFB PG × NFUFST DPOUBJOJOH QPJOUT UP DPNQBSF XJUI UIF NPVOUBJO EBUBTFU BOE BO BSFB PG CJMMJPO QPJOUT UP UFTU UIF TDBMBCJMJUZ PG PVS NFUIPE. M[S]. 'JHVSF ćF MPDBM QPJOU EFOTJUZ BOE UIF SBEJVT PG UIF TQMBU GPS Q B QPJOU TBNQMFE PO S BSF CBTFE PO UIF MPDBM GFBUVSF TJ[F G(Q) XIJDI JT UIF TIPSUFTU EJTUBODF GSPN Q UP UIF QPJOU BQQSPYJNBUJPO PG M[S] 'PS UIJT ĕHVSF ϵ ≈ . FBDI Q ∈ 1 XIFUIFS UIF CBMM #(Q, ϵG(Q)) DPOUBJOT BOZ QPJOU GSPN 1 PUIFS UIBO Q JUTFMG *G JU EPFT Q JT SFNPWFE GSPN 1 *O PVS WFSTJPO PG UIJT BMHP SJUIN XF QSPDFTT UIF QPJOUT JO B SBOEPN PSEFS CFDBVTF BO PSEFS JO XIJDI TVCTFRVFOU QPJOUT BSF UPP DMPTF NBZ DBVTF HBQT JO UIF TJNQMJĕFE QPJOU DMPVE ćF -'4 UIVT HJWFT B TJNQMF BOE FČFDUJWF EFĕOJUJPO PG UIF TJ[F PG B GFBUVSF BOE XJUI UIF ϵ TBNQMJOH DSJUFSJPO XF BSF BCMF UP SFMBUF UIF -'4 EJSFDUMZ UP UIF MPDBM TBNQMJOH EFOTJUZ 8F UIVT PCUBJO B HFPNFUSZ EFQFOEFOU TJNQMJĕDBUJPO PG UIF QPJOU DMPVE XIFSF BSFBT XJUI B MBSHF -'4 BSF SFQSFTFOUFE XJUI SFMBUJWFMZ GFXFS QPJOUT UIBO BS FBT XJUI B TNBMM -'4. 4JODF PVS CBMMTISJOLJOH BMHPSJUIN SFRVJSFT B OPSNBM WFDUPS GPS FWFSZ JOQVU QPJOU XF DPNQVUF UIFTF CFGPSFIBOE VTJOH B QSJODJQBM DPNQPOFOU BOBMZTJT PG UIF OFBSFTU OFJHICPVST PG FWFSZ QPJOU 5P TQFFE VQ OFBSFTU OFJHICPVS RVFSJFT GPS UIF OPSNBM DPNQVUBUJPO UIF ."5 FTUJNB UJPO UIF -'4 FTUJNBUJPO BOE UIF ϵTBNQMF TJN QMJĕDBUJPO XF VTF B LEUSFF 'VSUIFSNPSF UP NBLF UIF DPNQVUBUJPO PG UIF -'4 NPSF SPCVTU XF DPNQVUF UIF -'4 UIF EJTUBODF GSPN B TVS GBDF QPJOU UP UIF ."5 BT UIF NFEJBO EJTUBODF PG UIF OFBSFTU ."5 QPJOUT 5P QFSGPSN TQMBU UJOH XF JNQMFNFOU UIF TQMBU SFOEFSJOH NFUIPE PG #PUTDI BOE ,PCCFMU VTJOH 0QFO(TIBEFST. 5P DPNQMFUF PVS WJTVBMJTBUJPO BQQSPBDI XF BMTP NBLF UIF TQMBU SBEJVT BEBQUJWF UP UIF -'4 &BDI QPJOU JT SFOEFSFE BT B TQMBU XJUI B SBEJVT TFU UP ϵG(Q) TFF 'JHVSF 1PJOUT JO BSFBT XJUI B MPXFS QPJOU EFOTJUZ JO UIF ϵTBNQMF BSF UIFSFGPSF ESBXO XJUI MBSHFS TQMBUT .PSFPWFS CFDBVTF CPUI UIF TQMBU SBEJJ BOE UIF ϵTBNQMF BSF CPUI CBTFE PO UIF EJTUBODF ϵG(Q) UIF SFTVMUJOH WJTV 8F IBWF SFMFBTFE PVS 1ZUIPO JNQMFNFOUBUJPO BMJTBUJPO JT TVDI UIBU B TVSGBDFMJLF JNQSFTTJPO JT VOEFS BO PQFOTPVSDF MJDFOTF BOE NBEF JU QVC PCUBJOFE XIFSF IPMFT BSF NJOJNJTFE EFTQJUF UIF MJDMZ BWBJMBCMF 1FUFST . .

(11) QPJOUT BSF OP MPOHFS DMPTF UP UIF NBJO ."5 CSBODIFT BT B SFTVMU PG UIF TNBMM QFSUVSCBUJPOT JO 1O GPS XIJDI UIF VONPEJĕFE CBMMTISJOLJOH JT QBSUJDVMBSMZ TFOTJUJWF 'JHVSF B JMMVTUSBUFT IPX UIFTF QPJOUT BSF NPWFE UPXBSET M[S] BT B SF TVMU PG PVS CBMMQSFTFSWBUJPO IFVSJTUJD UIFSFCZ PCUBJOJOH B EFOTFS ."5 BQQSPYJNBUJPO UIBO QPTTJCMF XIFO UIFTF QPJOUT XPVME IBWF CFFO SF NPWFE FH "NFOUB FU BM . 'SPN PVS FYQFSJNFOUT XJUI SFBMXPSME -J%"3 EBUBTFUT XF PCTFSWF UIF TBNF CFIBWJPVS TFF 'JHVSFT BOE 'JHVSF 4JNVMBUFE BFSJBM -J%"3 QPJOU DMPVE "T B SFTVMU PG UIF QMBOF EFUFDUJPO IFVSJT GPS B IPVTF NPEFM EJNFOTJPOT × UJDT PG UIF FYUFSJPS ."5 QPJOUT BSF ĕM UFSFE BU UQMBOBS = ○ TJNJMBS UP UIF QMBOF EFUFD × N QPJOUT UJPO JO 'JHVSF 'VSUIFSNPSF GSPN UIF QMPU PG ME [1] XF TFF UIBU PVS EFOPJTJOH IFVSJTUJDT ."5 BQQSPYJNBUJPO BOE OPJTF IBOEMJOH IBWF B OFHMJHJCMF FČFDU PO UIF BQQSPYJNBUJPO PG UIF ."5 PG UIF OPJTFGSFF QPJOUDMPVE 1 *O PSEFS UP RVBOUJUBUJWFMZ BOBMZTF UIF FČFDU PG UIF OPWFM EFOPJTJOH IFVSJTUJDT UIBU XF QSPQPTF *U DBO BMTP CF PCTFSWFE FTQFDJBMMZ GSPN 'JH JO 4FDUJPO XF VTF B QPJOU DMPVE UIBU XBT VSF UIBU TPNF PG UIF NJOPS TJEFCSBODIFT PCUBJOFE CZ TJNVMBUJOH BO BFSJBM -J%"3 TDBOƭ EJTBQQFBS PS TISJOL JO UIF EFOPJTFE ."5 8F PO B TVSGBDF NPEFM PG B IPVTF TFF 'JHVSF DBO DPODMVEF UIBU UIFSF JT B USBEFPČ CFUXFFO /PJTF JT TJNVMBUFE CZ BEEJOH B OPSNBMMZ EJT UIF BNPVOU PG EFUBJM DBQUVSFE CZ UIF ."5 BOE USJCVUFE OPJTF DPNQPOFOU XJUI B TUBOEBSE EF UIF SPCVTUOFTT UP UIF OPJTF QSFTFOU JO UIF JOQVU WJBUJPO PG DN BMPOH UIF TDBOOJOH EJSFDUJPO QPJOU DMPVE 8IFSF PVS EFOPJTJOH BQQSPBDI EJT BOE BO BEEJUJPOBM DN JO UIF QPTJUJPO PG UIF UJOHVJTIFT JUTFMG JT UIBU JU EFMJWFST B NVDI EFOTFS TDBOOFS 8F SBO UIF VONPEJĕFE CBMMTISJOLJOH ."5 BQQSPYJNBUJPO UIBO QSFWJPVT NFUIPET BMHPSJUIN .B FU BM EFOPUFE M BOE XIFSF OPJTZ QPJOUT BSF SFNPWFE "MTP EFTQJUF PVS JNQSPWFE WBSJBOU XJUI EFOPJTJOH IFVSJTUJDT UIF EFQFOEFODF PG UXP VTFSEFĕOFE UISFTI EFOPUFE ME PO UIJT QPJOU DMPVE XJUI BEEFE PMET XF ĕOE UIBU UIF TBNF QBSBNFUFS WBMVFT OPJTF EFOPUFE 1O BOE XJUIPVU BEEFE OPJTF EF UQSFTFSWF = ○ BOE UQSFTFSWF = ○ BSF BEFRVBUF OPUFE 1 5P RVBOUJGZ UIF RVBMJUZ PG UIF SFTVMU GPS CPUI UIF VSCBO BOE UIF NPVOUBJO EBUBTFU JOH ."5 BQQSPYJNBUJPO XF NFBTVSF UIF EJT UBODF UP B SFGFSFODF ."5 UIBU JT EJSFDUMZ EF 6MUJNBUFMZ B NVDI NPSF EJTUJODUJWF BOE NPSF SJWFE GSPN UIF TVSGBDF NPEFM HFPNFUSZ EF VTFGVM ."5 BQQSPYJNBUJPO JT PCUBJOFE 8JUI OPUFE M[S] 'SPN 'JHVSF C XF PCTFSWF PVU PVS EFOPJTJOH IFVSJTUJDT DPNQVUJOH UIF -'4 BO PWFSBMM EFDSFBTF PG BU B DPOTFSWBUJWF XPVME OPU CF GFBTJCMF TJODF UIFSF BSF NBOZ ."5 UQSFTFSWF = ○ JO UIF FSSPS PG M[1O ] BT B SFTVMU QPJOUT OFBS UIF TBNQMFE TVSGBDF UIBU EJTUPSU UIF PG PVS CBMMQSFTFSWBUJPO EFOPJTJOH IFVSJTUJD UIBU -'4 DPNQVUBUJPO TFF 'JHVSF ćFSFGPSF JU BČFDUFE PG UIF JOUFSJPS ."5 QPJOUT ćFTF XPVME OPU CF QPTTJCMF UP TVDDFTTGVMMZ QFSGPSN PVS HFPNFUSZEFQFOEFOU QPJOU DMPVE TJNQMJĕ ƭ6TJOH JOEVTUSZ TUBOEBSE QBSBNFUFST ĘJHIU IFJHIU N ĘJHIU MJOF TQBDJOH N BOE UIF #MFO4PS TPęXBSF DBUJPO BOE TQMBUSBEJVT EFUFSNJOBUJPO GPS WJTV (TDIXBOEUOFS FU BM . BMJTBUJPO. .

(12) Top-view. Without denoising. With denoising. Front-view. ball preservation. B $SPTT TFDUJPO PG TJNVMBUFE QPJOU DMPVE XJUI OPJTF 1O. SFGFSFODF ."5 M[S] BOE M[1O ] 3FE MJOFT JOEJDBUF IPX QPJOUT ."5 BSF NPWFE BT B SFTVMU PG PVS EFOPJTJOH IFVSJTUJDT UQSFTFSWF = ○ BOE UQSFTFSWF = ○. plane detection. 'JHVSF .FEJBM BYJT BQQSPYJNBUJPO EBUBTFU # XJUIPVU EFOPJTJOH .B FU BM . BOE XJUI EFOPJTJOH PVS NFUIPE GPS UQSFTFSWF = ○ BOE UQMBOBS = ○ 4IPXO BSF UPQ WJFXT UPQ BOE TJEF WJFXT CPUUPN *OUFSJPS QPJOUT JO SFE &YUFSJPS QPJOUT JO QVSQMF. C 0WFSBMM FSSPS JO ."5 BQQSPYJNBUJPO XJUI SFTQFDU UP SFG FSFODF ."5 XJUI BOE XJUIPVU PVS EFOPJTJOH IFVSJTUJDT. 'JHVSF 4FRVFODF PG TISJOLJOH CBMMT PO B % EBUBTFU GPS B TJOHMF JOQVU QPJOU. .

(13) With denoising. Front-view. Top-view. Without denoising. 'JHVSF &YUFSJPS ."5 BQQSPYJNBUJPO XJUI PVS NFUIPE BOE XJUIPVU EFOPJTJOH .B FU BM . IFVSJTUJDT UQSFTFSWF = ○ BOE UQMBOBS = ○ 'JHVSF JMMVTUSBUFT WFSZ DMFBSMZ UIF TLFMFUPO MJLF OBUVSF PG UIF ."5 PG UIF TBNQMF EBUBTFU XIFSF UIF SJEHFT BOE WBMMFZT PG UIF NPVOUBJO SBOHF BSF USBOTMBUFE UP CSBODIFT PG UIF JOUFSJPS BOE FYUFSJPS ."5 SFTQFDUJWFMZ. 4DBMJOH. MJNJUFE ćF BNPVOU PG SFRVJSFE NFNPSZ JT OPX CPVOEFE CZ UIF MBSHFTU OVNCFS PG QPJOUT JOTJEF B UJMF SBUIFS UIBO UIF UPUBM QPJOU DPVOU PG UIF EBUBTFU "T B SFTVMU XF BSF BCMF UP QSPDFTT NBT TJWF EBUBTFUT 'VSUIFSNPSF PUIFST IBWF TIPXO UIBU IJHIMZ FďDJFOU QBSBMMFM JNQMFNFOUBUJPOT PG UIF CBMM TISJOLJOH BMHPSJUIN BSF GFBTJCMF .B FU BM +BMCB FU BM PVS BEEFE EFOPJTJOH IFVSJT UJDT EP OP BMUFS UIF NFNPSZ SFRVJSFNFOUT PG UIF CBMMTISJOLJOH BMHPSJUIN 8F UIFSFGPSF DPO DMVEF UIBU PVS BQQSPBDI UP PCUBJO B SPCVTU ."5 BQQSPYJNBUJPO PG B -J%"3 QPJOU DMPVE JT XFMM TDBMBCMF JO UFSNT PG CPUI NFNPSZ BOE DPNQV UBUJPOBM DPTU. 'JHVSF TIPXT UIF NFNPSZ VTBHF PG PVS TJN QMF QBSUJUJPOJOH TDIFNF EVSJOH UIF ."5 DPN QVUBUJPO PG B CJMMJPO QPJOU EBUBTFU 8F NVTU OPUF UIBU PVS DVSSFOU JNQMFNFOUBUJPO IBT JOFG ĕDJFOU NFNPSZ NBOBHFNFOU XIJDI JT JOIFSFOU UP UIF VTFE QSPHSBNNJOH MBOHVBHF 1ZUIPO ćF NFNPSZ NFBTVSFNFOUT JO 'JHVSF BSF UIFSFGPSF FYIJCJUJOH B TMJHIUMZ JODSFBTJOH USFOE PWFS UJNF JF BT UIF OVNCFS PG QSPDFTTFE QPJOUT ."5CBTFE TJNQMJöDBUJPO BOE TQMBUCBTFE WJTVBMJTBUJPO JODSFBTFT UIBU JT JOEFQFOEFOU GPS UIF UIFPSFU JDBM NFNPSZ SFRVJSFNFOUT PG UIF BMHPSJUIN /POFUIFMFTT XF NBLF UIF LFZ PCTFSWBUJPO UIBU 'JHVSFT BOE EFNPOTUSBUF UIF FČFDU PG UIF UIF BNPVOU PG SFRVJSFE NFNPSZ JT TVDDFTTGVMMZ ."5CBTFE TJNQMJĕDBUJPO BOE TQMBUT XJUI UIFJS. .

(14) C 8JUI EFOPJTJOH UQSFTFSWF = ○ UQMBOBS = ○. B 8JUIPVU EFOPJTJOH. 'JHVSF -PDBM GFBUVSF TJ[F BQQSPYJNBUJPOT GPS BO VSCBO EBUBTFU XJUI BOE XJUIPVU EFOPJTJOH 3FE JOEJDBUFT MPX MPDBM GFBUVSF TJ[F CMVF IJHI MPDBM GFBUVSF TJ[F. B 3BX QPJOU DMPVE. C *OUFSJPS BOE FYUFSJPS ."5. D *OUFSJPS ."5. E &YUFSJPS ."5. 'JHVSF ćF EFOPJTFE ."5 GPS UIF NPVOUBJO EBUBTFU $PMPST JOEJDBUF FMFWBUJPO. .

(15) ćF ĘBU SFHJPO IBT GFXFS TBNQMFT EVF UP UIF SFM BUJWFMZ IJHI -'4 #VU JO UIF DBTF PG -'4TJ[FE TQMBUT UIF MBSHFS TQMBU SBEJJ FČFDUJWFMZ DPNQFO TBUF GPS UIF DPBSTFS QPJOU EJTUSJCVUJPO MFBEJOH UP B WJSUVBMMZ IPMFGSFF WJTVBMJTBUJPO. %JTDVTTJPO 'JHVSF .FNPSZ DPOTVNQUJPO BT B GVODUJPO PG UIF OVNCFS PG QSPDFTTFE QPJOUT GPS UIF ."5 BQQSPYJNBUJPO PG UIF CJMMJPO QPJOU VSCBO EBUBTFU SBEJJ BEBQUFE UP UIF -'4 *O CPUI DBTFT UIF TJNQMJĕDBUJPO SFNPWFE PG UIF PSJHJOBM QPJOUT ϵ = . ZFU XIFO SFO EFSFE XJUI -'4TJ[FE TQMBUT UIF SFTVMUJOH WJTV BMJTBUJPO JT TJNJMBS UP UIF PSJHJOBM TQMBUUFE QPJOU DMPVE XJUI ĕYFE TQMBUSBEJJ 8IJMF UIF TJNQMJ ĕFE -'4TQMBUUFE SFOEFSJOH JT OPU BCTPMVUFMZ GSFF GSPN IPMFT GPS UIF VSCBO EBUBTFU TFF NBSL JO 'JHVSF XF BMTP PCTFSWF UIBU EFTQJUF UIF SFEVDUJPO JO QPJOUT UIF TQBSTFMZ TBNQMFE WFS UJDBM TVSGBDFT XBMMT OPX BQQFBS BT TPMJE TVS GBDFT TFF NBSL JO 'JHVSF E ćJT JT B OP UBCMF JNQSPWFNFOU PWFS ĕYFETJ[FE TQMBUT CF DBVTF UIJT BNQMJĕFE UIF WJFXFST TFOTF PG TUSVD UVSF BOE EFQUI BU BMM WJFXJOH EJTUBODFT 'JHVSF JMMVTUSBUFT GVSUIFS IPX UIF EJTUSJCVUJPO PG QPJOUT JO UIF TJNQMJĕFE QPJOU DMPVE SFTQFDUT UIF HFPNFUSZ PG UIF TBNQMFE TVSGBDF 4QMBUT BSF ESBXO UIFSF XJUI B EFDSFBTFE SBEJVT TP UIBU JU JT DMFBSMZ WJTJCMF UIBU NPSF QPJOUT BSF ESBXO JO BSFBT XJUI B SFMBUJWFMZ IJHI DVSWBUVSF TVDI BT UIF DSFBTFT JO UIF WBMMFZT BOE UIF DPSSFTQPOE JOH TQMBUT IBWF B TNBMMFS SBEJVT XIFO DPNQBSFE UP UIF QMBOBS BSFBT XJUI GFXFS BOE MBSHFS TQMBUT BMTP BQQBSFOU JO 'JHVSF E 'JOBMMZ JO 'JH VSF B XF DPNQBSF ĕYFETJ[FE TQMBUT XJUI -'4 TJ[FE TQMBUT GPS UIF TJNQMJĕFE NPVOUBJO EBUBTFU. 8F IBWF TIPXO UIBU B SPCVTU ."5 DBO CF DPN QVUFE GSPN -J%"3 QPJOU DMPVET BOE UIBU XF DBO VTF UIF ."5 UP QFSGPSN HFPNFUSZEFQFOEFOU TJNQMJĕDBUJPO BOE UP JNQSPWF TQMBUCBTFE WJTV BMJTBUJPO PG -J%"3 QPJOU DMPVET " LFZ NP UJWBUJPO GPS UIF EFWFMPQNFOU PG UIJT BQQSPBDI XBT UIF EJTUPSUFE EFQUIQFSDFQUJPO BOE TFOTF PG TUSVDUVSF EVF UP UIF QSFTFODF PG IPMFT XIFO B QPJOU DMPVE JT WJFXFE VQ DMPTF TFF 'JHVSF 'SPN 'JHVSF JU JT PCWJPVT UIBU TQMBUCBTFE SFOEFSJOH HJWFT TJHOJĕDBOUMZ MFTT BOE TNBMMFS IPMFT 'VSUIFSNPSF XJUI PVS ."5CBTFE QPJOU TJNQMJĕDBUJPO BOE TQMBUSBEJJ XF DBO NBJOUBJO UIJT CFOFĕU XIJMF SFUBJOJOH POMZ PG UIF QPJOUT )PXFWFS OPU BMM IPMFT BSF FMJNJOBUFE BOE TPNF OFX IPMFT BSF FWFO JOUSPEVDFE ćJT IBT UXP DBVTFT 'JSTU UIF MBSHFTU IPMFT BSF NBJOMZ B SFTVMU PG JOTVďDJFOU TBNQMJOH PG UIF -J%"3 QPJOU DMPVE EVF UP PDDMVTJPO FH NJTTJOH XBMMT. BOE UIF NBUFSJBM BOE PSJFOUBUJPO PG UIF TVSGBDF XJUI SFTQFDU UP UIF MBTFS TDBOOFS BMTP OPUFE CZ ,PWBʊ BOE ÇBMJL . 8JUIPVU FJUIFS NBL JOH FYQMJDJU BTTVNQUJPOT PO UIF TIBQF PG PC KFDUT PS BDRVJSJOH NPSF TBNQMFT JO UIF ĕFME MJU UMF DBO CF EPOF BCPVU UIJT JO PVS PQJOJPO 4FD POE XIJMF PVS TJNQMJĕDBUJPO QSPDFEVSF HVBSBO UFFT UIBU BSFBT BSF OPU PWFSTBNQMFE BDDPSEJOH UP UIF ϵTBNQMF DSJUFSJPO JU EPFT OPU HVBSBOUFF UIBU TVďDJFOU QPJOUT BSF QSFTFSWFE JO BMM DBTFT ćVT BMUIPVHI UIF MPDBM QPJOU EFOTJUJFT BSF BQQSPYJ NBUFMZ UIF TBNF EFQFOEJOH PO UIF SBOEPN J[FE PSEFS JO XIJDI QPJOUT BSF QSPDFTTFE EVS JOH UIF TJNQMJĕDBUJPO IPMFT NBZ BQQFBS JO TPNF QMBDFT 5P TPMWF UIJT JTTVF UIF MPDBM QPJOU EJTUSJ. .

(16) B 'VMM QPJOU DMPVE XJUI TJNQMF QPJOUT. C 'VMM QPJOU DMPVE XJUI ĕYFESBEJVT TQMBUT. D 4JNQMJĕFE QPJOU PG QPJOUT SFNPWFE DMPVE XJUI TJNQMF QPJOUT. E 4JNQMJĕFE QPJOU PG QPJOUT SFNPWFE DMPVE XJUI MGTSBEJVT TQMBUT. 'JHVSF 7JTVBMJTBUJPO SFTVMUT GPS UIF .PVOUBJO EBUBTFU ϵ = . . .

(17) B 'VMM QPJOU DMPVE XJUI TJNQMF QPJOUT. C 'VMM QPJOU DMPVE XJUI ĕYFESBEJVT TQMBUT. D 4JNQMJĕFE QPJOU PG QPJOUT SFNPWFE DMPVE XJUI TJNQMF QPJOUT. E 4JNQMJĕFE QPJOU PG QPJOUT SFNPWFE DMPVE XJUI MGTSBEJVT TQMBUT. 'JHVSF 7JTVBMJTBUJPO SFTVMUT GPS UIF 6SCBO EBUBTFU ϵ = . /PUF UIF EJČFSFOU QPJOUEFOTJUJFT PO WFSUJDBM BOE IPSJ[POUBM TVSGBDFT NBSLFE BOE . 'JHVSF ."5CBTFE TJNQMJĕDBUJPO BOE TQMBUSBEJJ ćF TQMBU SBEJJ JO UIJT JNBHF BSF SFEVDFE GPS JMMVTUSBUJWF QVSQPTFT. .

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