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ÐÀÇÍÎÅ<br>7.3. Ïîâîðîò 3D âåêòîðà çà øåñòü óìíîæåíèé

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<p>Îáû÷íî ïîâîðîò 3D âåêòîðà äåëàþò óìíîæåíèåì ìàòðèöû ïîâîðîòà íà ýòîò âåêòîð.
Ýòà îïåðàöèÿ òðåáóåò 9 óìíîæåíèé è 6 ñëîæåíèé. Íî ñ èñïîëüçîâàíèåì íåáîëüøîãî
precalculation (ïðåäâàðèòåëüíîãî ðàñ÷åòà) åå ìîæíî íåñêîëüêî óñêîðèòü.

<p>Ïóñòü íàì íàäî óìíîæèòü êàêóþ-òî ñòðîêó ìàòðèöû (a,b,c) íà âåêòîð (x,y,z).
Ðåçóëüòàò äîëæåí áûòü ðàâåí

<p class=expression>r = a*x+b*y+c*z.<br>

<p>Òî åñòü êàê ðàç ïî 3 óìíîæåíèÿ è 2 ñëîæåíèÿ íà îäíó ñòðîêó. Íî ñ äðóãîé
ñòîðîíû,

<p class=expression>r = a*x+b*y+c*z = (a*x+b*y+a*b+x*y)+c*z-a*b-x*y =
(a+y)*(b+x)+c*z-a*b-x*y.<br>

<p>Ïðîùå ýòà ôîðìóëà ÿâíî íå âûãëÿäèò. Íî äåëî â òîì, ÷òî x*y - ýòî ïîñòîÿííàÿ
âåëè÷èíà, òàê êàê x, y - ýòî êîîðäèíàòû âåðøèíû íåïîâåðíóòîãî îáúåêòà, à îíè
îáû÷íî íå ìåíÿþòñÿ. À a*b äîñòàòî÷íî ïîñ÷èòàòü ïðè ðàñ÷åòå ìàòðèöû ïîâîðîòà,
è ýòî òîæå ïîñòîÿííàÿ âåëè÷èíà äëÿ êàæäîé ìàòðèöû. Ò.î.

<p class=expression>r = (a+y)*(b+x)+c*z-c1-c2.<br>

<p>Â ðåçóëüòàòå èìååì 2 óìíîæåíèÿ è 4 ñëîæåíèÿ íà îäíó ñòðîêó, òî åñòü òå ñàìûå
6 óìíîæåíèé è 12 ñëîæåíèé íà âåêòîð. Âûèãðàëè 3 óìíîæåíèÿ öåíîé 6 ñëîæåíèé.

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