2010.04.24 NyARToolkit project nyatla@nyatla.jp 8x8 2 4x4

x = GX +HY +1 y=DX +EY +F GX +HY +1 AX +BY +C

(1)

(1) AX + BY + C GXx HYx = x (2) DX + EY + F GXy HYy = y

(2) 4 X1 X2 X3 X4 X1 X2 X3 X4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 1 X1 x1 1 X 2 x2 1 X 3 x3 1 X 4 x4 1 X1 y1 1 X 2 y2 1 X 3 y3 1 X 4 y4 Y1 x1 A x1 x2 Y2 x2 B = x3 Y3 x3 C G x4 Y4 x4 H Y1 y1 D y1 y2 Y2 y2 E F = y3 Y3 y3 G y4 Y4 y4 H

(3)

5*4 4*4 X1 X2 X3 X4 X1 X2 X3 X4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 X1 x1 X 2 x2 X 3 x3 X 4 x4 X1 y1 X 2 y2 X 3 y3 X 4 y4 Y1 x1 Y2 x2 Y3 x3 Y4 x4 Y1 y1 Y2 y2 Y3 y3 Y4 y4 x1 C A x2 C B = G x3 C H x4 C (4) y1 F D E = y2 F G y3 F H y4 F

X1 X2 Tx = X3 X4 X1 X2 Ty = X3 X4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 X1 x1 X 2 x2 X 3 x3 X 4 x4 X1 y1 X 2 y2 X 3 y3 X 4 y4 Y1 x1 Y2 x2 Y3 x3 Y4 x4 Y1 y1 Y2 y2 Y3 y3 Y4 y4 t x11 t Tx 1 = x21 t x31 t x41 t y11 t Ty 1 = y21 t y31 t y41 t x12 t x22 t x32 t x42 t y12 t y22 t y32 t y42 t x13 t x23 t x33 t x43 t y13 t y23 t y33 t y43 t x14 t x24 t x34 t x44 t y14 t y24 t y34 t y44

(5)

(6)

A B = T x G H D E = T y G H x1 C x2 C x3 C x4 C y1 F y2 F y3 F y4 F t x11 A B = t x21 t x31 G t x41 H t y11 D E = t y21 t y31 G H t y41 t x12 t x22 t x32 t x42 t y12 t y22 t y32 t y42 t x13 t x23 t x33 t x43 t y13 t y23 t y33 t y43 t x14 t x24 t x34 t x44 t y14 t y24 t y34 t y44 x1 C x2 C x3 C x4 C (8) y1 F y2 F y3 F y4 F

1

(7)

1

(8) A = t x11 x1 + t x12 x2 +t x13 x3 + t x14 x4 C(t x11 + t x12 +t x13 + t x14 ) B = t x21 x1 + t x22 x2 +tx32 x3 + t x42 x4 C(tx21 + t x22 +t x23 + t x24 ) G = t x31 x1 + t x23 x2 +tx33 x3 + t x43 x4 C(tx31 + t x32 +t x33 + t x34 ) H = t x41 x1 + t x24 x2 +t x34 x3 + t x44 x4 C(t x14 + t x24 +t x34 + t x44 ) D = t y11 y1 + t y12 y2 +t y13 y3 + t y14 y4 F(t y11 + t y12 +t y13 + t y14 ) (11) E = t y21 y1 + t y22 y2 +t y23 x3 + t y24 y4 F(ty21 + t y22 +t y23 + t y24 ) G = t y31 y1 + t y32 y2 +t y33 y3 + t y34 y4 F(t y31 + t y32 +t y33 + t y34 ) H = t y41 y1 + t y42 y2 +t y43 y3 + t y44 y4 F(ty41 + t y42 +t y43 + t y44 )

G,H C,F C,F A,B,D,E,G,H