OpenGL绘图基本矩阵运算讲解

时间：2025-03-30 08:10:29

OpenGL绘图基本矩阵运算讲解

  • 一、平移矩阵
  • 二、缩放矩阵
  • 三、旋转矩阵
  • 四、投影矩阵
  • 五、剪切矩阵
  • 六、矩阵分解
  • 七、视图矩阵

一、平移矩阵

坐标平移 ( d x ， d y ， d z ) (dx，dy，dz) (dx，dy，dz)：
[ 1 0 0 d x 0 1 0 d y 0 0 1 d z 0 0 0 1 ] [100dx010dy001dz0001]

⎣⎢⎢⎡​1000​0100​0010​dx​dy​dz​1​⎦⎥⎥⎤​
例：
[ 1 0 0 d x 0 1 0 d y 0 0 1 d z 0 0 0 1 ] ⋅ [ x y z 1 ] = [ ( 1 ⋅ x ) + ( 0 ⋅ y ) + ( 0 ⋅ z ) + ( d x ⋅ 1 ) ( 0 ⋅ x ) + ( 1 ⋅ y ) + ( 0 ⋅ z ) + ( d y ⋅ 1 ) ( 0 ⋅ x ) + ( 0 ⋅ y ) + ( 1 ⋅ z ) + ( d z ⋅ 1 ) ( 0 ⋅ x ) + ( 0 ⋅ y ) + ( 0 ⋅ z ) + ( 1 ⋅ 1 ) ] = [ x + d x y + d y z + d z 1 ] [100dx010dy001dz0001]
\cdot [xyz1]
= [(1⋅x)+(0⋅y)+(0⋅z)+(dx⋅1)(0⋅x)+(1⋅y)+(0⋅z)+(dy⋅1)(0⋅x)+(0⋅y)+(1⋅z)+(dz⋅1)(0⋅x)+(0⋅y)+(0⋅z)+(1⋅1)]
= [x+dxy+dyz+dz1]
⎣⎢⎢⎡​1000​0100​0010​dx​dy​dz​1​⎦⎥⎥⎤​⋅⎣⎢⎢⎡​xyz1​⎦⎥⎥⎤​=⎣⎢⎢⎡​(1⋅x)+(0⋅y)+(0⋅z)+(dx​⋅1)(0⋅x)+(1⋅y)+(0⋅z)+(dy​⋅1)(0⋅x)+(0⋅y)+(1⋅z)+(dz​⋅1)(0⋅x)+(0⋅y)+(0⋅z)+(1⋅1)​⎦⎥⎥⎤​=⎣⎢⎢⎡​x+dx​y+dy​z+dz​1​⎦⎥⎥⎤​

二、缩放矩阵

缩放 ( s x , s y , s z ) (s_x,s_y,s_z) (sx​,sy​,sz​) ：
[ s x 0 0 0 0 s y 0 0 0 0 s z 0 0 0 0 1 ] [sx0000sy0000sz00001]

⎣⎢⎢⎡​sx​000​0sy​00​00sz​0​0001​⎦⎥⎥⎤​
例：
[ s x 0 0 0 0 s y 0 0 0 0 s z 0 0 0 0 1 ] ⋅ [ x y z 1 ] = [ ( s x ⋅ x ) + ( 0 ⋅ y ) + ( 0 ⋅ z ) + ( 0 ⋅ 1 ) ( 0 ⋅ x ) + ( s y ⋅ y ) + ( 0 ⋅ z ) + ( 0 ⋅ 1 ) ( 0 ⋅ x ) + ( 0 ⋅ y ) + ( s z ⋅ z ) + ( 0 ⋅ 1 ) ( 0 ⋅ x ) + ( 0 ⋅ y ) + ( 0 ⋅ z ) + ( 1 ⋅ 1 ) ] = [ x ⋅ s x y ⋅ s y z ⋅ s z 1 ] [sx0000sy0000sz00001]
\cdot [xyz1]
= [(sx⋅x)+(0⋅y)+(0⋅z)+(0⋅1)(0⋅x)+(sy⋅y)+(0⋅z)+(0⋅1)(0⋅x)+(0⋅y)+(sz⋅z)+(0⋅1)(0⋅x)+(0⋅y)+(0⋅z)+(1⋅1)]
= [x⋅sxy⋅syz⋅sz1]
⎣⎢⎢⎡​sx​000​0sy​00​00sz​0​0001​⎦⎥⎥⎤​⋅⎣⎢⎢⎡​xyz1​⎦⎥⎥⎤​=⎣⎢⎢⎡​(sx​⋅x)+(0⋅y)+(0⋅z)+(0⋅1)(0⋅x)+(sy​⋅y)+(0⋅z)+(0⋅1)(0⋅x)+(0⋅y)+(sz​⋅z)+(0⋅1)(0⋅x)+(0⋅y)+(0⋅z)+(1⋅1)​⎦⎥⎥⎤​=⎣⎢⎢⎡​x⋅sx​y⋅sy​z⋅sz​1​⎦⎥⎥⎤​

三、旋转矩阵

绕 n ⃗ = ( n x , n y , n z ) \vec{n} = (n_x,n_y,n_z) n =(nx​,ny​,nz​) 旋转 θ θ θ
[ n x 2 ( 1 − c o s θ ) + c o s θ n x n y ( 1 − c o s θ ) + n z s i n θ n z n x ( 1 − c o s θ ) − n y s i n θ 0 n x n y ( 1 − c o s θ ) − n z s i n θ n y 2 ( 1 − c o s θ ) + c o s θ n y n z ( 1 − c o s θ ) + n x s i n θ 0 n z n x ( 1 − c o s θ ) + n y s i n θ n y n z ( 1 − c o s θ ) − n x s i n θ n z 2 ( 1 − c o s θ ) + c o s θ 0 0 0 0 1 ] [n2x(1−cosθ)+cosθnxny(1−cosθ)+nzsinθnznx(1−cosθ)−nysinθ0nxny(1−cosθ)−nzsinθn2y(1−cosθ)+cosθnynz(1−cosθ)+nxsinθ0nznx(1−cosθ)+nysinθnynz(1−cosθ)−nxsinθn2z(1−cosθ)+cosθ00001]

⎣⎢⎢⎡​nx2​(1−cosθ)+cosθnx​ny​(1−cosθ)−nz​sinθnz​nx​(1−cosθ)+ny​sinθ0​nx​ny​(1−cosθ)+nz​sinθny2​(1−cosθ)+cosθny​nz​(1−cosθ)−nx​sinθ0​nz​nx​(1−cosθ)−ny​sinθny​nz​(1−cosθ)+nx​sinθnz2​(1−cosθ)+cosθ0​0001​⎦⎥⎥⎤​

四、投影矩阵

n n n近裁剪面
f f f 远裁剪面
l ， r l，r l，r近裁剪面左右边界
t ， b t，b t，b近裁剪面上下边界

透视矩阵：
其他参数转换方法:
r = t a n ( f o v y ) ∗ a s p e c t ∗ n r = tan(fovy)* aspect * n r=tan(fovy)∗aspect∗n
l = − r l = -r l=−r
t = t a n ( f o v y ) ∗ n t = tan(fovy)* n t=tan(fovy)∗n
b = − t b = -t b=−t

矩阵内容:
[ 2 n r − l 0 r + l r − l 0 0 2 n t − b t + b t − b 0 0 0 − ( f + n ) f − n − 2 f n f − n 0 0 − 1 0 ] [2nr−l0r+lr−l002nt−bt+bt−b000−(f+n)f−n−2fnf−n00−10]

⎣⎢⎢⎡​r−l2n​000​0t−b2n​00​r−lr+l​t−bt+b​f−n−(f+n)​−1​00f−n−2fn​0​⎦⎥⎥⎤​

正交矩阵:

[ 2 n r − l 0 0 − r + l r − l 0 2 t − b 0 − t + b t − b 0 0 − 2 f − n − f + n f − n 0 0 0 1 ] [2nr−l00−r+lr−l02t−b0−t+bt−b00−2f−n−f+nf−n0001]

⎣⎢⎢⎡​r−l2n​000​0t−b2​00​00f−n−2​0​−r−lr+l​−t−bt+b​−f−nf+n​1​⎦⎥⎥⎤​

正交矩阵2D
f = 1 f=1 f=1
n = − 1 n=-1 n=−1
[ 2 n r − l 0 0 − r + l r − l 0 2 t − b 0 − t + b t − b 0 0 − 1 0 0 0 0 1 ] [2nr−l00−r+lr−l02t−b0−t+bt−b00−100001]

⎣⎢⎢⎡​r−l2n​000​0t−b2​00​00−10​−r−lr+l​−t−bt+b​01​⎦⎥⎥⎤​

五、剪切矩阵

h n m h_{nm} hnm​ 由n轴向m轴剪切值
剪切矩阵：
[ 1 h y x h z x 0 h x y 1 h z y 0 h x z h y z 1 0 0 0 0 1 ] [1hyxhzx0hxy1hzy0hxzhyz100001]

⎣⎢⎢⎡​1hxy​hxz​0​hyx​1hyz​0​hzx​hzy​10​0001​⎦⎥⎥⎤​

六、矩阵分解

记Model矩阵 M = T ∗ R ∗ S M = T * R * S M=T∗R∗S
其中 T为平移矩阵，R为旋转矩阵，S为缩放矩阵
记 R ′ [ R 11 R 12 R 13 0 R 21 R 22 R 23 0 R 31 R 32 R 33 0 0 0 0 1 ] = [ n x 2 ( 1 − c o s θ ) + c o s θ n x n y ( 1 − c o s θ ) + n z s i n θ n z n x ( 1 − c o s θ ) − n y s i n θ 0 n x n y ( 1 − c o s θ ) − n z s i n θ n y 2 ( 1 − c o s θ ) + c o s θ n y n z ( 1 − c o s θ ) + n x s i n θ 0 n z n x ( 1 − c o s θ ) + n y s i n θ n y n z ( 1 − c o s θ ) − n x s i n θ n z 2 ( 1 − c o s θ ) + c o s θ 0 0 0 0 1 ] 记R' [R11R12R130R21R22R230R31R32R3300001]

= [n2x(1−cosθ)+cosθnxny(1−cosθ)+nzsinθnznx(1−cosθ)−nysinθ0nxny(1−cosθ)−nzsinθn2y(1−cosθ)+cosθnynz(1−cosθ)+nxsinθ0nznx(1−cosθ)+nysinθnynz(1−cosθ)−nxsinθn2z(1−cosθ)+cosθ00001]
记R′⎣⎢⎢⎡​R11​R21​R31​0​R12​R22​R32​0​R13​R23​R33​0​0001​⎦⎥⎥⎤​=⎣⎢⎢⎡​nx2​(1−cosθ)+cosθnx​ny​(1−cosθ)−nz​sinθnz​nx​(1−cosθ)+ny​sinθ0​nx​ny​(1−cosθ)+nz​sinθny2​(1−cosθ)+cosθny​nz​(1−cosθ)−nx​sinθ0​nz​nx​(1−cosθ)−ny​sinθny​nz​(1−cosθ)+nx​sinθnz2​(1−cosθ)+cosθ0​0001​⎦⎥⎥⎤​
则 [ 1 0 0 d x 0 1 0 d y 0 0 1 d z 0 0 0 1 ] ⋅ [ R 11 R 12 R 13 0 R 21 R 22 R 23 0 R 31 R 32 R 33 0 0 0 0 1 ] ⋅ [ s x 0 0 0 0 s y 0 0 0 0 s z 0 0 0 0 1 ] = 则 [100dx010dy001dz0001]
\cdot [R11R12R130R21R22R230R31R32R3300001]
\cdot [sx0000sy0000sz00001]
= 则⎣⎢⎢⎡​1000​0100​0010​dx​dy​dz​1​⎦⎥⎥⎤​⋅⎣⎢⎢⎡​R11​R21​R31​0​R12​R22​R32​0​R13​R23​R33​0​0001​⎦⎥⎥⎤​⋅⎣⎢⎢⎡​sx​000​0sy​00​00sz​0​0001​⎦⎥⎥⎤​=
[ s x ∗ R 11 s x ∗ R 12 s x ∗ R 13 d x s y ∗ R 21 s y ∗ R 22 s y ∗ R 23 d y s z ∗ R 31 s z ∗ R 32 s z ∗ R 33 d z 0 0 0 1 ] [sx∗R11sx∗R12sx∗R13dxsy∗R21sy∗R22sy∗R23dysz∗R31sz∗R32sz∗R33dz0001]
⎣⎢⎢⎡​sx​∗R11​sy​∗R21​sz​∗R31​0​sx​∗R12​sy​∗R22​sz​∗R32​0​sx​∗R13​sy​∗R23​sz​∗R33​0​dx​dy​dz​1​⎦⎥⎥⎤​

七、视图矩阵

N ⃗ = E y e ⃗ − A t ⃗ \vec{N} = \vec{Eye} - \vec{At} N =Eye ​−At
N ⃗ = N ⃗ ∣ N ⃗ ∣ \vec{N} = \frac{\vec{N}}{|\vec{N}|} N =∣N ∣N ​

U ⃗ = U p ⃗ × N ⃗ \vec{U} = \vec{Up} \times\vec{N} U =Up ​×N
U ⃗ = U ⃗ ∣ U ⃗ ∣ \vec{U} = \frac{\vec{U}}{|\vec{U}|} U =∣U ∣U ​

V ⃗ = N ⃗ × U ⃗ \vec{V} = \vec{N}\times \vec{U} V =N ×U
V ⃗ = V ⃗ ∣ V ⃗ ∣ \vec{V} = \frac{\vec{V}}{|\vec{V}|} V =∣V ∣V ​
视图矩阵
[ U x U y U z − N ⃗ × E y e ⃗ V x V y V z − V ⃗ × E y e ⃗ N x N y N z − U ⃗ × E y e ⃗ 0 0 0 1 ] [UxUyUz−→N×→EyeVxVyVz−→V×→EyeNxNyNz−→U×→Eye0001]

⎣⎢⎢⎡​Ux​Vx​Nx​0​Uy​Vy​Ny​0​Uz​Vz​Nz​0​−N ×Eye ​−V ×Eye ​−U ×Eye ​1​⎦⎥⎥⎤​

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