Computer Graphics Homogenous Coordinate

平移和齐次坐标

Translation：

平移矩阵

E.g. move x by +5 units, leave y, z unchanged We need appropriate matrix. What is it?

平移矩阵中不包含 X y z 所以是一个很有挑战的问题

图形学中用齐次坐标来表达，加入多一个轴

Add a fourth homogeneous coordinate (w=l) 4x4 matrices very common in graphics, hardware • Last row always 0 00 1 (until next lecture) x' z' 100 010 001 5 o x z z

最后一行都是0001

w=0 表示一个向量，无穷远点

齐次坐标的有点是，只需要在渲染管线最后一步做一次除法就可以将齐次坐标转化为非齐次坐标。中间没有人恶化除法

GeneraI TransIation Matrix 1 --—3ъЭ= о 1 1 1 х 1 Х+Тх

-3下수= 1 0 0 0 0 1 0 0 0 0 1 0 T T T 1 3

组合 RT TR 不同，平移旋转不可以交换顺序

Combining Translations, Rotations P' = (TR)P = MP = RP + T

)죄 5 `

P' = (TR)P 1 0 o 0 0 1 1 R 21 R31 o R12 R32 o RI 3 R R33 O 0 o 1 Ril R21 R31 0 23 1 0 1

先平移后旋转

+ d& " (u + d)H " dl"V " d(ÆH) 』 d

P' = (RT)P = — + T) — RII R21 R 31 o R12 R R o R13 R23 R33 o o 1 o o 1 o R3x3 1 3x1 1 1

import numpy as np

def rotation_matrix_2d(deg):

cosi = np.cos(deg*np.pi/180)

sini = np.sin(deg*np.pi/180)

rotation = np.matrix([[cosi,-sini],[sini, cosi]])

return rotation

i = np.matrix([0,0]).T

translation_matrix = np.matrix([1.0,1.0]).T

rotation_matrix = rotation_matrix_2d(215.0)

print(rotation_matrix*i + translation_matrix)

print(rotation_matrixi + rotation_matrixtranslation_matrix)