Wannan takarda ta zayyana aikin kera octahedron na yau da kullun ta amfani da na'urar bugawa 3D. Tana haɗa lissafin jiki mai ra'ayi da kera dijital na aiki. Tsarin ya haɗa da lissafin ƙarshen (vertices) da fuskar polyhedron, ƙirar samfurin 3D na zahiri a cikin OpenSCAD, samar da fayil ɗin STL, da kuma samar da abu na zahiri a ƙarshe. Aikin yana ɗaukar sanin ƙa'idodin bugawa 3D na asali.
Octahedron na yau da kullun wani tsari ne na Platonic mai fuskar triangular masu daidaitaccen gefe takwas da ƙarshen (vertices) shida. Ƙirar lissafi ta farko ita ce tushen ƙirar dijital.
Ana iya gina octahedron a cikin $\mathbb{R}^3$ ta hanyar farawa da murabba'i mai tsayin gefe $s$ a cikin filin xy. Wani layi daidai da filin yana wucewa ta tsakiyar murabba'in. Wurare biyu akan wannan layin (ɗaya sama, ɗaya ƙasa da filin) an sanya su ta yadda nisan su zuwa duk kusurwoyi huɗu na murabba'in ya yi daidai da $s$. Waɗannan wurare shida su ne ƙarshen (vertices).
Ana saita $s = 1$, kusurwoyin murabba'in an ayyana su kamar haka: $p_0 = (0,0,0)$, $p_1 = (1,0,0)$, $p_2 = (1,1,0)$, $p_3 = (0,1,0)$. Layin daidai shi ne axis na z ta hanyar $(0.5, 0.5, 0)$. Ƙarshen sama da ƙasa $p_4$ da $p_5$ ana samun su ta hanyar warware ma'aunin nisa daga $(0.5, 0.5, \hat{z})$ zuwa kowane kusurwa: $(0.5)^2 + (0.5)^2 + \hat{z}^2 = 1^2$. Wannan yana haifar da $\hat{z} = \pm\sqrt{0.5} \approx \pm 0.707$. Don haka, $p_4 = (0.5, 0.5, 0.707)$ da $p_5 = (0.5, 0.5, -0.707)$.
An ayyana ƙarshen (vertices) da fuska a cikin lambar OpenSCAD don samar da samfurin 3D. Ana ayyana fuska ta hanyar jera lambobin ƙarshen (vertex indices) cikin tsari na agogo.
polyhedron(
points = [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [1.0, 1.0, 0.0],
[0.0, 1.0, 0.0], [0.5, 0.5, 0.707], [0.5, 0.5, -0.707]],
triangles = [[4, 1, 0], [4, 2, 1], [4, 3, 2], [4, 0, 3],
[5, 0, 1], [5, 1, 2], [5, 2, 3], [5, 3, 0]]
);Wannan yana ƙirƙirar samfuri mai daidaitaccen lissafi amma ba samfurin da za a iya bugawa nan da nan ba (Hoto na 1 a cikin PDF).
Daidaituwa da ƙirar lissafi don kera zahiri yana buƙatar magance ƙuntatawa na aiki na fasahar bugawa 3D.
An gano batutuwa biyu masu mahimmanci: 1) Girman naúrar samfurin (naúra 1) ya yi ƙanƙanta ga na'urorin bugawa 3D na yau da kullun waɗanda suka dogara da milimita, yana buƙatar sikelin. 2) Dole ne abubuwa su kasance da tushe mai ƙarfi, lebur akan farantin gini (filin xy). Kawai fassara samfurin don ƙarshen (vertex) ya taɓa farantin ba ya isa, saboda madaidaicin batu ba ya ba da kwanciyar hankali.
Maganin ya ƙunshi jujjuya octahedron a kan axis na x (wanda ya ƙunshi $p_0$ da $p_1$) ta kusurwa $\alpha$ ta yadda ƙarshen (vertex) $p_4$ ya motsa zuwa filin xy, yana tabbatar da duk $z \ge 0$. Matrix ɗin juyawa shine: $$R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha \\ 0 & \sin\alpha & \cos\alpha \end{bmatrix}$$ Yin amfani da shi ga $p_4 = (0.5, 0.5, \sqrt{0.5})$ da saita sakamakon z-coordinate zuwa sifili yana ba da sharadi: $\frac{1}{2}\sin\alpha + \frac{\sqrt{2}}{2}\cos\alpha = 0$. Wannan yana sauƙaƙa zuwa $\tan\alpha = -\sqrt{2}$, yana haifar da $\alpha \approx -54.74^\circ$.
Yin amfani da juyawar $R$ ga duk ƙarshen (vertices) (kuma daga baya sikelin) yana samar da octahedron mai kwanciyar hankali, mai yiwuwar bugawa wanda ke zaune a filin xy. Ƙarshen da aka canza (zuwa uku bayan decimal) sune: $\hat{p}_0=(0.0,0.0,0.0)$, $\hat{p}_1=(1.0,0.0,0.0)$, $\hat{p}_2=(1.0,-0.577,0.816)$, $\hat{p}_3=(0.0,-0.577,0.816)$, $\hat{p}_4=(0.5,-0.865,0.0)$, $\hat{p}_5=(0.5,0.288,0.816)$. An nuna wannan samfurin a Hoto na 2 na PDF.
Takardar ta gabatar da sakamako guda biyu masu mahimmanci na gani (hotuna):
Samun nasarar samar da waɗannan samfuran daban-daban guda biyu yana tabbatar da fitar da lissafi da wajibcin matakin canji.
Tsarin don Binciken "Ƙira-don-3D-Bugawa":
Wannan takarda ta yi amfani da tsarin da ake iya amfani da shi a ɓoye don canza kowane samfurin lissafi don ƙari na masana'antu. Ana iya tsara matakan kamar haka:
Misalin Lamari (Aiwatar da Tsarin):
Matsala: Buga tetrahedron na yau da kullun mai tsayin gefe 10mm.
Mataki na 1 & 2: Ayyana ƙarshen (vertices), misali, (0,0,0), (10,0,0), (5, 8.66, 0), (5, 2.89, 8.16). Samfurin a cikin CAD.
Mataki na 3 Bincike: Samfurin yana kwance akan fuskar triangular ɗaya (kyakkyawan kwanciyar hankali). Duk da haka, ƙarshen (vertices) na fuskar suna da z=0, amma maki na ciki na fuskar suma suna da z=0, suna ƙirƙirar tushe mai kyau. Sikelin daidai yake (10mm).
Mataki na 4 Canji: A wannan yanayin, fuskantarwar farko ta riga ta fi dacewa. Ba a buƙatar juyawa, kawai watakila fassara don tsakiyar farantin gini.
Wannan misalin yana nuna yadda tsarin ke jagorantar yanke shawara, yana iya adana lokaci da kayan aiki idan aka kwatanta da gwaji da kuskure.
Ƙa'idodin da aka nuna suna da fa'ida mai faɗi fiye da polyhedron guda ɗaya:
Fahimta ta Tsaki: Aikin Aboufadel wani babban darasi ne a cikin tazarar da aka yi watsi da ita sau da yawa tsakanin ƙirar lissafi mai tsabta da kera dijital na aiki. Yana fallasa gaskiya mai mahimmanci: cikakkiyar ƙirar CAD ta lissafi sau da yawa gazawar masana'antu ce. Ƙimar gaske ta takarda ba ta cikin fitar da ƙarshen (vertices) octahedron ba—wanda aka warware matsalar—amma a cikin rubuta cikakken bayani game da aikin sarrafa bayanai (juyawa, sikelin) da ake buƙata don haɗa rarrabuwar dijital da zahiri. Wannan ya yi daidai da binciken daga Cibiyar Bits da Atoms ta MIT, wadda ta jaddada "ƙira don kera" a matsayin fanni daban daga ƙirar lissafi.
Kwararar Hankali: Takardar tana bin tsarin aikin injiniya mara kyau: 1) Ma'anar (ƙuntatawa na lissafi), 2) Maganin (lissafin matsayi), 3) Aiwatarwa (lambar OpenSCAD), da 4) Daidaituwa (don masana'antu). Wannan yayi daidai da tsarin da aka saba yi a cikin binciken ƙari na masana'antu, kamar yadda aka zayyana a cikin bita kamar waɗanda ke cikin mujallar Ƙari na Masana'antu. Duk da haka, kwararar ta nuna cewa Mataki na 4 ba shi da sasantawa kuma sau da yawa ya fi rikitarwa fiye da ƙirar farko.
Ƙarfi & Kurakurai: Ƙarfinsa shine bayyananniyar koyarwa da aiki na hannu. Yana ba da cikakkiyar girke-girke, mai maimaitawa. Kurakurai, daga mahangar masana'antu, shine yanayin hannu, na ɗaya-lokaci. An warware kusurwar juyawa $\alpha$ ta hanyar bincike don wannan takamaiman lamari. A cikin software na ƙwararrun CAD/CAE, za a sarrafa wannan ta hanyar masu warware ƙuntatawa ko algorithms na ƙira waɗanda ke la'akari da fuskantar bugawa da rage tallafi ta atomatik, kamar yadda ake gani a kayan aiki kamar Autodesk Netfabb ko Siemens NX. Hanyar takardar ba ta da girma zuwa sifofi masu rikitarwa, waɗanda ba na yau da kullun ba.
Fahimta masu Aiki: Ga malamai, wannan cikakkiyar module ce don darussan STEM waɗanda ke haɗa lissafi da injiniya. Ga masu aiki, abin da ya fi mahimmanci shine koyaushe a yi la'akari da axis na masana'antu da kwanciyar hankali na tushe tun daga farko. Ya kamata tsarin ya sanar da zaɓin tsarin matsayi na farko. Bugu da ƙari, wannan binciken lamari yana jayayya don haɓaka kayan haɗin "binciken yiwuwar bugawa" don kayan aikin buɗe ido kamar OpenSCAD, yana sarrafa irin binciken da aka yi da hannu a nan. Nan gaba yana cikin saka ƙuntatawa na masana'antu kai tsaye cikin madauki na ƙira.