Computer graphics - lecture notes - 10/14/09

Reflection:

$Rf_{xy} =$

$1xxx$1xxx
$x1xx$x1xx
$xx-1x$xx-1x
$xxx1$xxx1

For yz, -1 is at 1,1. For xz, -1 is at 2, 2 $Rf_{z=5} = T(0, 0, -5) Rf_{xy}(0, 0, 5)$

3D Shear $SH_{xy} =$

$1xxx$1xxx
$x1xx$x1xx
$sh_x sh_y xx$sh_x sh_y xx
$xxx1$xxx1

$SH_{xz} =$

$1xxx$1xxx
$sh_x 1 sh_z x$sh_x 1 sh_z x
$xx1x$xx1x
$xxx1$xxx1

$SH_{yz} =$

$1 sh_y sh_z 0$1 sh_y sh_z 0
$x1xx$x1xx
$xx1x$xx1x
$xxx1$xxx1

Rotate a line around its akis by $\beta$ degrees: $M = R_x^{90 - \theta} R_z^\beta R_x^{-90+\theta}$

Given $P_1 (x_1, y_1, z_1), P_2(x_2, y_2, z_2), \theta = 30$. Find: M to rotate $\theta$ around $P_1P_2$ $M = T_{orig} R_x^{\alpha} R_y^{\beta} R_z^{\theta} R_y^{-\beta} R_x^{-\alpha} T_{-orig}$

2D: (S - scale, T - translate, R - rotate, SH - shear) $T_1 T_2 = T_2 T_1 YES$ $S_1 S_2 = S_2 S_1 YES$ $R_1 R_2 = R_2 R_1 NO$ $SH_{x1} SH_{x2} = SH_{x2} SH_{x1} YES$

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