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旋转

绕轴旋转​

  • 定义向量：vec1→=(x1,y1,z1,1)\overrightarrow{vec_1} = (x_1,y_1,z_1,1)
  • 定义旋转后的向量：vec2→=(x2,y2,z2,1)\overrightarrow{vec_2} = (x_2,y_2,z_2,1)
  • 定义向量长度为：len=∣vec→∣len = |\overrightarrow{vec}|
  • 定义向量与起始轴的夹角为α∘\alpha^\circ
  • 定义向量绕轴旋转β∘\beta^\circ
  • c=cos⁡(β)c = \cos(\beta)
  • s=sin⁡(β)s = \sin(\beta)

旋转方向​

  • 当物体绕 z 轴，从 x 轴正半轴向 y 轴正半轴逆时针旋转时，是正向旋转，反之为负。
  • 当物体绕 x 轴，从 y 轴正半轴向 z 轴正半轴逆时针旋转时，是正向旋转，反之为负。
  • 当物体绕 y 轴，从 z 轴正半轴向 x 轴正半轴逆时针旋转时，是正向旋转，反之为负。

定义起始轴(弧度为 0)​

  • 当物体绕 z 轴，x 轴正半轴为起始轴，往 y 轴正半轴为正
  • 当物体绕 x 轴，y 轴正半轴为起始轴，往 z 轴正半轴为正
  • 当物体绕 y 轴，z 轴正半轴为起始轴，往 x 轴正半轴为正

绕 Z 轴旋转​

/*
   Y
   │
   │
   └──────── X
*/
x2=len×cos⁡(α+β)=len×(cos⁡(α)cos⁡(β)−sin⁡(α)sin⁡(β))=len×cos⁡(α)×cos⁡(β)−len×sin⁡(α)×sin⁡(β)=x1×cos⁡(β)−y1×sin⁡(β)x_2 = len \times \cos(\alpha+\beta) = len \times (\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)) = len \times \cos(\alpha) \times \cos(\beta) - len \times \sin(\alpha) \times \sin(\beta) = x_1 \times \cos(\beta) - y_1 \times \sin(\beta) y2=len×sin⁡(α+β)=len×(sin⁡(α)cos⁡(β)+cos⁡(α)sin⁡(β))=len×sin⁡(α)×cos⁡(β)+len×cos⁡(α)×sin⁡(β)=y1×cos⁡(β)+x1×sin⁡(β)y_2 = len \times \sin(\alpha+\beta) = len \times (\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)) = len \times \sin(\alpha) \times \cos(\beta) + len \times \cos(\alpha) \times \sin(\beta) = y_1 \times \cos(\beta) + x_1 \times \sin(\beta) vec2→=[cs00−sc0000100001]×vec1→\overrightarrow{vec_2} = \begin{bmatrix} {c} & {s} & 0 & 0 \\ {-s} & {c} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \times \overrightarrow{vec_1}

绕 X 轴旋转​

/*
         Y
         │
         │
Z ────────┘
*/
y2=len×cos⁡(α+β)=len×(cos⁡(α)cos⁡(β)−sin⁡(α)sin⁡(β))=len×cos⁡(α)×cos⁡(β)−len×sin⁡(α)×sin⁡(β)=y1×cos⁡(β)−z1×sin⁡(β)y_2 = len \times \cos(\alpha+\beta) = len \times (\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)) = len \times \cos(\alpha) \times \cos(\beta) - len \times \sin(\alpha) \times \sin(\beta) = y_1 \times \cos(\beta) - z_1 \times \sin(\beta) z2=len×sin⁡(α+β)=len×(sin⁡(α)cos⁡(β)+cos⁡(α)sin⁡(β))=len×sin⁡(α)×cos⁡(β)+len×cos⁡(α)×sin⁡(β)=z1×cos⁡(β)+y1×sin⁡(β)z_2 = len \times \sin(\alpha+\beta) = len \times (\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)) = len \times \sin(\alpha) \times \cos(\beta) + len \times \cos(\alpha) \times \sin(\beta) = z_1 \times \cos(\beta) + y_1 \times \sin(\beta) vec2→=[10000cs00−sc00001]×vec1→\overrightarrow{vec_2} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & {c} & {s} & 0 \\ 0 & {-s} & {c} & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \times \overrightarrow{vec_1}

绕 Y 轴旋转​

/*
   ┌──────── X
   │
   │
   Z
*/
z2=len×cos⁡(α+β)=len×(cos⁡(α)cos⁡(β)−sin⁡(α)sin⁡(β))=len×cos⁡(α)×cos⁡(β)−len×sin⁡(α)×sin⁡(β)=z1×cos⁡(β)−x1×sin⁡(β)z_2 = len \times \cos(\alpha+\beta) = len \times (\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)) = len \times \cos(\alpha) \times \cos(\beta) - len \times \sin(\alpha) \times \sin(\beta) = z_1 \times \cos(\beta) - x_1 \times \sin(\beta) x2=len×sin⁡(α+β)=len×(sin⁡(α)cos⁡(β)+cos⁡(α)sin⁡(β))=len×sin⁡(α)×cos⁡(β)+len×cos⁡(α)×sin⁡(β)=x1×cos⁡(β)+z1×sin⁡(β)x_2 = len \times \sin(\alpha+\beta) = len \times (\sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)) = len \times \sin(\alpha) \times \cos(\beta) + len \times \cos(\alpha) \times \sin(\beta) = x_1 \times \cos(\beta) + z_1 \times \sin(\beta) vec2→=[c0−s00100s0c00001]×vec1→\overrightarrow{vec_2} = \begin{bmatrix} {c} & 0 & {-s} & 0 \\ 0 & 1 & 0 & 0 \\ {s} & 0 & {c} & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \times \overrightarrow{vec_1}