3D 電腦圖學筆記(1)

3D Computer Graphics—Alan Watt

將mathematical or geometric數學和幾何的graphics model – into 可視化—two-dimensional projection

Transformation是重要的工具in generating 3D scenes

2D view of the environment表示物件的方法是
polygon mesh model

3D geometry affine transformation

V'=Vt*D translation平移

V'=S*V scaling 縮小放大

V'=R*V rotation 旋轉

translation

v'=Tv

x' [1 0 0 Tx] [x]

[y']=[0 1 0 Ty] [y]

z' [0 0 1 Tz] [z]

1 [0 0 0 1] [1]

x'=x+Tx

y'=y+Ty

z'=z+Tz

scaling

v'=sv

s=[ Sx 0 0 0]

[ 0 Sy 0 0]

[ 0 0 Sz 0]

[ 0 0 0 1]

x'=Sx*x

y'=Sy*y

z'=Sz*z

Rx=[ 1 0 0 0 ]

[ 0 cosθ -sinθ 0]

[0 sinθ cosθ 0]

[0 0 0 1]

Ry=[ cosθ 0 sinθ 0 ]

[ 0 1 0 0]

[-sinθ 0 cosθ 0]

[ 0 0 0 1]

Rz=[cosθ -sinθ 0 0]

[sinθ cosθ 0 0]

[0 0 1 0]

[0 0 0 1]

對z軸旋轉

x'=xcosθ-ysinθ

y'=xsinθ+ycosθ

z'=z

M3=M2M1

先旋轉後平移Rotation followed by translation

Vt*R

local coordinate system

1)平移至原點 T2

2)利用所需軸旋轉 R

3)平移物件至他的原點 T1

T2RT1=

[ 1 0 0 -Tx][cosθ -sinθ 0 0][1 0 0 Tx]

[ 0 1 0 -Ty][sinθ cosθ 0 0][0 1 0 Ty]

[ 0 0 1 0 ][0 0 1 0][0 0 1 0]

[ 0 0 0 1 ][0 0 0 1][0 0 0 1]

[cosθ -sinθ 0 -Txcosθ+Tysinθ+Tx]

[sinθ cosθ 0 -Txsinθ-Tycosθ+Ty]

[0 0 1 0 ]

[0 0 0 1 ]