【越飞越高讲堂2】用LISP论矩阵
本帖最后由 Highflybird 于 2013-5-21 18:11 编辑用LISP论矩阵矩阵的LISP程序，国内讨论的比较少， 而国外的研究比较深入。
经过长时间的收藏和探索，我综合成了这篇帖子。
这个帖子里面的函数主要是跟CAD 相关。附件包含了本帖完整的lisp代码，还有测试样例。
是一个比较完整的矩阵库。
另外，本帖附上了一些矩阵相关链接。错误和纰漏之处请大家多多指教。样例及演示可以参考附件。
一、向量的运算
向量，矩阵是息息相关的。这里列出了向量的一些基本运算。
;;;-----------------------------------------------------------;;
;;; 两向量相加 addition ;;
;;; Input: v1,v2 -vectors in R^n ;;
;;; OutPut: A vector ;;
;;;-----------------------------------------------------------;;
(defun MAT:v+v (v1 v2)
(mapcar '+ v1 v2)
)
;;;-----------------------------------------------------------;;
;;; 两向量相减subtraction ;;
;;; Input: v1,v2 -vectors in R^n ;;
;;; OutPut: A vector ;;
;;;-----------------------------------------------------------;;
(defun MAT:v-v (v1 v2)
(mapcar '- v1 v2)
)
;;;-----------------------------------------------------------;;
;;; 两向量相乘multiplication ;;
;;; Input: v1,v2 -vectors in R^n ;;
;;; OutPut: A vector ;;
;;;-----------------------------------------------------------;;
(defun MAT:v*v (v1 v2)
(mapcar '* v1 v2)
)
;;;-----------------------------------------------------------;;
;;; 两向量相除division ;;
;;; Input: v1,v2 -vectors in R^n ;;
;;; OutPut: A vector ;;
;;;-----------------------------------------------------------;;
(defun MAT:v/v (v1 v2)
(mapcar '/ v1 v2)
)
;;;-----------------------------------------------------------;;
;;; 向量乘标量(系数) ;;
;;; Vector x Scalar - Lee Mac ;;
;;; Args: v - vector in R^n, s - real scalar ;;
;;;-----------------------------------------------------------;;
(defun MAT:vxs ( v s )
(mapcar (function (lambda ( n ) (* n s))) v)
)
;;;-----------------------------------------------------------;;
;;; 两向量的点积 ;;
;;; Vector Dot Product ;;
;;; Input: v1,v2 -vectors in R^n ;;
;;;-----------------------------------------------------------;;
(defun MATot (v1 v2)
(apply '+ (mapcar '* v1 v2))
)
;;;-----------------------------------------------------------;;
;;; 两向量的叉积 ;;
;;; Vector Cross Product ;;
;;; Args: u,v - vectors in R^3 ;;
;;;-----------------------------------------------------------;;
(defun MAT:vxv ( u v )
(list
(- (* (cadr u) (caddr v)) (* (cadr v) (caddr u)))
(- (* (carv) (caddr u)) (* (caru) (caddr v)))
(- (* (caru) (cadrv)) (* (carv) (cadru)))
)
)
;;;-----------------------------------------------------------;;
;;; 线性组合标量组乘向量组 ;;
;;; Linear combination - highflybird ;;
;;; Input: Vectors - vectors, Scalars, - a real number list ;;
;;; Output: a vector ;;
;;;-----------------------------------------------------------;;
(defun MAT:SxVs (Vectors Scalars)
(apply 'mapcar (cons '+ (mapcar 'MAT:vxs Vectors Scalars)))
)
;;;-----------------------------------------------------------;;
;;; 向量的模(长度) ;;
;;; Vector Norm - Lee Mac ;;
;;; Args: v - vector in R^n ;;
;;;-----------------------------------------------------------;;
(defun MAT:norm ( v )
(sqrt (apply '+ (mapcar '* v v)))
)
;;;-----------------------------------------------------------;;
;;; 向量的模(长度) ;;
;;; Vector Norm - highflybird ;;
;;; Args: v - vector in R^3 ;;
;;;-----------------------------------------------------------;;
(defun MAT:Norm3D ( v )
(distance '(0 0 0) v)
)
;;;-----------------------------------------------------------;;
;;; 单位向量 ;;
;;; Unit Vector - Lee Mac ;;
;;; Args: v - vector in R^n ;;
;;;-----------------------------------------------------------;;
(defun MAT:Unitization (v)
( (lambda (n)
(if (equal 0.0 n 1e-14)
nil
(MAT:vxs v (/ 1.0 n))
)
)
(MAT:norm v)
)
)
;;;-----------------------------------------------------------;;
;;; 单位向量 ;;
;;; Unit Vector - highflybird ;;
;;; Args: v - vector in R^3 ;;
;;;-----------------------------------------------------------;;
(defun MAT:unit ( v / l)
(cond
( (= (setq l (MAT:Norm3D v)) 1.0 ) v)
( (> l 1e-14) (MAT:vxs v (/ 1.0 l)))
)
)
;;;-----------------------------------------------------------;;
;;; 两个2d向量的叉积的数值 ;;
;;; 输入: 两个点（或者两个向量） ;;
;;; 输出: 一个数值.如果为正则是逆时针,两向量形成的平面法线向量;;
;;; 向上，为负则是顺时针，为零则两向量共线或平行。 ;;
;;; 这个数值也为原点,P1,P2三点面积的两倍。 ;;
;;;-----------------------------------------------------------;;
(defun MATet2V (v1 v2)
(- (* (car v1) (cadr v2)) (* (car v2) (cadr v1)))
)
二、向量的旋转
;;;-----------------------------------------------------------;;
;;; 旋转一个向量或者点90度 ;;
;;; 输入: 一个向量 ;;
;;; 输出: 被旋转90度后的向量 ;;
;;;-----------------------------------------------------------;;
(defun MAT:Rot90 (vec)
(vl-list* (- (cadr vec)) (car vec) (cddr vec))
)
;;;-----------------------------------------------------------;;
;;; 旋转向量到指定角度 ;;
;;; 输入: 一个向量和指定的角度 ;;
;;; 输出: 被旋转后的向量 ;;
;;;-----------------------------------------------------------;;
(defun MAT:Rot2D (v a / c s x y)
(setq c (cos a) s (sin a))
(setq x (car v) y (cadr v))
(list (- (* x c) (* y s)) (+ (* x s) (* y c)))
)
三、 行列式
这里只讨论二阶和三阶的行列式
;;;-----------------------------------------------------------;;
;;; 2d行列式 determinant in R^2 ;;
;;; Args: 4 numbers ;;
;;;-----------------------------------------------------------;;
(defun MATet2 (x1 y1 x2 y2)
(- (* x1 y2) (* x2 y1))
)
;;;-----------------------------------------------------------;;
;;; 3d行列式determinant in R^3 ;;
;;; Args: 9 numbers ;;
;;;-----------------------------------------------------------;;
(defun MATet3 (a1 b1 c1 a2 b2 c2 a3 b3 c3)
(+ (* a1 (- (* b2 c3) (* b3 c2)))
(* a2 (- (* b3 c1) (* b1 c3)))
(* a3 (- (* b1 c2) (* b2 c1)))
)
)
四、 矩阵的基本运算
;;;-----------------------------------------------------------;;
;;; 矩阵转置 ;;
;;; MAT:trp Transpose a matrix -Doug Wilson- ;;
;;; 输入：矩阵 ;;
;;; 输出：转置后的矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:trp (m)
(apply 'mapcar (cons 'list m))
)
;;;-----------------------------------------------------------;;
;;; 矩阵相加 ;;
;;; Matrix + Matrix - Lee Mac ;;
;;; Args: m,n - nxn matrices ;;
;;;-----------------------------------------------------------;;
(defun MAT:m+m ( m n )
(mapcar '(lambda ( r s ) (mapcar '+ r s)) m n)
)
;;;-----------------------------------------------------------;;
;;; 矩阵相减 ;;
;;; Matrix - Matrix - Lee Mac ;;
;;; Args: m,n - nxn matrices ;;
;;;-----------------------------------------------------------;;
(defun MAT:m-m ( m n )
(mapcar '(lambda ( r s ) (mapcar '- r s)) m n)
)
;;;-----------------------------------------------------------;;
;;; 矩阵相乘 ;;
;;; MAT:mxm Multiply two matrices -Vladimir Nesterovsky- ;;
;;;-----------------------------------------------------------;;
(defun MAT:mxm (m q)
(mapcar (function (lambda (r) (MAT:mxv (MAT:trp q) r))) m)
)
五、 矩阵与向量的运算
;;;-----------------------------------------------------------;;
;;; 向量或点的矩阵变换(向量乘矩阵) ;;
;;; Matrix x Vector - Vladimir Nesterovsky ;;
;;; Args: m - nxn matrix, v - vector in R^n ;;
;;;-----------------------------------------------------------;;
(defun MAT:mxv (m v)
(mapcar (function (lambda (r) (apply '+ (mapcar '* r v)))) m)
)
;;;-----------------------------------------------------------;;
;;; 点的矩阵(4x4 matrix) 变换 ;;
;;; 输入：矩阵m和一个三维点p ;;
;;; 输出：点变换后的位置 ;;
;;;-----------------------------------------------------------;;
(defun MAT:mxp (m p)
(reverse (cdr (reverse (MAT:mxv m (append p '(1.0))))))
)
;;;-----------------------------------------------------------;;
;;; 矩阵乘标量 ;;
;;; Matrix x Scalar - Lee Mac ;;
;;; Args: m - nxn matrix, n - real scalar ;;
;;;-----------------------------------------------------------;;
(defun MAT:mxs ( m s )
(mapcar (function (lambda ( v )(MAT:VxS v s))) m)
)
六、矩阵的平面和空间变换
以下是一些矩阵变换的函数。平移变换
;;;-----------------------------------------------------------;;
;;; 平移变换矩阵方式1 ;;
;;; 参数: ;;
;;; v - 位移矢量 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ Translate by Matrix }==-----------------;;
;;; ;;
;;; Translation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012 ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; v- Displacement vector by which to translate ;;
;;;-----------------------------------------------------------;;
(defun MAT:Translation ( v )
(list
(list 1. 0. 0. (car v))
(list 0. 1. 0. (cadr v))
(list 0. 0. 1. (caddr v))
(list 0. 0. 0. 1.)
)
)
;;;-----------------------------------------------------------;;
;;; 平移变换矩阵方式2 ;;
;;; 参数: ;;
;;; p1 - 基点 ;;
;;; p2 - 目标点 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ Translate by Matrix }==-----------------;;
;;; ;;
;;; Translation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012 ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; p1, p2 - Points representing vector by which to translate ;;
;;;-----------------------------------------------------------;;
(defun MAT:TranslateBy2P ( p1 p2 )
(MAT:Translation (mapcar '- p2 p1))
)
等比缩放变换
;;;-----------------------------------------------------------;;
;;; 比例缩放矩阵 ;;
;;; 参数: ;;
;;; Cen - 基点 ;;
;;; scale - 缩放比例 ;;
;;;-----------------------------------------------------------;;
;;;-----------------=={ Scale by Matrix }==-------------------;;
;;; ;;
;;; Scaling Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012 ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; Cen - Base Point for Scaling Transformation ;;
;;; scale- Scale Factor by which to scale object ;;
;;;-----------------------------------------------------------;;
(defun MAT:Scaling ( Cen scale / s)
(setq s (- 1 scale))
(list
(list scale 0. 0. (* s (car Cen)))
(list 0. scale 0. (* s (cadr Cen)))
(list 0. 0. scale (* s (caddr Cen)))
'(0. 0. 0. 1.)
)
)
二维旋转变换
;;;-----------------------------------------------------------;;
;;; 二维旋转变换矩阵 ;;
;;; 参数: ;;
;;; Cen - 基点 ;;
;;; ang - 旋转角度 ;;
;;;-----------------------------------------------------------;;
;;;-----------------=={ Rotate by Matrix }==------------------;;
;;; ;;
;;; Rotation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012 ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; Cen - Base Point for Rotation Transformation ;;
;;; ang - Angle through which to rotate object ;;
;;;-----------------------------------------------------------;;
(defun MAT:Rotation ( Cen ang / c s x y)
(setq c (cos ang) s (sin ang))
(setq x (car Cen) y (cadr Cen))
(list
(list c (- s) 0. (- x (- (* c x) (* s y))))
(list s c0. (- y (+ (* s x) (* c y))))
'(0. 0. 1. 0.)
'(0. 0. 0. 1.)
)
)
三维旋转变换
;;;-----------------------------------------------------------;;
;;; 三维旋转变换矩阵 ;;
;;; 参数: ;;
;;; Cen- 基点 ;;
;;; Axis - 旋转轴 ;;
;;; ang- 旋转角 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ 3D Rotate by Matrix }==-----------------;;
;;; Author: highflybird. ;;
;;; Arguments: ;;
;;; Cen ---Input origin point of rotation ;;
;;; Axis---Input axis vector of rotation ;;
;;; Ang ---Input angle of rotation ;;
;;;-----------------------------------------------------------;;
(defun MAT:Rotation3D (Cen Axis Ang / A B C D M N P x y z)
(setq D (distance '(0 0 0) Axis))
(if (or (< D 1e-8) (zerop ang))
'((1. 0. 0. 0.) (0. 1. 0. 0.) (0. 0. 1. 0.) (0. 0. 0. 1.))
(setq N (mapcar '/ Axis (list D D D))
x (car N)
y (cadr N)
z (caddr N)
A (cos Ang)
B (sin Ang)
C (- 1 A)
M (list (list (+ A (* x x C))
(- (* x y C) (* z B))
(+ (* y B) (* x z C))
)
(list (+ (* z B) (* x y C))
(+ A (* y y C))
(- (* y z C) (* x B))
)
(list (- (* x z C) (* y B))
(+ (* x B) (* y z C))
(+ A (* z z C))
)
)
p (mapcar '- Cen (Mat:mxv M Cen))
M (MatispToMatrix M p)
)
)
)
;;;-----------------------------------------------------------;;
;;; 三维旋转变换矩阵（通过两点和旋转角） ;;
;;; 参数: ;;
;;; p1,p2- 两点定义的旋转轴 ;;
;;; ang - 旋转角度 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ 3D Rotate by Matrix }==-----------------;;
;;; Rotation matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012 ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; p1,p2- Two 3D points defining the axis of rotation ;;
;;; ang - Rotation Angle ;;
;;;-----------------------------------------------------------;;
(defun MAT:RotateBy2P ( p1 p2 ang )
(MAT:Rotation3D P1 (mapcar '- p2 p1) ang)
)
二维镜像变换
;;;-----------------------------------------------------------;;
;;; 二维镜像变换矩阵 ;;
;;; 参数: ;;
;;; p1 - 镜像向量第一点 ;;
;;; p2 - 镜像向量第二点 ;;
;;;-----------------------------------------------------------;;
;;;----------------=={ Reflect by Matrix }==------------------;;
;;; ;;
;;; Reflects a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to transform ;;
;;; p1, p2 - Points representing vector in which to reflect ;;
;;;-----------------------------------------------------------;;
(defun MAT:Reflect ( p1 p2 / a c s x y)
(setq a (angle p1 p2) a (+ a a))
(setq c (cos a) s (sin a))
(setq x (car p1) y (cadr p1))
(list
(list c s0. (- x (+ (* c x) (* s y))))
(list s (- c) 0. (- y (- (* s x) (* c y))))
'(0. 0. 1. 0.)
'(0. 0. 0. 1.)
)
)
三维镜像变换
;;;-----------------------------------------------------------;;
;;; 三维镜像变换矩阵 ;;
;;; 参数: ;;
;;; p1,p2,p3 - 三点定义的镜像平面 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ 3D Reflect by Matrix }==----------------;;
;;; ;;
;;; Reflection matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: highflybird, Copyright ? 2012- ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; p1,p2,p3 - Three 3D points defining the reflection plane;;
;;;-----------------------------------------------------------;;
(defun MAT:Reflect3D (p1 p2 p3 / m ux uy uz)
(mapcar
'set
'(ux uy uz)
(MAT:unit (MAT:vxv (mapcar '- p2 p1) (mapcar '- p3 p1)))
)
(setq m (list (list (- 1. (* 2. ux ux)) (* -2. uy ux) (* -2. ux uz))
(list (* -2. ux uy) (- 1. (* 2. uy uy)) (* -2. uy uz))
(list (* -2. ux uz) (* -2. uy uz) (- 1. (* 2. uz uz)))
)
)
(Mat:DispToMatrix m (mapcar '- p1 (MAT:mxv m p1)))
)
三维环形阵列变换
;;;-----------------------------------------------------------;;
;;; 三维环形阵列 ;;
;;; 输入：Objlst -- 物体集 ;;
;;; Number -- 要阵列的个数（包含自身在内） ;;
;;; FillAngle -- 旋转角度 ;;
;;; IsCCW -- 是否逆时针 ;;
;;; P1 -- 阵列中心点 ;;
;;; P2 -- 阵列轴线的另一点 ;;
;;; 输出：阵列的物体列表 ;;
;;;-----------------------------------------------------------;;
(defun Mat:3dPolarArray (Objlst Number FillAngle IsCCW P1 P2 / lst1 lst2 ANG MAT NEW)
(if (and (= (type number) 'INT) (> number 1))
(progn
(if IsCCW
(setq FillAngle (float FillAngle))
(setq FillAngle (- FillAngle pi pi))
)
(setq ang (/ FillAngle (1- Number)))
(setq mat (vlax-tmatrix (MAT:RotateBy2P P1 P2 ang)))
(repeat (1- Number)
(setq lst1 nil)
(foreach obj ObjLst
(setq new (vla-copy obj))
(vla-transformby new mat)
(setq lst1 (cons new lst1))
)
(setq objLst (reverse lst1))
(setq lst2 (cons objLst lst2))
)
(reverse lst2)
)
(list ObjLst)
)
)
附件中包含Lee-mac的一些算法，与我的大同小异。区别在于，我的矩阵是为大量运算准备，
Lee-mac的为单次次数不多时运用。
;;;---------------=={ 二维变换 }==-----------------;;
;;;-----------------------------------------------------------;;
;;;-----------------------------------------------------------;;
;;; 比例缩放矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1 - 基点 ;;
;;; scale- 缩放比例 ;;
;;;-----------------------------------------------------------;;
;;;-----------------=={ Scale by Matrix }==-------------------;;
;;; ;;
;;; Scales a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to transform ;;
;;; p1 - Base Point for Scaling Transformation ;;
;;; scale- Scale Factor by which to scale object ;;
;;;-----------------------------------------------------------;;
(defun LM:ScaleByMatrix ( target p1 scale / m )
(LM:ApplyMatrixTransformation target
(setq m
(list
(list scale 0. 0.)
(list 0. scale 0.)
(list 0. 0. scale)
)
)
(mapcar '- p1 (MAT:mxv m p1))
)
)
;;;-----------------------------------------------------------;;
;;; 平移变换矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1 - 基点 ;;
;;; p2 - 目标点 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ Translate by Matrix }==-----------------;;
;;; ;;
;;; Translates a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to transform ;;
;;; p1, p2 - Points representing vector by which to translate ;;
;;;-----------------------------------------------------------;;
(defun LM:TranslateByMatrix ( target p1 p2 )
(LM:ApplyMatrixTransformation target
(list
(list 1. 0. 0.)
(list 0. 1. 0.)
(list 0. 0. 1.)
)
(mapcar '- p2 p1)
)
)
;;;-----------------------------------------------------------;;
;;; 旋转变换矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1 - 基点 ;;
;;; ang - 旋转角度 ;;
;;;-----------------------------------------------------------;;
;;;-----------------=={ Rotate by Matrix }==------------------;;
;;; ;;
;;; Rotates a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to transform ;;
;;; p1 - Base Point for Rotation Transformation ;;
;;; ang - Angle through which to rotate object ;;
;;;-----------------------------------------------------------;;
(defun LM:RotateByMatrix ( target p1 ang / m)
(LM:ApplyMatrixTransformation target
(setq m
(list
(list (cos ang) (- (sin ang)) 0.)
(list (sin ang) (cos ang)0.)
(list 0. 0. 1.)
)
)
(mapcar '- p1 (MAT:mxv m p1))
)
)
;;;-----------------------------------------------------------;;
;;; 镜像变换矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1 - 镜像向量第一点 ;;
;;; p2 - 镜像向量第二点 ;;
;;;-----------------------------------------------------------;;
;;;----------------=={ Reflect by Matrix }==------------------;;
;;; ;;
;;; Reflects a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to transform ;;
;;; p1, p2 - Points representing vector in which to reflect ;;
;;;-----------------------------------------------------------;;
(defun LM:ReflectByMatrix ( target p1 p2 / m)
(
(lambda ( a / m )
(LM:ApplyMatrixTransformation target
(setq m
(list
(list (cos a) (sin a)0.)
(list (sin a) (- (cos a)) 0.)
(list 0. 0. 1.)
)
)
(mapcar '- p1 (MAT:mxv m p1))
)
)
(* 2. (angle p1 p2))
)
)
;;;-----------------------------------------------------------;;
;;; 变换函数 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; matrix - 3x3 矩阵 ;;
;;; vector - 移动向量 ;;
;;;-----------------------------------------------------------;;
;;;----------=={ Apply Matrix Transformation }==--------------;;
;;; ;;
;;; Transforms a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to Transform ;;
;;; matrix - 3x3 Matrix by which to Transform object ;;
;;; vector - 3D translation vector ;;
;;;-----------------------------------------------------------;;
(defun LM:ApplyMatrixTransformation ( target matrix vector )
(cond
( (eq 'VLA-OBJECT (type target))
(vla-TransformBy target
(vlax-tMatrix
(append (mapcar (function (lambda ( x v ) (append x (list v)))) matrix vector)
'((0. 0. 0. 1.))
)
)
)
)
( (listp target)
(mapcar
(function
(lambda ( point ) (mapcar '+ (MAT:mxv matrix point) vector))
)
target
)
)
)
)
;;;---------------=={ 三维变换 }==-----------------;;
;;;-----------------------------------------------------------;;
;;;-----------------------------------------------------------;;
;;; 三维旋转变换矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1,p2- 两点定义的旋转轴 ;;
;;; ang - 旋转角度 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ 3D Rotate by Matrix }==-----------------;;
;;; ;;
;;; Rotates a VLA-Object or Point List about a 3D axis using;;
;;; a Transformation matrix. ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2011 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to Rotate ;;
;;; p1,p2- Two 3D points defining the axis of rotation ;;
;;; ang - Rotation Angle ;;
;;;-----------------------------------------------------------;;
(defun LM:Rotate3D ( target p1 p2 ang / ux uy uz u m)
(mapcar 'set '(ux uy uz) (setq u (MAT:unit (mapcar '- p2 p1))))
(LM:ApplyMatrixTransformation target
(setq m
(MAT:m+m
(list
(list (cos ang) 0. 0.)
(list 0. (cos ang) 0.)
(list 0. 0. (cos ang))
)
(MAT:m+m
(MAT:mxs
(list
(list 0. (- uz) uy)
(list uz 0. (- ux))
(list (- uy) ux 0.)
)
(sin ang)
)
(MAT:mxs (mapcar (function (lambda ( e ) (MAT:vxs u e))) u) (- 1. (cos ang)))
)
)
)
(mapcar '- p1 (MAT:mxv m p1))
)
)
;;;-----------------------------------------------------------;;
;;; 三维镜像变换矩阵 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; p1,p2,p3 - 三点定义的镜像平面 ;;
;;;-----------------------------------------------------------;;
;;;---------------=={ 3D Reflect by Matrix }==----------------;;
;;; ;;
;;; Reflects a VLA-Object or Point List in a plane using a ;;
;;; Transformation matrix. ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2011 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to Reflect ;;
;;; p1,p2,p3 - Three 3D points defining the reflection plane;;
;;;-----------------------------------------------------------;;
(defun LM:Reflect3D ( target p1 p2 p3 / m u ux uy uz )
(mapcar 'set '(ux uy uz) (setq u (MAT:unit (MAT:vxv (mapcar '- p2 p1) (mapcar '- p3 p1)))))
(LM:ApplyMatrixTransformation target
(setq m
(list
(list (- 1. (* 2. ux ux)) (* -2. uy ux) (* -2. ux uz))
(list (* -2. ux uy) (- 1. (* 2. uy uy)) (* -2. uy uz))
(list (* -2. ux uz) (* -2. uy uz) (- 1. (* 2. uz uz)))
)
)
(mapcar '- p1 (MAT:mxv m p1))
)
)
;;;-----------------------------------------------------------;;
;;; 变换函数 ;;
;;; 参数: ;;
;;; target - vla-object 或者点 ;;
;;; matrix - 3x3 矩阵 ;;
;;; vector - 移动向量 ;;
;;;-----------------------------------------------------------;;
;;;----------=={ Apply Matrix Transformation }==--------------;;
;;; ;;
;;; Transforms a VLA-Object or Point List using a ;;
;;; Transformation Matrix ;;
;;;-----------------------------------------------------------;;
;;; Author: Lee Mac, Copyright ? 2010 - www.lee-mac.com ;;
;;;-----------------------------------------------------------;;
;;; Arguments: ;;
;;; target - VLA-Object or Point List to Transform ;;
;;; matrix - 3x3 Matrix by which to Transform object ;;
;;; vector - 3D translation vector ;;
;;;-----------------------------------------------------------;;
(defun LM:ApplyMatrixTransformation ( target matrix vector )
(cond
( (eq 'VLA-OBJECT (type target))
(vla-TransformBy target
(vlax-tMatrix
(append (mapcar (function (lambda ( x v ) (append x (list v)))) matrix vector)
'((0. 0. 0. 1.))
)
)
)
)
( (listp target)
(mapcar
(function
(lambda ( point ) (mapcar '+ (MAT:mxv matrix point) vector))
)
target
)
)
)
)
七、块参照，属性的变换矩阵和逆矩阵
;;;-----------------------------------------------------------;;
;;; 块参照的变换矩阵和逆矩阵 ;;
;;;-----------------------------------------------------------;;
;;;-----------------------------------------------------------;;
;;; 功能： 某点在块内坐标系统和世界或者用户坐标系统的转换 ;;
;;; 参数： pt 要变换的点。 ;;
;;; rlst 用 nentselp或者nentsel得到的表的最后一项 ;;
;;; from坐标系：0,WCS; 1,当前UCS; 2,块参照坐标系RCS;;
;;; to 坐标系：0,WCS; 1,当前UCS; 2,块参照坐标系RCS;;
;;;-----------------------------------------------------------;;
;;; MAT:TransNested (gile) ;;
;;; Translates a point coordinates from WCS or UCS to RCS ;;
;;; -coordinates system of a ;;
;;; reference (xref or block) whatever its nested level- ;;
;;; ;;
;;; Arguments ;;
;;; pt : the point to translate ;;
;;; rlst : the parents entities list from the deepest nested;;
;;; to the one inserted in current space -same as ;;
;;; (last (nentsel)) or (last (nentselp)) ;;
;;; from to : as with trans function: 0.WCS, 1.UCS, 2.RCS ;;
;;;-----------------------------------------------------------;;
(defun MAT:TransNested (pt rlst from to / GEOM)
(and (= 1 from) (setq pt (trans pt 1 0)))
(and (= 2 to) (setq rlst (reverse rlst)))
(and (or (= 2 from) (= 2 to))
(while rlst
(setq geom (if (= 2 to)
(MAT:RevRefGeom (car rlst))
(MAT:RefGeom (car rlst))
)
rlst (cdr rlst)
pt (mapcar '+ (MAT:mxv (car geom) pt) (cadr geom))
)
)
)
(if (= 1 to)
(trans pt 0 1)
pt
)
)
;;;-----------------------------------------------------------;;
;;; 功能：图块的变换矩阵 ;;
;;; 输入：块参照的图元名 ;;
;;; 输出：块参照的变换矩阵 ;;
;;;-----------------------------------------------------------;;
;;; MAT:RefGeom (gile) ;;
;;; Returns a list which first item is a 3x3 transformation ;;
;;; matrix(rotation,scales normal) and second item the object ;;
;;; insertion point in its parent(xref, bloc or space) ;;
;;; ;;
;;; Argument : an ename ;;
;;;-----------------------------------------------------------;;
(defun MAT:RefGeom (ename / elst ang norm mat)
(setq elst (entget ename)
ang(cdr (assoc 50 elst))
norm (cdr (assoc 210 elst))
)
(list
(setq mat
(MAT:mxm
(mapcar (function (lambda (v) (trans v 0 norm T)))
'((1.0 0.0 0.0) (0.0 1.0 0.0) (0.0 0.0 1.0))
)
(MAT:mxm
(list (list (cos ang) (- (sin ang)) 0.0)
(list (sin ang) (cos ang) 0.0)
'(0.0 0.0 1.0)
)
(list (list (cdr (assoc 41 elst)) 0.0 0.0)
(list 0.0 (cdr (assoc 42 elst)) 0.0)
(list 0.0 0.0 (cdr (assoc 43 elst)))
)
)
)
)
(mapcar
'-
(trans (cdr (assoc 10 elst)) norm 0)
(MAT:mxv mat
(cdr (assoc 10 (tblsearch "BLOCK" (cdr (assoc 2 elst)))))
)
)
)
)
;;;-----------------------------------------------------------;;
;;; 功能：图块的变换矩阵的逆矩阵 ;;
;;;-----------------------------------------------------------;;
;;; MAT:RevRefGeom (gile) ;;
;;; MAT:RefGeom inverse function ;;
;;; 输入：块参照的图元名 ;;
;;; 输出：块参照的变换矩阵的逆矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:RevRefGeom (ename / entData ang norm mat)
(setq entData (entget ename)
ang (- (cdr (assoc 50 entData)))
norm (cdr (assoc 210 entData))
)
(list
(setq mat
(MAT:mxm
(list (list (/ 1 (cdr (assoc 41 entData))) 0.0 0.0)
(list 0.0 (/ 1 (cdr (assoc 42 entData))) 0.0)
(list 0.0 0.0 (/ 1 (cdr (assoc 43 entData))))
)
(MAT:mxm
(list (list (cos ang) (- (sin ang)) 0.0)
(list (sin ang) (cos ang) 0.0)
'(0.0 0.0 1.0)
)
(mapcar (function (lambda (v) (trans v norm 0 T)))
'((1.0 0.0 0.0) (0.0 1.0 0.0) (0.0 0.0 1.0))
)
)
)
)
(mapcar '-
(cdr (assoc 10 (tblsearch "BLOCK" (cdr (assoc 2 entData)))))
(MAT:mxv mat (trans (cdr (assoc 10 entData)) norm 0))
)
)
)
;;;-----------------------------------------------------------;;
;;; 属性的变换矩阵Attrib Transformation Matrix. -highflybird;;
;;; 输入：Ename 属性的图元名 ;;
;;; 输出：属性的变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:AttGeom (ename / ang norm mat elst)
(setq elst (entget ename)
ang(cdr (assoc 50 elst))
norm (cdr (assoc 210 elst))
)
(list
(setq mat
(mxm
(mapcar (function (lambda (v) (trans v 0 norm T)))
'((1.0 0.0 0.0) (0.0 1.0 0.0) (0.0 0.0 1.0))
)
(list (list (cos ang) (- (sin ang)) 0.0)
(list (sin ang) (cos ang) 0.0)
'(0.0 0.0 1.0)
)
)
)
(trans (cdr (assoc 10 elst)) norm 0)
)
)
八、三点变换矩阵，UCS变换矩阵，图元变换矩阵和通用变换矩阵
;;;-----------------------------------------------------------;;
;;; Append displacement vector to a matrix -Highflybird- ;;
;;; 把位移矢量添加到矩阵中 ;;
;;; 输入：mat -- 矩阵(3x3)，disp -- 位移矢量 ;;
;;; 输出：一个4X4的变换CAD的标准变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun Mat:DispToMatrix (mat disp)
(append
(mapcar 'append mat (mapcar 'list disp))
'((0. 0. 0. 1.))
)
)
;;;-----------------------------------------------------------;;
;;; 从一个坐标系统到另一个坐标系统的变换矩阵 ;;
;;; 输入：from - 源坐标系；to - 目的坐标系 ;;
;;; 输出：一个4X4的变换CAD的标准变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:Trans (from to)
(Mat:DispToMatrix
(mapcar
(function (lambda (v) (trans v from to t)))
'((1. 0. 0.) (0. 1. 0.) (0. 0. 1.))
)
(trans '(0 0 0) to from)
)
)
;;;-----------------------------------------------------------;;
;;; wcs到ucs矩阵，也可称UCS的变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:w2u () (MAT:Trans 0 1))
;;;-----------------------------------------------------------;;
;;; ucs到wcs矩阵，也可称UCS的逆变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:u2w () (MAT:Trans 1 0))
;;;-----------------------------------------------------------;;
;;; 通用变换矩阵 by highflybird ;;
;;; 输入：from - 原坐标系， ;;
;;; to - 目的坐标系， ;;
;;; Org- 目的坐标系的原点相对原坐标系的位置 ;;
;;; Ang- 相对于原坐标系的旋转角度 ;;
;;; 输出：两个矩阵，一个是从原坐标系变换到目的坐标系的变换矩阵;;
;;; 一个是从目的坐标系变换到原坐标系的变换矩阵 ;;
;;;-----------------------------------------------------------;;
(defun MAT:Trans1 (from to Org Ang / Mat Rot Inv Cen)
(setq Mat (mapcar (function (lambda (v) (trans v from to T)))
'((1. 0. 0.) (0. 1. 0.) (0. 0. 1.))
)
)
(if (not (equal ang 0 1e-14))
(setq Rot (list (list (cos ang) (- (sin ang)) 0.)
(list (sin ang) (cos ang) 0.)
(list 0. 0. 1.)
)
mat (MAT:mxm mat Rot)
)
)
(setq Cen (trans Org to from))
(setq Inv (mat:trp mat))
(list
(Mat:DispToMatrix Inv (mat:mxv Inv (mapcar '- Cen))) ;from->to (trans pt from to)
(Mat:DispToMatrix mat Cen) ;to->from (trans pt to from)
)
)
;;;-----------------------------------------------------------;;
;;; 通过两个坐标轴和坐标原点定义的变换矩阵-by highflybird ;;
;;; 输入：Org- 坐标系原点， ;;
;;; Vx - 坐标系X 方向， ;;
;;; Vy - 坐标系y 方向 ;;
;;; 输出：两个矩阵，一个是该坐标系的变换矩阵，一个是其逆矩阵;;
;;;-----------------------------------------------------------;;
(defun MAT:2VMatrix (Org Vx Vy / Vz Rot)
(if (or (equal Vx '(0 0 0) 1e-14) (equal Vy '(0 0 0) 1e-14))
'((1. 0. 0. 0.) (0. 1. 0. 0.) (0. 0. 1. 0.) (0. 0. 0. 1.))
(progn
(setq Vx (Mat:Unit Vx)) ;X Axis
(setq Vy (Mat:Unit Vy)) ;Y Axis
(setq Vz (Mat:unit (MAT:vxv Vx Vy))) ;Z Axis
(setq Vy (Mat:unit (MAT:vxv Vz Vx))) ;Y Axis
(setq Rot (list Vx Vy Vz)) ;Rotation matrix
(list ;Inverse Rotation matrix
(Mat:DispToMatrix (MAT:trp Rot) Org) ;The transformation matrix from UCS to WCS
(Mat:DispToMatrix Rot (MAT:mxv Rot (mapcar '- Org))) ;The transformation matrix from WCS to UCS
)
)
)
)
;;;-----------------------------------------------------------;;
;;; Mat:3PMatrix-Highflybird- ;;
;;; 通过两个坐标轴和坐标原点定义的变换矩阵-by highflybird ;;
;;; 输入：P1 - 坐标系原点， ;;
;;; P2 - 坐标系的第2点 ;;
;;; P3 - 坐标系的第3点 ;;
;;; 输出：两个矩阵，一个是该坐标系的变换矩阵，一个是其逆矩阵;;
;;;-----------------------------------------------------------;;
(defun Mat:3PMatrix (p1 p2 p3 / v1 v2 v3)
(MAT:2VMatrix P1 (mapcar '- p2 p1) (mapcar '- p3 p1))
)
;;;-----------------------------------------------------------;;
;;; 平齐实体的变换矩阵-by highflybird ;;
;;; 输入：Ent - 实体名 ;;
;;; 输出：平齐这个实体的变换矩阵和它的逆矩阵 ;;
;;;-----------------------------------------------------------;;
(defun Mat:EntityMatrix (Ent / z dxf Cen obj an m1 mat Inv org)
(setq dxf (entget ent))
(if (setq Cen (cdr (assoc 10 dxf))) ;Insertpoint,center or startpoint,etc.
(if (null (caddr Cen))
(setq Cen (append Cen '(0.0)))
)
(setq Cen '(0 0 0))
)
(setq obj (vlax-ename->vla-object Ent))
(if (and (vlax-property-available-p obj 'elevation) ;If it has elevation value.
(wcmatch (vla-get-objectname obj) "*Polyline") ;It's a "AcDb2dPolyline" or "AcDbPolyline" object
)
(setq z (vla-get-elevation obj)
Cen (list (car Cen) (cadr Cen) (+ (caddr Cen) z)) ;add elevation value
)
)
(if (vlax-property-available-p obj 'rotation) ;if it has a rotaion angle
(setq an (vla-get-rotation obj))
(setq an 0)
)
(MAT:Trans1 0 Ent Cen an) ;return two matrices, the first is WCS->OCS,the second is OCS->WCS
)
;;;-----------------------------------------------------------;;
;;; 点变换1 ;;
;;; 输入: 要变换的点和原点及变换向量 ;;
;;; 输出: 点变换后的位置 ;;
;;;-----------------------------------------------------------;;
(defun MAT:TransU2W (p p0 v / d x0 y0 x1 y1 dv rt)
(setq d (distance '(0 0) v))
(if (equal d 1e-14)
P0
(setq x1 (carp)
y1 (cadr p)
x0 (carv)
y0 (cadr v)
dv (list (/ (- (* x1 x0) (* y1 y0)) d)
(/ (+ (* y1 x0) (* x1 y0)) d)
)
rt (mapcar '+ P0 dv)
)
)
)
;;;-----------------------------------------------------------;;
;;; 点变换2 ;;
;;; 输入: 要变换的点和原点及变换向量 ;;
;;; 输出: 点变换后的位移向量 ;;
;;;-----------------------------------------------------------;;
(defun MAT:TransW2U (p p0 v / d x0 y0 x1 y1 dv)
(setq d (distance '(0 0) v))
(if (equal d 1e-14)
(list 0 0)
(setq x1 (- (car p) (car p0))
y1 (- (cadr p) (cadr p0))
x0 (car v)
y0 (cadr v)
dv (list (/ (+ (* x1 x0) (* y1 y0)) d)
(/ (- (* y1 x0) (* x1 y0)) d)
)
)
)
)
九、轴测变换矩阵
;;;-----------------------------------------------------------;;
;;;通用的轴测变换矩阵 highflybird2012.12 ;;
;;;Axonometric projections Rotation matrices ;;
;;;Isometric projection: a = (/ pi 4),b = (atan (- (sqrt 2))) ;;
;;;Input: a - Rotation angle about the vertical axis ;;
;;; b - Rotation angle about the horizontal axis ;;
;;;Output: transforamtion matrix of this projection ;;
;;;-----------------------------------------------------------;;
(defun MAT:ISO (a b / ca sa cb sb)
(setq ca (cos a))
(setq sa (sin a))
(setq cb (cos b))
(setq sb (sin b))
(list (list ca (- sa) 0 0)
(list (* sa cb) (* ca cb) (- sb) 0)
(list (* sa sb) (* ca sb) cb 0)
(list 0 0 0 1)
)
)
十、矩阵的特征值和特征向量
chlh_jd讨论比较深入，参见如下链接：
http://bbs.mjtd.com/thread-99908-1-1.html
http://www.theswamp.org/index.php?topic=43453.0十一、 矩阵的分解、求逆和解方程组
http://www.theswamp.org/index.php?topic=32478.0
http://www.theswamp.org/index.php?topic=22638.msg439381#msg439381
http://www.theswamp.org/index.php?topic=13505.msg163485#msg163485
关于上面的几个链接地址的源代码已经录入下面的附件中了
真是精彩！收藏学习！ 精辟！多谢分享！支持！ 大作只有深深的思考后才敢发表见解，先拜读。 用Lisp解矩阵问题，值得学习。 精品啊好好学习一下 确实分析的不错:):) PLgis 发表于 2013-4-17 15:31 static/image/common/back.gif
用Lisp解矩阵问题，值得学习。
计算机图形学算法是关键，用什么语言写出来就不是那么重要了，包括用其他语言二次开发ACAD，ARX,VBA等等，矩阵变换是ACAD图形实体各种编辑命令实现的基石。 基础还是要掌握的，至于实现（比如借鉴Cal）没必要较真，工程师和程序员的区别要把握度 真的很不错的讲义！！！！！！！！:D 真的不错，像楼主这样钻研精神和无私精神致敬。 首页留名，讲得很不错:) {:soso_e113:}神帖留名{:soso__6151841889465315901_1:} 有一个未定义函数 MatispToMatrix Free-Lancer 发表于 2013-4-27 14:57 static/image/common/back.gif
有一个未定义函数 MatispToMatrix
建议下载附件:matrix-lib.lsp
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