用齐次坐标绘制正12面体

ashleytgg 发表于 2018-3-14 16:33:07

本帖最后由 ashleytgg 于 2018-3-14 16:33 编辑

参照《计算机图形学原理教程》编著 ——李建平，电子科技大学出版社，中其次坐标的图形变换原理，写了几个lisp 函数。
绘制了正12面体，觉得非常有趣，在此和大家飞享下。其中也也引用了 high_fly_bird 的一个小函数矢量叉积。

;; 定义一个三维镜像的函数
;; group 为一个四维点集合 , point_A point_Bpoint_C 为镜像平面的三个点
(defun mirror_3d (group    point_A point_Bpoint_C /
   Cen    Axis    x_Ceny_Cen z_Cen
   x_Axis y_Axis z_AxisP_1 P_2
   cos_a    sin_a cos_bsin_b P1->P2
   Cen->Axis T_ry    T_rxT_XY_mirror
   T_ry-1 T_rx-1 T_m-1T_综合 Vector_0
   Vector_1
)

;; 求平面 ABC 的法向量
(setqVector_0 (mapcar '- point_B point_A)
Vector_1 (mapcar '- point_C point_A)
)
;; 求向量Vector_0 Vector_1 的叉积,为平面法向量normal_vector
(setqnormal_vector
(G:CrossProductor Vector_0 Vector_1)
)
;; 定义新的原点，和新的 Z轴
(setqCenpoint_A
Axis (mapcar '+ Cen normal_vector)
)
;; 把原点移动到Cen ，Z轴向量转变为 Cen->Axis
(setqT_m (list
   (list 1 0 0 (* (car Cen) -1))
   (list 0 1 0 (* (cadr Cen) -1))
   (list 0 0 1 (* (caddr Cen) -1))
   (list 0 0 0 1)
   )
)

;; 设置point_A point_B 的坐标 x_A y_A z_A ,x_B y_B z_B
(mapcar 'set
'(x_Cen y_Cen z_Cen x_Axis y_Axis z_Axis)
(append Cen Axis)
)
(setqP_1 (list y_Cen Z_Cen)
P_2 (list y_Axis Z_Axis)
P1->P2 (distance p_1 P_2)
)
(if (> P1->P2 1e-8)
(setq
   cos_a (/ (- z_Axis z_Cen) P1->P2)
   sin_a (/ (- y_Axis y_Cen) P1->P2)
)
(setq cos_a1
sin_a0
)
)
(setqT_rx (list
      (list 1 0 0 0)
      (list 0 cos_a (* -1 sin_a) 0)
      (list 0 sin_a cos_a 0)
      (list 0 0 0 1)
   )
)
(setq
Cen->Axis (distance Cen Axis)
cos_b (/ P1->P2 Cen->Axis)
sin_b (/ (- x_Axis x_Cen) Cen->Axis)
)
(setqT_ry (list
      (list cos_b 0 (* -1 sin_b) 0)
      (list 0 1 0 0)
      (list sin_b 0 cos_b 0)
      (list 0 0 0 1)
   )
)
;; 对XY平面进行镜像
(setqT_XY_mirror
(list
(list 1 0 0 0)
(list 0 1 0 0)
(list 0 0 -1 0)
(list 0 0 0 1)
)
)
(setqT_ry-1 (list
(list cos_b 0 sin_b 0)
(list 0 1 0 0)
(list (* -1 sin_b) 0 cos_b 0)
(list 0 0 0 1)
      )
)
(setqT_rx-1 (list
(list 1 0 0 0)
(list 0 cos_a sin_a 0)
(list 0 (* -1 sin_a) cos_a 0)
(list 0 0 0 1)
      )
)
(setqT_m-1 (list
(list 1 0 0 (car Cen))
(list 0 1 0 (cadr Cen))
(list 0 0 1 (caddr Cen))
(list 0 0 0 1)
   )
)
(setqT_综合
      (mapcar '(lambda(lst)
   (setqlst (mxv T_m lst)
   lst (mxv T_rx lst)
   lst (mxv T_ry lst)
   lst (mxv T_XY_mirror lst)
   lst (mxv T_ry-1 lst)
   lst (mxv T_rx-1 lst)
   lst (mxv T_m-1 lst)
   )
   )
      (list
   '(1 0 0 0)
   '(0 1 0 0)
   '(0 0 1 0)
   '(0 0 0 1)
      )
      )
;; 对T_综合进行装置
T_综合 (list (mapcar 'car T_综合)
      (mapcar 'cadr T_综合)
      (mapcar 'caddr T_综合)
      (mapcar 'cadddr T_综合)
      )
)
(mapcar '(lambda (u)
   (mxv T_综合 U)
)
group
)
)
;;(mirror_3d group point_A point_B point_C)
;;定义一个坐标系重置函数，把一个已知坐标系，原点重置为point_A ，Z轴设置方向point_A->point_B
(defun reset_coordinate_system (grouppoint_Apoint_B/x_A
   y_Az_Ax_By_Bz_B
   P_1P_2cos_asin_acos_b
   sin_bP1->P2A->BT_ryT_rx
   T_综合
         )
(setqT_m (list
   (list 1 0 0 (* (car point_A) -1))
   (list 0 1 0 (* (cadr point_A) -1))
   (list 0 0 1 (* (caddr point_A) -1))
   (list 0 0 0 1)
   )
)
(mapcar 'set
'(x_A y_A z_A x_B y_B z_B)
(append point_A point_B)
)
(setqP_1 (list y_A Z_A)
P_2 (list y_B Z_B)
P1->P2 (distance p_1 P_2)
)
(if (> P1->P2 1e-8)
(setq
   cos_a (/ (- z_B z_A) P1->P2)
   sin_a (/ (- y_B y_A) P1->P2)
)
(setq cos_a1
sin_a0
)
)
(setqT_rx (list
      (list 1 0 0 0)
      (list 0 cos_a (* -1 sin_a) 0)
      (list 0 sin_a cos_a 0)
      (list 0 0 0 1)
   )
)
(setq
A->B(distance point_A point_B)
cos_b (/ P1->P2 A->B)
sin_b (/ (- x_B x_A) A->B)
)
(setqT_ry (list
      (list cos_b 0 (* -1 sin_b) 0)
      (list 0 1 0 0)
      (list sin_b 0 cos_b 0)
      (list 0 0 0 1)
   )
)
(setqT_综合
      (mapcar '(lambda(lst)
   (setqlst (mxv T_m lst)
   lst (mxv T_rx lst)
   lst (mxv T_ry lst)
   )
   )
      (list
   '(1 0 0 0)
   '(0 1 0 0)
   '(0 0 1 0)
   '(0 0 0 1)
      )
      )
;; 对T_综合进行装置
T_综合 (list (mapcar 'car T_综合)
      (mapcar 'cadr T_综合)
      (mapcar 'caddr T_综合)
      (mapcar 'cadddr T_综合)
      )
)
(mapcar '(lambda (u)
   (mxv T_综合 U)
)
group
)
)
;;(reset_coordinate_systemgrouppoint_Apoint_B );;定义一个三维对齐函数，既把一个三维向量P_A->P_B P_A->P_C , 经过一系列平移、旋转操作，和另外一个
;;已经知道的向量target_A->target_B target_A->target_C , 重合
(defun align_3d(group point_A point_B
point_C target_A target_B
target_C / normal_vector
normal_vector_target axis_rotation_0
axis_rotation_1 T_m U_offset
Vector_0 Vector_1
)

;; 求平面 ABC 的平面法向量 normal_vector
(setqVector_0 (mapcar '- point_B point_A)
Vector_1 (mapcar '- point_C point_A)
)
(setqnormal_vector
(G:CrossProductor Vector_0 Vector_1)
)
;; 求平面target_A target_B target_C 的平面法向量 normal_vector_target
(setqVector_0 (mapcar '- target_B target_A)
Vector_1 (mapcar '- target_C target_A)
)
(setqnormal_vector_target
(G:CrossProductor Vector_0 Vector_1)
)
;; 求向量 normal_vector normal_vector_target 组成平面的法向量
(setqAxis_vector
(G:CrossProductor normal_vector normal_vector_target)
)

(if (= (apply '+ Axis_vector) 0)
;; 当Axis_vector 为0 时，表示normal_vector 和normal_vector_target 共线，则
;; 平面target_A target_B target_C和平面 ABC 共面
(setq ang 0)
;; 建立以 '(0 0 0)normal_vector normal_vector_target 位XY平面，'(0 0 0)->Axis_vector Z轴的坐标系
(setq lst    (reset_coordinate_system
      (list normal_vector normal_vector_target)
      '(0 0 0)
      Axis_vector
   )
ang_0    (angle '(0 0)
      (mapcar '+ '(0 0) (car lst))
   )
ang_1    (angle '(0 0)
      (mapcar '+ '(0 0) (cadr lst))
   )
;; ang为由平面 ABC 旋转到平面target_A target_B target_C 的旋转角度，
;;也是向量O->normal_vector 和O->normal_vector_target 之间的夹角
ang    (- ang_1 ang_0)
;; 求由平面 ABC 旋转到平面target_A target_B target_C 的旋转轴
axis_rotation_0 point_A
axis_rotation_1 (mapcar '+ point_A Axis_vector)
)
)
(if (/= ang 0)
;; 图像group 绕着旋转轴axis_rotation_0 axis_rotation_1 旋转ang度
(setq group(rotate_3d (append (list (append point_A (list 1))
      (append point_B (list 1))
      )
      group
      )
      axis_rotation_0
      axis_rotation_1
      ang
)
)
)
(setqU_offset (mapcar '- target_A point_A)
T_m (list
   (list 1 0 0 (car U_offset))
   (list 0 1 0 (cadr U_offset))
   (list 0 0 1 (caddr U_offset))
   (list 0 0 0 1)
)
)
(setqgroup (mapcar '(lambda (u)
   (mxv T_m U)
      )
      group
   )
)
;; 把point_A->point_B target_A->target_B 变成四维点
(setq
lst_四维 (mapcar
      '(lambda(x)
   (append x (list 1))
)
      (list (car group) (cadr group) target_A target_B)
   )
)
(setq
lst (reset_coordinate_system
   lst_四维
   target_A
   (mapcar '+ target_A normal_vector_target)
)
;; 计算向量 point_A->point_B的象限角
ang_0 (angle (mapcar '+ '(0 0) (car lst))
(mapcar '+ '(0 0) (cadr lst))
)
;;计算向量target_A->target_B 的象限角
ang_1 (angle (mapcar '+ '(0 0) (caddr lst))
(mapcar '+ '(0 0) (cadddr lst))
)
;; 向量 point_A->point_Btarget_A->target_B 在同一个平面内时的旋转角度 Δ
Δ (- ang_1 ang_0)
)
;; 让图像group 绕着 point_A->normal_vector 旋转Δ ，再绕着绕着旋转轴axis_rotation_0 axis_rotation_1 旋转ang度
;; 最后坐标原点移动到target_A
(rotate_3d (cddr group)
   target_A
   (mapcar '+ target_A normal_vector_target)
   Δ
)
)

;; group 为一个四维点集合 , point_A point_B 为旋转轴的两个点，φ 为旋转角度，单位弧度
;; 旋转轴point_A point_B 的距离必须大于0 ，本程序不考虑 point_A point_B 重合的情形
(defun rotate_3d (group point_A point_B φ / x_A
   y_A z_A x_B y_B z_B P_1
   P_2 cos_a sin_a cos_b sin_b P1->P2
   A->B T_ry T_rx T_rZ T_ry-1T_rx-1
   T_m-1 T_综合
)
(setqT_m (list
   (list 1 0 0 (* (car point_A) -1))
   (list 0 1 0 (* (cadr point_A) -1))
   (list 0 0 1 (* (caddr point_A) -1))
   (list 0 0 0 1)
   )
)

;; 设置point_A point_B 的坐标 x_A y_A z_A ,x_B y_B z_B
(mapcar 'set
'(x_A y_A z_A x_B y_B z_B)
(append point_A point_B)
)
(setqP_1 (list y_A Z_A)
P_2 (list y_B Z_B)
P1->P2 (distance p_1 P_2)
)
(if (> P1->P2 1e-8)
;; 当向量A->B 不平行于X 轴时
(setq
   cos_a (/ (- z_B z_A) P1->P2)
   sin_a (/ (- y_B y_A) P1->P2)
)
;; 当向量A->B 平行于X 轴时 ,既垂直于 YZ平面,此时A->B与X轴的夹角为0度
(setq cos_a1
sin_a0
)
)
(setqT_rx (list
      (list 1 0 0 0)
      (list 0 cos_a (* -1 sin_a) 0)
      (list 0 sin_a cos_a 0)
      (list 0 0 0 1)
   )
)
(setq
A->B(distance point_A point_B)
cos_b (/ P1->P2 A->B)
sin_b (/ (- x_B x_A) A->B)
)
(setqT_ry (list
      (list cos_b 0 (* -1 sin_b) 0)
      (list 0 1 0 0)
      (list sin_b 0 cos_b 0)
      (list 0 0 0 1)
   )
)
;; 绕AB顺时针旋转 φ 角度,就变成绕新的Z 轴逆时针旋转φ 角度
(setqT_rZ (list
      (list (cos φ) (* -1 (sin φ)) 0 0)
      (list (sin φ) (cos φ) 0 0)
      (list 0 0 1 0)
      (list 0 0 0 1)
   )
)
(setqT_ry-1 (list
(list cos_b 0 sin_b 0)
(list 0 1 0 0)
(list (* -1 sin_b) 0 cos_b 0)
(list 0 0 0 1)
      )
)
(setqT_rx-1 (list
(list 1 0 0 0)
(list 0 cos_a sin_a 0)
(list 0 (* -1 sin_a) cos_a 0)
(list 0 0 0 1)
      )
)
(setqT_m-1 (list
(list 1 0 0 (car point_A))
(list 0 1 0 (cadr point_A))
(list 0 0 1 (caddr point_A))
(list 0 0 0 1)
   )
)
(setqT_综合
      (mapcar '(lambda(lst)
   (setqlst (mxv T_m lst)
   lst (mxv T_rx lst)
   lst (mxv T_ry lst)
   lst (mxv T_rz lst)
   lst (mxv T_ry-1 lst)
   lst (mxv T_rx-1 lst)
   lst (mxv T_m-1 lst)
   )
   )
      (list
   '(1 0 0 0)
   '(0 1 0 0)
   '(0 0 1 0)
   '(0 0 0 1)
      )
      )
;; 对T_综合进行装置
T_综合 (list (mapcar 'car T_综合)
      (mapcar 'cadr T_综合)
      (mapcar 'caddr T_综合)
      (mapcar 'cadddr T_综合)
      )
)
(mapcar '(lambda (u)
   (mxv T_综合 U)
)
group
)
)
;;( rotate_3dgroup point_A point_B φ)

renhaitao_nice 发表于 2018-3-14 18:42:18

高手太多，我等惭愧跪拜

ashleytgg 发表于 2018-3-15 12:01:21

都是书本上抄来的，我只是翻译下而已。

434939575 发表于 2018-3-15 16:54:12

确实好用。学习了。

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用齐次坐标绘制正12面体