1° LEZIONE: punto, retta, piano, operazione proiettiva

Per poter entrare a fondo nel campo del disegno dell'architettura bisogna addentrarsi un secondino in un ambito che non alletta molto: la geometria proiettiva! Tranquilli, cercherò di essere più breve possibile! :) 

La geometria elementare riguarda le proprietà dimensionali delle figure, e cioè le dimensioni lineari, angolari, di superficie e volume, le congruenze e le equivalenze.
La Geometria Proiettiva, invece, studia le proprietà delle figure rispetto ad una serie di trasformazioni dette Trasformazioni Proiettive. Tali trasformazioni si ottengono sottoponendo le figure ad operazioni di proiezione (da un punto) e di sezione (con un piano), chiamate entrambe operazioni geometriche fondamentali, poichè sono alla base di qualsiasi operazione proiettiva.
Con tali operazioni la figura subisce alterazioni metriche ma rimangono invariate le proprietà proiettive.
Le proprietà proiettive o invarianti di Poncelet rappresentano quelle proprietà che le figure non perdono quando vengono proiettate da un punto su di un piano. Queste proprietà sono:

1)Appartenenza: se un punto P appartiene ad una retta "r" allora anche la proiezione di P apparterrà alla proiezione di "r".

2)Incidenza: se due rette "r" ed "s" sono incidenti in un punto, allora lo saranno anche le loro proiezioni.

3)Collinearità: se più punti sono allineati, allora lo saranno anche le loro proiezioni.

Andiamo a definire ogni ente geometrico: 

-PUNTO: è un ente adimensionale e si indica con una lettera maiuscola dell’alfabeto (P, Q, S…..)

-RETTA: è un ente ad una sola dimensione, quella della lunghezza. Ha una sola direzione, un verso e si indica con una lettera minuscola dell’alfabeto (r, s, t……). Può essere considerata come un insieme di infiniti punti allineati (retta punteggiata). Le rette si distinguono per una diversa direzione nel piano. Rette parallele hanno la stessa direzione

-PIANO: è un ente a due dimensioni, lunghezza e larghezza. Si indica con una lettera greca (α, β, γ..) e può essere considerato sia come l’insieme degli infiniti punti di un piano (piano punteggiato) o anche come l’insieme di infinite rette appartenenti ad uno stesso piano (piano rigato).
I piani si distinguono per una diversa giacitura. Le giaciture fondamentali sono quella orizzontale e verticale. Piani paralleli hanno la stessa giacitura.
 
Per estensione, lo spazio è un ente tridimensionale.

Ricordiamo, ora, alcuni postulati riguardanti gli enti geometrici:

1) Due punti distinti individuano una ed una sola retta.

2) Tre punti non allineati individuano un piano.

3) Per un punto non appartenente ad una retta "r" può passare una sola retta "s" parallela ad "r".



OPERAZIONE PROIETTIVA

Partendo da un fascio di rette proiettanti uscenti da un centro di proiezione C, un piano di proiezione π ed un punto P estraneo ad π, l'operazione di proiezione avviene attraverso 2 fasi:

1) Proiezione: si unisce il punto P al centro di proiezione C.

2) Sezione: si interpone tra P e C un piano π; il punto di sezione P' è la proiezione di P sul piano π.

Purtroppo, questo tipo di operazione è unidirezionale e non biunivoca, cioè non possiamo definire le qualità spaziali di un oggetto a partire dalla sua proiezione bidimensionale; per far ciò abbiamo bisogno di altri metodi che spiegherò nella prossima lezione! :)

Partenope

The Author

"Il nero è sempre stato il mio colore preferito.Ho sempre adorato vederlo nascondersi dietro i colori,per poter dare loro luce e portarli in vita,Generando Emozioni. Architettura,Pittura,Musica,Fotografia, Scrittura sono i colori di NERO CARBONE!"

4 commenti:

  1. HI. I'M AN ENORMOUS EX MAFIA'S MONEY LAUNDERER: STUART NEWBY FROM MONACO. EX KLEINWORT BENSON LONDON. THEN MONTANI MONACO. AS I JUST SAID, I'M A VERY BIG EX MAFIA'S MONEY LAUNDERER ( HERE IS WHILE I TRIED TO RENT A VILLA PURCHASED BY A FREQUENT ASSASSIN MAFIOSO FROM NAPLES, VERY CLOSE TO EQUALLY ASSASSIN MAFIOSO SILVIO BERLUSCONI, WHOSE KILLING MONEY I LAUNDERED FOR DECADES IN COTE D'AZUR https://www.stayonedegree.com/en/home/137/information/view
    ANYWAY, I WASHED MOUNTAINS OF EXTREMELY DIRTY MONEY, IN MY LIFE, ALSO IN LONDON, IN SINGAPORE AND ESPECIALLY IN HONG KONG WITH VERY WELL KNOWN CRIMINAL MICHAEL CHINNICK FROM MANDARIN CAPITAL, EX MORGAN STANLEY LONDON AND HONG KONG
    https://www.mdncapital.com/staff/hong-kong/michael-chinnick
    VERY VERY WELL KNOWN CRIMINAL MATTHEW HOLLAND EX MORGAN STANLEY LONDON AND HONG KONG
    https://www.livemint.com/Companies/reB6t5P0G6G1ZilD2CGSRO/Morgan-Stanleys-Matthew-Holland-retires-from-Asia-equities.html
    AND VERY VERY VERY VERY VERY VERY.. EXTREMELY WELL KNOWN CRIMINAL DANIEL HEGGLIN ALSO EX MORGAN STANLEY LONDON AND NEW YORK
    https://www.py.cz/pipermail/python/2017-March/012858.html
    https://groups.google.com/forum/#!topic/sci.med.cardiology/HnVbofbkrsY ).
    1
    SINCE MANY YEARS, THEN, AS I SAID, I HAVE BEEN WASHING MOUNTAINS OF ASSASSIN MAFIA' S MONEY IN COTE D'AZUR, IN MONACO, IN MONTECARLO, THRU MY EXTREMELY DIRTY COMPANY CALLED MONTANI ! I HAVE BEEN WASHING LOT OF MAFIA KILLING CASH ALSO WITH THIS PEDOPHILE FREEMASON BROTHER OF MINE, WITH THIS MALAVITOSO HOODED FREEMASON BROTHER OF MINE: VERY VERY VERY CRIMINAL PAOLO BARRAI ( FIRED BADLY BY CITIBANK MILAN EXACTLY AS I GOT FIRED BADLY BY KELINWORT BENSON LONDON). BORN IN MILAN ON 28.6.65. BOUT WHO I WANT TO TELL A LOT, NOW. I WANT TO MAKE SOME BIG REPENTANT MAFIOSO'S CONFESSIONS BOUT ALL THIS AND RIGHT NOW!!!!!!!!!!
    STAY VERY AWAY FROM THIS EXTREMELY DANGEROUS PEDOPHILE ALWAYS SODOMIZING CHILDREN: PAOLO PIETRO BARRAI (KNOWN ALL OVER THE WORLD AS " THE PEDOPHILE OF BITEOIN".... BITCOIN: THAT "IT" USES X MAFIA-MONEY-LAUNDERING THRU VERY CRIMINAL EIDOO, VERY CRIMINAL CRYPTOLAB S A, VERY CRIMINAL CRYPTOPOLIS, VERY CRIMINAL MEDICALCHAIN, VERY CRIMINAL BITMAX, VERY CRIMINAL WMO GROUP)! EXTREMELY DANGEROUS PEDERAST PAOLO PIETRO BARRAI WELL KNOWN ALL AROUND THE WORLD AS " THE PEDOPHILE OF BITCOIN"! MAFIA MONEY LAUNDERER PAOLO BARRAI BORN IN MILAN ON 28.6.65 USES BITCOIN TO ORGANIZE THE RAPING OF CHILDREN AND TO PROMOTE PORNOGRAPHY FOR PEDOPHILES LIKE HIM!
    http://noticiasdeportoseguro.blogspot.com/2011/03/quem-e-pietro-paolo-barrai.html
    https://oneway2day.files.wordpress.com/2019/01/4e793-barrai2bind-pag01.jpg
    https://pbs.twimg.com/media/CIB3862WcAA8F7c.png
    https://groups.google.com/forum/#!topic/alt.tv.homicide/KlPMYZEmGIY
    WE GOT LOT OF PROOVES BOUT IT!!! YES, IS HOMOSEXUAL OF PEDOPHILE KIND PAOLO BARRAI: BLOG "MERDATO" LIBERO ALIAS "STALKING SHIT SPREADED IN A FREE WAY WITH NAZIST JOSEPH GOEBBELS'S STYLE". STALKING SHIT BASED ON IMMENSE FAKE IN ORDER TO KILL EVERYONE IS NOT HITLERIAN & ANTI SEMITIC LIKE HIM! PAOLO BARRAI IS A BASTARD ASSASSIN, PAOLO BARRAI IS A PRINCIPAL OF MANY MURDERS! PAOLO BARRAI HAS ALREADY GONE 3 TIMES TO JAIL! PAOLO BARRAI HAS BEEN FIRED BADLY BY CITIBANK MILAN! PAOLO BARRAI HAS BEEN VERY HEAVILY FINED BY CONSOB! PAOLO BARRAI HAS ESCAPED FROM BRASIL AT NIGHT, GETTING A FIRST AVAILABLE FLIGHT, IN THE MIDDLE OF CARNAVAL DAYS, TO NOT FINISH IN JAIL: FOR EXTREMELY SERIOUS CRIMES LIKE HAVING RAPED AN 8 YEARS OLD CHILD, AS WELL AS FOR ENORMOUS FRAUDS, MAFIA MONEY LAUNDERING, NAZIST AND RACIST PROPAGANDA, EXECUTION OF KILLING THREATS, BLACKMAILS AND EXTORSIONS.... AND, LITTLE SWEET CHERRY ON CAKE: PEDERAST SEX HE HAD WITH AND 8 YEARS OLD CHILD.

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  2. 2
    THIS ASSASSIN WORM OF PAOLO BARRAI IS ALSO A MEGA SCAMMER, BURNING ALL SAVINGS OF EVERYONE FALLING IN HIS -CYBER-MEDIATIC-FINANCIAL TRAPS! AL CAPONE OF FINANCE: PAOLO BARRAI BORN IN MILAN ON 28.6.1965 ( LAUNDERING CASH FROM COSA NOSTRA, CAMORRA, NDRANGHETA, AS WELL AS CRIMINAL CASH FROM NAZIRACIST MOVEMENTS LEGA NORD AND FORZA ITALIA, IN EXTREMELY CRIMINAL EIDOO SWITZERLAND AND EXTREMELY CRIMINAL CRYPTOPOLIS SWITZERLAND)!!!
    AGAIN.. AND..AGAIN..AND..AGAIN...
    IS ONE OF MOST BASTARD CRIMINAL CONNECTED TO PANAMA PAPERS, STINKY WORM PAOLO BARRAI, ALREADY THREE TIMES IN JAIL, BORN IN MILAN ON 28.6.1965.
    YES, I AM WRITING BOUT CRIMINAL ASSASSIN PEDOPHILE PAOLO BARRAI FROM MEDICALCHAIN, CRIMINAL ASSASSIN PEDOPHILE PAOLO BARRAI FROM CRYPTOLAB S A,
    CRIMINAL ASSASSIN PEDOPHILE PAOLO BARRAI FROM CRYPTOPOLIS, CRIMINAL ASSASSIN PEDOPHILE PAOLO BARRAI FROM BITMAX, CRIMINAL ASSASSIN PEDOPHILE PAOLO BARRAI FROM VERY FAILING ICO EIDOO, ICO MEGA MAFIA MONEY LAUNDERER FOR NDRANGHETA, VIA STINKY DELINQUENT NETALE FERRARA FROM REGGIO CALABRIA.
    CAREFUL TO THIS LOUSY VERY NASTY, SAVAGE DELINQUENT: PAOLO PIETRO BARRAI EX CITIBANK MILAN, BORN IN MILAN ON 28.06.1965, CONDEMEND TO JAIL ALREADY MANY TIMES!
    ONE OF DIRTIEST MAFIA AND CRIMINAL COLLARS MONEY LAUNDERER IN PANAMA ( AND NOT ONLY IN PANAMA)... EXTREMELY DELINQUENT INDIVIDUAL, ALREADY THREE TIMES IN PRISON ( EITHER IN ITALY AND ESPECIALLY IN BRAZIL), STINKY FELONIOUS PAOLO BARRAI (OR STINKY FELONIOUS PAOLO "PIETRO" BARRAI), BORN IN MILAN ON 28.06.1965.

    FROM VERY VERY CRIMINAL WMO SA PANAMA, VERY VERY CRIMINAL WORLD MAN OPPORTUINITES VIA MAZZINI 14 LUGANO, VERY VERY CRIMINAL BSI ITALIA SRL VIA SOCRATE 26 MILAN, VERY VERY CRIMINAL BLOG MERCATO LIBERO ( NOT FOR NOTHING, EVERYONE IN ITALY AND INITALIAN SPEAKING PART OF SWITZERLAND, KNOWING HOW MANY CRIMES AND WHAT HORRIBLE KIND OF CRIMES, PAOLO BARRAI, ALWAYS DOES, CALLS THIS BLOG: "MERDATO" LIBERO).
    FINED VERY HEAVILY BY CONSOB FOR MEGA FRAUDS HE DID ON PV BUSINESS AND NOT ONLY
    http://www.bluerating.com/trading/179-promotori/28601-qmultaq-da-70mila-euro-per-un-ex-promotore-che-ha-violato-gli-obblighi-informativi.html
    https://www.adviseonly.com/blog/investire/mercati-finanziari/il-trading-dei-miracoli/
    http://www.advisoronline.it/albo/consob/22190-consob-sanziona-a-raffica.action
    FIRED BADLY BY CITIBANK MILAN FOR MEGA ROBERRIES, COLLECTION OF BRIBES AND FRAUDS "IT" USED TO ORGAINZE THERE. MADE CONDEMNED TO JAIL BY CITIBANK MILAN.
    CONDEMNED TO JAIL, THEN, EVEN IN BRAZIL, FOR OTHER FRAUDS, FOR MAFIA MONEY LAUNDERING, FOR ATTEMPTED EXTORTION, FOR DEATH THREATS DONE, FOR BLACKMAILS DONE, FOR INTERNET STALKING DONE AND ALSO ATTEMPTS OF HAVING PEDOPHILE SEX WITH VIEW UNDERAGE LOCAL BOYS ( INFACT, SINCE ALL THIS GOT SENTENCED, EXTREMELY CRIMINAL PAOLO BARRAI HAS NEVER PUT A FEET IN BRAZIL ANYMORE... HERE FOLLOWS A LINK CORRESPONDING TO THE BEGINNING OF ANY INVESTIGATION
    http://www.rotadosertao.com/noticia/10516-porto-seguro-policia-investiga-blogueiro-italiano-suspeito-de-estelionato )
    "IT" TOTALLY BURNED, TOTALLY NULLIFY 100% OF THE SAVINGS SOME CANDID PEOPLE HE SCRUBBED VIA INTERNET, GAVE HIM TO INVEST ( EVEN BECAUSE IN HAVING "NOSE FOR BUSINESS", IN HAVING "BUSINESS INTUITION", PAOLO BARRAI IS AN ABSOLUTE DONKEY: HIS PREVISIONS IN WORLDWIDE STOCKS, COMODITIES, CURRENCIES, REAL ESTATES, ARE ALWAYS WRONG, WRONG, INCREDIBLY WRONG... ALWAYS.... WITH ZERO EXCEPTIONS...EVERY ONE IN FINANCIAL BUSINESS CALLS HIM "REVERSAL INDICATOR"). "IT" HAS BEEN WASHING IN PANAMA ( IN CONNECTION WITH MOSSACK FONSECA AND NOT ONLY), SWITZERLAND ( BY FINTER BANK ZURICH LUGANO AND BANCA STATO LUGANO), GERMANY, SPAIN, ESTONIA AND DUBAI, LOT AND LOT OF
    -ASSASSIN ITALIAN COSA NOSTRA, NDRANGHETA, CAMORRA'S CASH
    -ASSASSIN COLOMBIAN MAFIA CASH

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  3. 3
    -NAZIRACIST LEGA NORD'S CASH ( COMING FROM MEGA CORRUPTION, MEGA ROBBERIES, DIRTY BUSINESSES OF THESE KUKLUKKLANIST MIXED TO NDRANGHETIST FROM LEGA NORD, IN CONNECTION WITH DICTATORS LIKE PUTIN, GHEDDAFI AND LE PEN'S FAMILY)
    HE CALLED MANY TIMES OBAMA BARACK A BLACK BASTARD
    http://ilpunto-borsainvestimenti.blogspot.com/2013/01/ecco-limpatto-fiscale-che-ha-fatto.html
    AN HISTERICAL FAIRY
    http://ilpunto-borsainvestimenti.blogspot.com/2014/03/mercato-libero-e-per-la-grande-madre.html
    ( "IT" CALLS OTHER PEOPLE "FAIRY" BUT THEN ALWAYS LOOKS FOR 8, 10, 12, 14 YEARS OLD GYPSY MALE PROSTITUTES TO MAKE PAEDERAST SEX WITH).
    FASCIST MAFIOSO DICTATOR, PRINCIPAL OF MANY MURDERS AND SLAUGHTERS, AS MUCH AS ASCERTAINED PEDOPHILE SILVIO BERLUSCONI'S CASH
    http://www.mercatolibero.info/panama-mercato-libero/
    INFACT, SILVIO BERLUSCONI, LEGA NORD, COSA NOSTRA, CAMORRA, NDRANGHETA, COLOMBIAN MAFIA, MEXICAN MAFIA, ARE, SINCE ALWAYS, SAME KILLING CHOLERIC SEWER!
    PS
    BE STRONG NOW PLS. AS JUST MENTIONED BEFORE, BUT BETTER EXPLAINED NOW, EXTREMELY CRIMINAL PAOLO BARRAI BORN IN MILAN ON 28.6.1965, HAS THE CARD OF WORLDWIDE ASSOCIATION OF PEDOPHILES: " PEDOPHILE PRIDE".
    "IT" SAID MANY TIMES, IN PRIVATE, "IT" HAS BECOME, SINCE VIEW YEARS, AN HOMOSEXUAL PEDOPHILE. A PAEDERAST. AND "IT" IS PROUD TO MAKE SEX WITH BOYS JUST A LITTLE BIT MORE THAN CHILDREN, BEING KIDS 8, 10, 12, 14 YEARS OLD. "IT" SAID THIS HAPPENS EVERY TIME "IT" GOES TO PANAMA, LATIN AMERICA IN GENERAL AND THAILAND ( AND OUT OF PANAMA PAPERS ARE COMING OUT VARIOUS CASES OF PEDOPHILES AND MAFIOSI HAVING EXTREMELY DIRTY CASH THERE: ONE OF THEIR MAIN CRIMINAL BANKING INTERMEDIARIES WAS THIS "SURPLUS OF PRISON" WHO IS PAOLO BARRAI )!!!!!!!!!

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