Coordinate System
The body coordinate system (right): In order to describe the basic body coordinate system introduced by
the user coordinate system (right): introduced describing the entire body of the user coordinate system
observation coordinate system (left): gesture introduced as an observation, the observer is located position, because the Z-axis direction of the eye is directed at the
device coordinate system: the coordinate system of the display
normalized device coordinates: related to specific device coordinate system
X-based hand, the hand switch from the X axis to the Y-axis, Z-axis corresponds to see if you can thumb
Convert
\[ p^{'}=\left[\begin{matrix}\end{matrix}\right] \]
平移变换T
\[ p^{'}=\left[\begin{matrix} x^{'}&y^{'}&1 \end{matrix}\right]=\left[\begin{matrix}x&y&1\end{matrix}\right]\left[\begin{matrix} 1&0&0\\ 0&1&0\\ T_x&T_y&1\\ \end{matrix}\right] \]
比例变换S
\[ p^{'}=\left[\begin{matrix} x^{'}&y^{'}&1 \end{matrix}\right]=\left[\begin{matrix}x&y&1\end{matrix}\right]\left[\begin{matrix} S_x&0&0\\ 0&S_y&0\\ 0&0&1\\ \end{matrix}\right] \]
旋转变换R
\[ p^{'}=\left[\begin{matrix} x^{'}&y^{'}&1 \end{matrix}\right]=\left[\begin{matrix}x&y&1\end{matrix}\right]\left[\begin{matrix} \cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1 \\ \ end {matrix} \ right] \] when connected by matrix multiplication from left to right
reading order memory, cos, sin, -sin, cos
Symmetry transform
\ [p ^ { '} = \ left [\ begin {matrix} x ^ {'} & y ^ { '} & 1 \ end {matrix} \ right] = \ left [\ begin {matrix} x & y & 1 \ end { matrix} \ right] \ left [
\ begin {matrix} a & d & 0 \\ b & e & 0 \\ 0 & 0 & 1 \\ \ end {matrix} \ right] \] About Y = -X symmetric
\ [\ left [\ begin { matrix} 0 & - 1 & 0 \\ -1 & 0 &
0 \\ 0 & 0 & 1 \\ \ end {matrix} \ right] \] shearing transformation, the X axis direction on the wrong Y cut
\ [p ^ { '} = \ left [\ begin {matrix} x ^ { '} & y ^ { '} & 1 \ end {matrix} \ right] = \ left [\ begin {matrix} x & y & 1 \ end {matrix} \ right] \ left [\ begin {matrix} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \\ \ end
{matrix} \ right] = \ left (\ begin {matrix} x + by & y & 1 \ end {matrix} \ right) \] on the wrong X cut
\ [\ left (\ begin { matrix} x + by & y & 1 \ end {
matrix} \ right) \] translation, rotation, scaling, shearing belong topological invariant geometric transformation.
Viewing transformation
I understand that the game is a window, then a block area of the screen is used to run the game, so it is necessary now to the origin, during zooming, then moved back, is the viewing transformation.
\ [H = T_1 (-wxl, -wyl) * S (S_x, S_y) * T_2 (vxl, vyl) \\ = \ left [\ begin {matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -wxl & -wyl & 1 \\ \ end {matrix} \ right] \ left [\ begin {matrix} \ frac {vxh-vxl} {wxh-wxl} & 0 & 0 \\ 0 & \ frac {vxh-vxl} {wxh-wxl} & 0 \\ 0 & 0 & 1 \\ \ end {matrix} \ right] \
left [\ begin {matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ vxl & vyl & 1 \\ \ end {matrix} \ right] \] three-dimensional rotational conversion
mnemonics: about which axis of rotation, which lower right one grid (if it reaches the limit cycle) as a starting point, cos sin -sin cos This allows the
inversion of rotation of sin just put the digital inverted position on the line, cos unchanged
Isometric projection
normal vector and the angle between the projection plane of the three axes are equal.