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ÒÅÊÑÒÓÐÈÐÎÂÀÍÈÅ<br>4.4. Ïàðàáîëè÷åñêîå

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<p>Ýòîò ìåòîä îñíîâàí íà ïðèáëèæåíèè u, v êâàäðàòè÷íûìè ôóíêöèÿìè - ïàðàáîëàìè,
òî åñòü. Äëÿ êàæäîé ñòðîêè ñòðîèòñÿ ïðèáëèæàþùèå u, v êâàäðàòè÷íûå ôóíêöèè,
äàëüøå ñ èõ ïîìîùüþ îíè èíòåðïîëèðóþòñÿ ïî ñòðîêå. Äëÿ ýòîãî íàì ïîíàäîáÿòñÿ
òî÷íûå çíà÷åíèÿ u, v â òðåõ òî÷êàõ - íà÷àëå, ñåðåäèíå è êîíöå ñòðîêè. Èõ
ñ÷èòàåì òî÷íî òàê æå, êàê â 4.3.

<p>Èòàê, ïóñòü ó íàñ åñòü òî÷íûå çíà÷åíèÿ u â íà÷àëå, ñåðåäèíå è êîíöå ñòðîêè,
òî åñòü íà ðàññòîÿíèè 0, length/2 è length ïèêñåëîâ îò íà÷àëà ýòîé ñòðîêè,
îáîçíà÷èì èõ êàê ua, ub, è uc ñîîòâåòñòâåííî. Ìû ïûòàåìñÿ ïðèáëèçèòü u
êâàäðàòè÷íîé ôóíêöèåé, òî åñòü ïîëàãàåì, ÷òî

<p class=expression>u = A*x*x + B*x + C,<br>

<p>ãäå x - ðàññòîÿíèå îò òåêóùåé òî÷êè äî íà÷àëà ñòðîêè. Òîãäà, ïîäñòàâèâ â
ôîðìóëó ua, ub, uc è ñîîòâåòñòâóþùèå èì x, ïîëó÷àåì:

<p class=expression>ua = C,<br>
ub = A*(length*length)/4 + B*length/2 + C,<br>
uc = A*(length*length)   + B*length   + C.<br>

<p>Ò.î. C = ua, à äëÿ A è B èìååì ñèñòåìó óðàâíåíèé:

<p class=expression>A*(length*length)/4 + B*length/2 = ub - ua,<br>
A*(length*length)   + B*length   = uc - ua.<br>

<p>Óìíîæèì ïåðâîå óðàâíåíèå íà ÷åòûðå, âû÷òåì èç íåãî âòîðîå:

<p class=expression>4*A*(length*length)/4 + 4*B*length/2 -
A*(length*length    -   B*length =<br> 4*(ub - ua) - (uc - ua),<br><br>
B*length = 4*(ub - ua) - (uc - ua),<br>
B = (4*(ub - ua) - (uc - ua)) / length.<br>

<p>Óìíîæèì ïåðâîå óðàâíåíèå íà äâà, âû÷òåì åãî èç âòîðîãî:

<p class=expression>A*(length*length)   +   B*length -
2*A*(length*length)/4 - 2*B*length/2 =<br> (uc - ua) - 2*(ub - ua),<br><br>
A*(length*length)/2 = (uc - ua) - 2*(ub - ua),<br>
A = (2*(uc - ua) - 4*(ub - ua)) / (length*length).<br>

<p>Ïîëó÷èëè ôîðìóëû äëÿ A, B, C. Íàéäåì òåïåðü du è ddu. Äëÿ òåêóùåé òî÷êè x
èìååì:

<p class=expression>du(x) = u(x+1) - u(x),<br>
du = (A*(x+1)*(x+1)+B*(x+1)+C) - (A*x*x+B*x+C) = A*(2*x+1) + B,<br>
<br>
ddu(x) = du(x+1) - du(x),<br>
ddu = (A*(2*(x+1)+1)+B) - (A*(2*x+1)+B) = 2*A.<br>

<p>Ò.î., íà÷àëüíûå çíà÷åíèÿ u, du, ddu áóäóò ðàâíû

<p class=expression>u   = C,<br>
du  = A + B,<br>
ddu = 2*A.<br>

<p>À ïî èçâåñòíûì íà÷àëüíûì çíà÷åíèÿì u, du, ddu ïîñëåäîâàòåëüíî âû÷èñëÿåì
çíà÷åíèÿ u â ëþáîé òî÷êå:

<p class=expression>u(0), du(0), ddu - èçâåñòíû<br>
u(1) = u(0) + du(0), du(1) = du(0) + ddu<br>
u(2) = u(1) + du(1), du(2) = du(1) + ddu<br>
u(3) = u(2) + du(2), du(3) = du(2) + ddu<br>
...<br>

<p>Äëÿ v âñå äåëàåòñÿ ïîëíîñòüþ àíàëîãè÷íî.

<p>Òàêèì îáðàçîì, ðèñîâàíèå ñòðîêè áóäåò âûãëÿäåòü ïðèìåðíî òàê:

<pre class=source>
// ...

// ñ÷èòàåì u, v äëÿ íà÷àëà, ñåðåäèíû è êîíöà ñòðîêè
ua = uz_start / z1_start;
va = vz_start / z1_start;
ub = (uz_start + uz_end) / (z1_start + z1_end);
vb = (vz_start + vz_end) / (z1_start + z1_end);
uc = uz_end / z1_end;
vc = vz_end / z1_end;

// ñ÷èòàåì íà÷àëüíîå du è ddu
pa = 2 * ((uc - ua) - 2 * (ub - ua)) / (length * length);
pb = (4 * (ub - ua) - (uc - ua)) / length;
pc = ua;
u   = pc;
du  = pa + pb;
ddu = 2 * pa;

// ñ÷èòàåì íà÷àëüíîå dv è ddv
pa = 2 * ((vc - va) - 2 * (vb - va)) / (length * length);
pb = (4 * (vb - va) - (vc - va)) / length;
pc = v_a;
v   = pc;
dv  = pa + pb;
ddv = 2 * pa;

// ðèñóåì êóñîê
while (length--) {
  putpixel(current_sx, current_sy, texture[v][u]);
  u += du;
  v += dv;
  du += ddu;
  dv += ddv;
}
// ...
</pre>

<p>Ïî ñðàâíåíèþ ñ ïåðñïåêòèâíî-êîððåêòíûì òåêñòóðèðîâàíèåì èìååì áîëåå ìåäëåííûé
âíóòðåííèé öèêë, íî ìåíüøåå äëÿ äëèííûõ ñòðîê êîëè÷åñòâî äåëåíèé. Ðàñ÷åò ua,
va è èæå ñ íèìè ìîæíî ñäåëàòü ñ ïîìîùüþ òðåõ äåëåíèé, äåëåíèÿ íà length è
(length*length) ìîæíî çàìåíèòü óìíîæåíèÿìè íà 1/length è 1/(length*length),
áåðÿ ýòè çíà÷åíèÿ èç çàðàíåå ïîñ÷èòàííîé òàáëè÷êè. Ò.î., íà ñòðîêó ïðèõîäèòñÿ
òðè äåëåíèÿ íåçàâèñèìî îò åå äëèíû. Êà÷åñòâî áîëåå-ìåíåå ïðèåìëåìîå; à äëÿ
êîðîòêèõ ñòðîê ìîæíî èñïîëüçîâàòü îáû÷íóþ ëèíåéíóþ èíòåðïîëÿöèþ, òî÷íî òàê æå,
êàê è â ñëó÷àå ñ ïåðñïåêòèâíî-êîððåêòíûì òåêñòóðèðîâàíèåì. Ïîëó÷àåì âïîëíå
êîíêóðåíòîñïîñîáíûé ìåòîä òåêñòóðèðîâàíèÿ.

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