matrix transform for luastg

Another time copy from css (first attempt is ease function)

Theory

From css3 animation, we have matrix, matrix3d method to describe transform of a image affinely.

For 2d image: 0. translate, skew and scale according to (0, 0)

$$begin{bmatrix} x \ y \ 1 end{bmatrix} begin{bmatrix} text{scaleX} & text{skewX} & text{translateX} \ text{skewY} & text{scaleY} & text{translateY} \ 0 & 0 & 1 end{bmatrix}$$

1. rotation according to (0, 0)

$$begin{bmatrix} cos{theta} & -sin{theta} & 0 \ sin{theta} & cos{theta} & 0 \ 0 & 0 & 1 end{bmatrix}$$

Matrix is kinda linear transformation, so rotation matrix ($R$) + sketching matrix ($S$) = $T(=RS)$

If we have matrix $A$ (invertiable), we can use eigenvalue decomposition to tween, $A=VDV^{-1}$, if separated into 10 steps, $T = VD^{1/10}V^{-1}$

Also for matrix 3d on 2d image:

o. translate, skew and scale according to (0, 0, 0)

$$begin{bmatrix} x \ y \ 1 end{bmatrix} begin{bmatrix} text{scaleX} & text{skewX} & text{translateX} \ text{skewY} & text{scaleY} & text{translateY} \ text{skewZ} & text{translateZ} & text{scaleZ} \ 0 & 0 & 1 end{bmatrix}$$

o.rotation according to (x, y, z) rotation wiki

$$mathcal{M}(hat{mathbf{v}},theta) = begin{bmatrix} cos theta + (1 - cos theta) x^2 & (1 - cos theta) x y - (sin theta) z & (1 - cos theta) x z + (sin theta) y & 0 \ (1 - cos theta) y x + (sin theta) z & cos theta + (1 - cos theta) y^2 & (1 - cos theta) y z - (sin theta) x & 0 \ (1 - cos theta) z x - (sin theta) y & (1 - cos theta) z y + (sin theta) x & cos theta + (1 - cos theta) z^2 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}$$

Similarily, we can have eigenvalue decomposiiton to make tweening animation. One point is remained that for 3d object projected to xy-plane, we should make parameter equation for each surface.

Finally we can use projection of surface to create beautiful bullets.

Realization

Use pyqt as gui, matplotlib.pylab, mpl_toolkits.mplot3d as drawer. To simplify this problem, we only draw edges of convex polyhedron on xy-plane.

Place points at anticlockwise direction and add edges

anticlockwise sort

Calculate the mass center, $vec{OA}, vec{OB}$ as vector from mass center to point A, B. If $vec{OA} times vec{OB} < 0$, then B is anticlockwise to A. Quick sort the whole point set. Add edges AB in the edges set

Solve the convex

Accordint to anticlockwise points set and edges set, judge one edge AB and one point C. If the $CBA < 90^{circ}$ then it’s concave, error.

Example

Remain to be done

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            <div class="ds-share" style="text-align: right" data-thread-key="/2016/05/13/LuaSTG_Matrix_Transform" data-title="Matrix Transform for LuaSTG" data-url="http://airtnp.github.io/2016/05/13/LuaSTG_Matrix_Transform/" data-images="http://airtnp.github.io/img/post-matrix.jpg" data-content="Another time copy from css (first attempt is ease function)

Theory

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