Extra:Cg Math Functions

常用Cg函数

数学函数

abs(x):绝对值

// float类型的实现
float abs(float x) {
    return max(-a, a);
}

sin(x):正弦,输入为弧度

// float类型的实现
float sin(float a) {
  /* C simulation gives a max absolute error of less than 1.8e-7 */
  float4 c0 = float4( 0.0,            0.5,
                      1.0,            0.0            );
  float4 c1 = float4( 0.25,          -9.0,
                      0.75,           0.159154943091 );
  float4 c2 = float4( 24.9808039603, -24.9808039603,
                     -60.1458091736,  60.1458091736  );
  float4 c3 = float4( 85.4537887573, -85.4537887573,
                     -64.9393539429,  64.9393539429  );
  float4 c4 = float4( 19.7392082214, -19.7392082214,
                     -1.0,            1.0            );

  /* r0.x = sin(a) */
  float3 r0, r1, r2;

  r1.x  = c1.w * a - c1.x;                // only difference from cos!
  r1.y  = frac( r1.x );                   // and extract fraction
  r2.x  = (float) ( r1.y < c1.x );        // range check: 0.0 to 0.25
  r2.yz = (float2) ( r1.yy >= c1.yz );    // range check: 0.75 to 1.0
  r2.y  = dot( r2, c4.zwz );              // range check: 0.25 to 0.75
  r0    = c0.xyz - r1.yyy;                // range centering
  r0    = r0 * r0;
  r1    = c2.xyx * r0 + c2.zwz;           // start power series
  r1    =     r1 * r0 + c3.xyx;
  r1    =     r1 * r0 + c3.zwz;
  r1    =     r1 * r0 + c4.xyx;
  r1    =     r1 * r0 + c4.zwz;
  r0.x  = dot( r1, -r2 );                 // range extract

  return r0.x;
}

cos(x):正弦,输入为弧度

// float类型的实现
float cos(float a) {
  /* C simulation gives a max absolute error of less than 1.8e-7 */
  const float4 c0 = float4( 0.0,            0.5,
                            1.0,            0.0            );
  const float4 c1 = float4( 0.25,          -9.0,
                            0.75,           0.159154943091 );
  const float4 c2 = float4( 24.9808039603, -24.9808039603,
                           -60.1458091736,  60.1458091736  );
  const float4 c3 = float4( 85.4537887573, -85.4537887573,
                           -64.9393539429,  64.9393539429  );
  const float4 c4 = float4( 19.7392082214, -19.7392082214,
                           -1.0,            1.0            );

  /* r0.x = cos(a) */
  float3 r0, r1, r2;

  r1.x  = c1.w * a;                       // normalize input
  r1.y  = frac( r1.x );                   // and extract fraction
  r2.x  = (float) ( r1.y < c1.x );        // range check: 0.0 to 0.25
  r2.yz = (float2) ( r1.yy >= c1.yz );    // range check: 0.75 to 1.0
  r2.y  = dot( r2, c4.zwz );              // range check: 0.25 to 0.75
  r0    = c0.xyz - r1.yyy;                // range centering
  r0    = r0 * r0;
  r1    = c2.xyx * r0 + c2.zwz;           // start power series
  r1    =     r1 * r0 + c3.xyx;
  r1    =     r1 * r0 + c3.zwz;
  r1    =     r1 * r0 + c4.xyx;
  r1    =     r1 * r0 + c4.zwz;
  r0.x  = dot( r1, -r2 );                 // range extract

  return r0.x;
}

sincos(x,out s,out c):s=sin(x),c=cos(x)

ceil(x):向上取整

floor(x):向下取整(floor(-1.3)= -2)

round(x):四舍五入

frac(x):取x的小数部分

clamp(x,a,b):把x截取到[a,b]

saturate(x):把x截取到[0,1]

lerp(a,b,f):(1-f) * a + f * b

step(a,x):返回x>=a

smoothstep(min,max,x):x=min时返回0,x=max时返回1;否则返回下式的值
$$
-2\left(x-min\over max-min\right)^3+3\left(x-min\over max-min\right)^2
$$
pow(x,y):计算x的y次方

sqrt(x):计算x的算术平方根

noise(x):返回根据x生成的伪随机数,范围[0,1]

min(a,b):取最小

max(a,b):取最大

normalize(x):把x化为单位向量

length(x):返回向量x的模

distance(x,y):计算x,y的欧氏距离

dot(a,b):点积

cross(a,b):叉积

mul(a,b):乘法

光照函数

光照函数的输入向量都必须归一化,入射光方向均指从外指向顶点的方向

reflect(I,N):反射函数,I为入射光向量,N为反射表面的法向量,返回反射光向量

float3 reflect(float3 i, float3 n) {
  return i - 2.0 * n * dot(n, i);
}

refract(I,N,eta):折射函数,I为入射光向量,N为反射表面的法向量,eta是介质折射率,返回折射光向量

float3 refract(float3 i, float3 n, float eta) {
  float cosi = dot(-i, n);
  float cost2 = 1.0f - eta * eta * (1.0f - cosi * cosi);
  float3 t = eta * i + ((eta * cosi - sqrt(abs(cost2))) * n);
  return t * (float3)(cost2 > 0);
}

纹理采样函数

tex2D(sampler2D samp,float2 s):samp是待采样的纹理,s是纹理坐标

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