FFT与游戏开发（六）

FFT与游戏开发（六）

先上成果，简单的漫反射光照，不过已经可以体现出法线了。

法线

有了高度场，还需要法线信息，法线可以通过对高度场求梯度得到，这里默认z轴朝上。

1. 高度场(Height)
   [P(x,y,t) = (x,y,H(x,y,t)) ]

2. 副切线(BiTangent)
   [egin{aligned} B(x,y,t) &= left( frac{partial x}{partial x}, frac{partial y}{partial x}, frac{partial H(x,y,t)}{partial x} ight) \ &= left( 1, 0, frac{partial H(x,y,t)}{partial x} ight) \ end{aligned} ]

3. 切线(Tangent)
   [egin{aligned} T(x,y,t) &= left( frac{partial x}{partial y}, frac{partial y}{partial y}, frac{partial H(x,y,t)}{partial y} ight) \ &= left( 0, 1, frac{partial H(x,y,t)}{partial y} ight) \ end{aligned} ]

4. 法线(Normal)
   [egin{aligned} N(x,y,t) &= B(x,y,t) imes T(x,y,t) \ &= left( -frac{partial H(x,y,t)}{partial x}, -frac{partial H(x,y,t)}{partial y}, 1 ight) end{aligned} ]

高度场的全微分（梯度）

[egin{aligned} abla h(overrightarrow x, t) &= left(frac{partial h}{partial x}, frac{partial h}{partial y} ight) \ &= abla sum_{overrightarrow k} ilde h (overrightarrow k, t) e^{j overrightarrow k cdot overrightarrow x} \ &= sum_{overrightarrow k} ilde h (overrightarrow k, t) abla e^{j overrightarrow k cdot overrightarrow x} \ &= sum_{overrightarrow k} ilde h (overrightarrow k, t) abla e^{j(k_x x + k_z z)} \ &= sum_{overrightarrow k} ilde h (overrightarrow k, t) left( e^{j(k_x x + k_z z) }jk_x, e^{j(k_x x + k_z z) }jk_z ight) \ &= sum_{overrightarrow k} ilde h (overrightarrow k, t) j overrightarrow k e^{j overrightarrow k cdot overrightarrow x} \ end{aligned} ]

由此可以套用之前计算高度场的那套iFFT，只不过$$ ilde h $$变成了$$ ilde h j overrightarrow k $$