RenderHelp.h

//=====================================================================
//
// RenderHelp.h - 可编程渲染管线实现，渲染器教学，着色程序学习
//
// By skywind3000 (at) gmail.com, 2020/08/08
//
// Features:
//
// - 单个头文件的渲染器实现，没有任何依赖
// - 模型标准，计算精确，类 Direct3D 接口
// - 包含一套精简何理的矢量/矩阵库
// - 包含一套位图 Bitmap 库，方便画点/画线，加载纹理，保存渲染结果
// - 支持二次线性插值纹理采样器
// - 支持深度缓存
// - 支持多种数据类型的 varying
// - 支持顶点着色器 (Vertex Shader) 和像素着色器 (Pixel Shader)
// - 支持加载 24 位和 32 位的 bmp 图片纹理
//
//=====================================================================
#ifndef _RENDER_HELP_H_
#define _RENDER_HELP_H_

#include <stddef.h>
#include <stdint.h>
#include <string.h>
#include <math.h>
#include <assert.h>

#include <vector>
#include <map>
#include <initializer_list>
#include <stdexcept>
#include <functional>
#include <ostream>
#include <sstream>
#include <iostream>


//---------------------------------------------------------------------
// 数学库：矢量定义
//---------------------------------------------------------------------

// 通用矢量：N 是矢量维度，T 为数据类型
template <size_t N, typename T> struct Vector {
	T m[N];    // 元素数组
	inline Vector() { for (size_t i = 0; i < N; i++) m[i] = T(); }
	inline Vector(const T *ptr) { for (size_t i = 0; i < N; i++) m[i] = ptr[i]; }
	inline Vector(const Vector<N, T> &u) { for (size_t i = 0; i < N; i++) m[i] = u.m[i]; }
	inline Vector(const std::initializer_list<T> &u) { 
		auto it = u.begin(); for (size_t i = 0; i < N; i++) m[i] = *it++; }
	inline const T& operator[] (size_t i) const { assert(i < N); return m[i]; }
	inline T& operator[] (size_t i) { assert(i < N); return m[i]; }
	inline void load(const T *ptr) { for (size_t i = 0; i < N; i++) m[i] = ptr[i]; }
	inline void save(T *ptr) { for (size_t i = 0; i < N; i++) ptr[i] = m[i]; }
};


// 特化二维矢量
template <typename T> struct Vector<2, T> {
	union {
		struct { T x, y; };    // 元素别名
		struct { T u, v; };    // 元素别名
		T m[2];                // 元素数组
	};
	inline Vector(): x(T()), y(T()) {}
	inline Vector(T X, T Y): x(X), y(Y) {}
	inline Vector(const Vector<2, T> &u): x(u.x), y(u.y) {}
	inline Vector(const T *ptr): x(ptr[0]), y(ptr[1]) {}
	inline const T& operator[] (size_t i) const { assert(i < 2); return m[i]; }
	inline T& operator[] (size_t i) { assert(i < 2); return m[i]; }
	inline void load(const T *ptr) { for (size_t i = 0; i < 2; i++) m[i] = ptr[i]; }
	inline void save(T *ptr) { for (size_t i = 0; i < 2; i++) ptr[i] = m[i]; }
	inline Vector<2, T> xy() const { return *this; }
	inline Vector<3, T> xy1() const { return Vector<3, T>(x, y, 1); }
	inline Vector<4, T> xy11() const { return Vector<4, T>(x, y, 1, 1); }
};


// 特化三维矢量
template <typename T> struct Vector<3, T> {
	union {
		struct { T x, y, z; };    // 元素别名
		struct { T r, g, b; };    // 元素别名
		T m[3];                   // 元素数组
	};
	inline Vector(): x(T()), y(T()), z(T()) {}
	inline Vector(T X, T Y, T Z): x(X), y(Y), z(Z) {}
	inline Vector(const Vector<3, T> &u): x(u.x), y(u.y), z(u.z) {}
	inline Vector(const T *ptr): x(ptr[0]), y(ptr[1]), z(ptr[2]) {}
	inline const T& operator[] (size_t i) const { assert(i < 3); return m[i]; }
	inline T& operator[] (size_t i) { assert(i < 3); return m[i]; }
	inline void load(const T *ptr) { for (size_t i = 0; i < 3; i++) m[i] = ptr[i]; }
	inline void save(T *ptr) { for (size_t i = 0; i < 3; i++) ptr[i] = m[i]; }
	inline Vector<2, T> xy() const { return Vector<2, T>(x, y); }
	inline Vector<3, T> xyz() const { return *this; }
	inline Vector<4, T> xyz1() const { return Vector<4, T>(x, y, z, 1); }
};


// 特化四维矢量
template <typename T> struct Vector<4, T> {
	union {
		struct { T x, y, z, w; };    // 元素别名
		struct { T r, g, b, a; };    // 元素别名
		T m[4];                      // 元素数组
	};
	inline Vector(): x(T()), y(T()), z(T()), w(T()) {}
	inline Vector(T X, T Y, T Z, T W): x(X), y(Y), z(Z), w(W) {}
	inline Vector(const Vector<4, T> &u): x(u.x), y(u.y), z(u.z), w(u.w) {}
	inline Vector(const T *ptr): x(ptr[0]), y(ptr[1]), z(ptr[2]), w(ptr[3]) {}
	inline const T& operator[] (size_t i) const { assert(i < 4); return m[i]; }
	inline T& operator[] (size_t i) { assert(i < 4); return m[i]; }
	inline void load(const T *ptr) { for (size_t i = 0; i < 4; i++) m[i] = ptr[i]; }
	inline void save(T *ptr) { for (size_t i = 0; i < 4; i++) ptr[i] = m[i]; }
	inline Vector<2, T> xy() const { return Vector<2, T>(x, y); }
	inline Vector<3, T> xyz() const { return Vector<3, T>(x, y, z); }
	inline Vector<4, T> xyzw() const { return *this; }
};


//---------------------------------------------------------------------
// 数学库：矢量运算
//---------------------------------------------------------------------

// = (+a)
template <size_t N, typename T> 
inline Vector<N, T> operator + (const Vector<N, T>& a) {
	return a;
}

// = (-a)
template <size_t N, typename T> 
inline Vector<N, T> operator - (const Vector<N, T>& a) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) b[i] = -a[i];
	return b;
}

// = (a == b) ? true : false
template <size_t N, typename T>
inline bool operator == (const Vector<N, T>& a, const Vector<N, T>& b) {
	for (size_t i = 0; i < N; i++) if (a[i] != b[i]) return false;
	return true;
}

// = (a != b)? true : false
template <size_t N, typename T>
inline bool operator != (const Vector<N, T>& a, const Vector<N, T>& b) {
	return !(a == b);
}

// = a + b
template <size_t N, typename T> 
inline Vector<N, T> operator + (const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = a[i] + b[i];
	return c;
}

// = a - b
template <size_t N, typename T> 
inline Vector<N, T> operator - (const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = a[i] - b[i];
	return c;
}

// = a * b，不是点乘也不是叉乘，而是各个元素分别相乘，色彩计算时有用
template <size_t N, typename T> 
inline Vector<N, T> operator * (const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = a[i] * b[i];
	return c;
}

// = a / b，各个元素相除
template <size_t N, typename T> 
inline Vector<N, T> operator / (const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = a[i] / b[i];
	return c;
}

// = a * x
template <size_t N, typename T> 
inline Vector<N, T> operator * (const Vector<N, T>& a, T x) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) b[i] = a[i] * x;
	return b;
}

// = x * a
template <size_t N, typename T> 
inline Vector<N, T> operator * (T x, const Vector<N, T>& a) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) b[i] = a[i] * x;
	return b;
}

// = a / x
template <size_t N, typename T> 
inline Vector<N, T> operator / (const Vector<N, T>& a, T x) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) b[i] = a[i] / x;
	return b;
}

// = x / a
template <size_t N, typename T> 
inline Vector<N, T> operator / (T x, const Vector<N, T>& a) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) b[i] = x / a[i];
	return b;
}

// a += b
template <size_t N, typename T>
inline Vector<N, T>& operator += (Vector<N, T>& a, const Vector<N, T>& b) {
	for (size_t i = 0; i < N; i++) a[i] += b[i];
	return a;
}

// a -= b
template <size_t N, typename T>
inline Vector<N, T>& operator -= (Vector<N, T>& a, const Vector<N, T>& b) {
	for (size_t i = 0; i < N; i++) a[i] -= b[i];
	return a;
}

// a *= b
template <size_t N, typename T>
inline Vector<N, T>& operator *= (Vector<N, T>& a, const Vector<N, T>& b) {
	for (size_t i = 0; i < N; i++) a[i] *= b[i];
	return a;
}

// a /= b
template <size_t N, typename T>
inline Vector<N, T>& operator /= (Vector<N, T>& a, const Vector<N, T>& b) {
	for (size_t i = 0; i < N; i++) a[i] /= b[i];
	return a;
}

// a *= x
template <size_t N, typename T>
inline Vector<N, T>& operator *= (Vector<N, T>& a, T x) {
	for (size_t i = 0; i < N; i++) a[i] *= x;
	return a;
}

// a /= x
template <size_t N, typename T>
inline Vector<N, T>& operator /= (Vector<N, T>& a, T x) {
	for (size_t i = 0; i < N; i++) a[i] /= x;
	return a;
}


//---------------------------------------------------------------------
// 数学库：矢量函数
//---------------------------------------------------------------------

// 不同维度的矢量转换
template<size_t N1, size_t N2, typename T> 
inline Vector<N1, T> vector_convert(const Vector<N2, T>& a, T fill = 1) {
	Vector<N1, T> b;
	for (size_t i = 0; i < N1; i++) 
		b[i] = (i < N2)? a[i] : fill;
	return b;
}

// = |a| ^ 2
template<size_t N, typename T>
inline T vector_length_square(const Vector<N, T>& a) {
	T sum = 0;
	for (size_t i = 0; i < N; i++) sum += a[i] * a[i];
	return sum;
}

// = |a|
template<size_t N, typename T>
inline T vector_length(const Vector<N, T>& a) {
	return sqrt(vector_length_square(a));
}

// = |a| , 特化 float 类型，使用 sqrtf
template<size_t N>
inline float vector_length(const Vector<N, float>& a) {
	return sqrtf(vector_length_square(a));
}

// = a / |a|
template<size_t N, typename T>
inline Vector<N, T> vector_normalize(const Vector<N, T>& a) {
	return a / vector_length(a);
}

// 矢量点乘
template<size_t N, typename T>
inline T vector_dot(const Vector<N, T>& a, const Vector<N, T>& b) {
	T sum = 0;
	for (size_t i = 0; i < N; i++) sum += a[i] * b[i];
	return sum;
}

// 二维矢量叉乘，得到标量
template<typename T>
inline T vector_cross(const Vector<2, T>& a, const Vector<2, T>& b) {
	return a.x * b.y - a.y * b.x;
}

// 三维矢量叉乘，得到新矢量
template<typename T>
inline Vector<3, T> vector_cross(const Vector<3, T>& a, const Vector<3, T>& b) {
	return Vector<3, T>(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x);
}

// 四维矢量叉乘：前三维叉乘，后一位保留
template<typename T>
inline Vector<4, T> vector_cross(const Vector<4, T>& a, const Vector<4, T>& b) {
	return Vector<4, T>(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x, a.w);
}

// = a + (b - a) * t
template<size_t N, typename T>
inline Vector<N, T> vector_lerp(const Vector<N, T>& a, const Vector<N, T>& b, float t) {
	return a + (b - a) * t;
}

// 各个元素取最大值
template<size_t N, typename T>
inline Vector<N, T> vector_max(const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = (a[i] > b[i])? a[i] : b[i];
	return c;
}

// 各个元素取最小值
template<size_t N, typename T>
inline Vector<N, T> vector_min(const Vector<N, T>& a, const Vector<N, T>& b) {
	Vector<N, T> c;
	for (size_t i = 0; i < N; i++) c[i] = (a[i] < b[i])? a[i] : b[i];
	return c;
}

// 将矢量的值控制在 minx/maxx 范围内
template<size_t N, typename T>
inline Vector<N, T> vector_between(const Vector<N, T>& minx, const Vector<N, T>& maxx, const Vector<N, T>& x) {
	return vector_min(vector_max(minx, x), maxx);
}

// 判断矢量是否接近
template<size_t N, typename T>
inline bool vector_near(const Vector<N, T>& a, const Vector<N, T>& b, T dist) {
	return (vector_length_square(a - b) <= dist);
}

// 判断两个单精度矢量是否近似
template<size_t N>
inline bool vector_near_equal(const Vector<N, float>& a, const Vector<N, float>& b, float e = 0.0001) {
	return vector_near(a, b, e);
}

// 判断两个双精度矢量是否近似
template<size_t N>
inline bool vector_near_equal(const Vector<N, double>& a, const Vector<N, double>& b, double e = 0.0000001) {
	return vector_near(a, b, e);
}

// 矢量值元素范围裁剪
template<size_t N, typename T>
inline Vector<N, T> vector_clamp(const Vector<N, T>& a, T minx = 0, T maxx = 1) {
	Vector<N, T> b;
	for (size_t i = 0; i < N; i++) {
		T x = (a[i] < minx)? minx : a[i];
		b[i] = (x > maxx)? maxx : x;	
	}
	return b;
}

// 输出到文本流
template<size_t N, typename T>
inline std::ostream& operator << (std::ostream& os, const Vector<N, T>& a) {
	os << "[";
	for (size_t i = 0; i < N; i++) {
		os << a[i];
		if (i < N - 1) os << ", ";
	}
	os << "]";
	return os;
}

// 输出成字符串
template<size_t N, typename T>
inline std::string vector_repr(const Vector<N, T>& a) {
	std::stringstream ss;
	ss << a;
	return ss.str();
}


//---------------------------------------------------------------------
// 数学库：矩阵
//---------------------------------------------------------------------
template<size_t ROW, size_t COL, typename T> struct Matrix {
	T m[ROW][COL];

	inline Matrix() {}

	inline Matrix(const Matrix<ROW, COL, T>& src) {
		for (size_t r = 0; r < ROW; r++) {
			for (size_t c = 0; c < COL; c++) 
				m[r][c] = src.m[r][c];
		}
	}

	inline Matrix(const std::initializer_list<Vector<COL, T>> &u) { 
		auto it = u.begin(); 
		for (size_t i = 0; i < ROW; i++) SetRow(i, *it++);
	}

	inline const T* operator [] (size_t row) const { assert(row < ROW); return m[row]; }
	inline T* operator [] (size_t row) { assert(row < ROW); return m[row]; }
	
	// 取一行
	inline Vector<COL, T> Row(size_t row) const {
		assert(row < ROW);
		Vector<COL, T> a;
		for (size_t i = 0; i < COL; i++) a[i] = m[row][i];
		return a;
	}

	// 取一列
	inline Vector<ROW, T> Col(size_t col) const {
		assert(col < COL);
		Vector<ROW, T> a;
		for (size_t i = 0; i < ROW; i++) a[i] = m[i][col];
		return a;
	}

	// 设置一行
	inline void SetRow(size_t row, const Vector<COL, T>& a) {
		assert(row < ROW);
		for (size_t i = 0; i < COL; i++) m[row][i] = a[i];
	}

	// 设置一列
	inline void SetCol(size_t col, const Vector<ROW, T>& a) {
		assert(col < COL);
		for (size_t i = 0; i < ROW; i++) m[i][col] = a[i];
	}

	// 取得删除某行和某列的子矩阵：子式
	inline Matrix<ROW-1, COL-1, T> GetMinor(size_t row, size_t col) const {
		Matrix<ROW-1, COL-1, T> ret;
		for (size_t r = 0; r < ROW - 1; r++) {
			for (size_t c = 0; c < COL - 1; c++) {
				ret.m[r][c] = m[r < row? r : r + 1][c < col? c : c + 1];
			}
		}
		return ret;
	}

	// 取得转置矩阵
	inline Matrix<COL, ROW, T> Transpose() const {
		Matrix<COL, ROW, T> ret;
		for (size_t r = 0; r < ROW; r++) {
			for (size_t c = 0; c < COL; c++)
				ret.m[c][r] = m[r][c];
		}
		return ret;
	}

	// 取得 0 矩阵
	inline static Matrix<ROW, COL, T> GetZero() {
		Matrix<ROW, COL, T> ret;
		for (size_t r = 0; r < ROW; r++) {
			for (size_t c = 0; c < COL; c++) 
				ret.m[r][c] = 0;
		}
		return ret;
	}

	// 取得单位矩阵
	inline static Matrix<ROW, COL, T> GetIdentity() {
		Matrix<ROW, COL, T> ret;
		for (size_t r = 0; r < ROW; r++) {
			for (size_t c = 0; c < COL; c++) 
				ret.m[r][c] = (r == c)? 1 : 0;
		}
		return ret;
	}
};



//---------------------------------------------------------------------
// 数学库：矩阵运算
//---------------------------------------------------------------------
template<size_t ROW, size_t COL, typename T>
inline bool operator == (const Matrix<ROW, COL, T>& a, const Matrix<ROW, COL, T>& b) {
	for (size_t r = 0; r < ROW; r++) {
		for (size_t c = 0; c < COL; c++) {
			if (a.m[r][c] != b.m[r][c]) return false;
		}
	}
	return true;
}

template<size_t ROW, size_t COL, typename T>
inline bool operator != (const Matrix<ROW, COL, T>& a, const Matrix<ROW, COL, T>& b) {
	return !(a == b);
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator + (const Matrix<ROW, COL, T>& src) {
	return src;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator - (const Matrix<ROW, COL, T>& src) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) 
			out.m[j][i] = -src.m[j][i];
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator + (const Matrix<ROW, COL, T>& a, const Matrix<ROW, COL, T>& b) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) 
			out.m[j][i] = a.m[j][i] + b.m[j][i];
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator - (const Matrix<ROW, COL, T>& a, const Matrix<ROW, COL, T>& b) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) 
			out.m[j][i] = a.m[j][i] - b.m[j][i];
	}
	return out;
}

template<size_t ROW, size_t COL, size_t NEWCOL, typename T>
inline Matrix<ROW, NEWCOL, T> operator * (const Matrix<ROW, COL, T>& a, const Matrix<COL, NEWCOL, T>& b) {
	Matrix<ROW, NEWCOL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < NEWCOL; i++) {
			out.m[j][i] = vector_dot(a.Row(j), b.Col(i));
		}
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator * (const Matrix<ROW, COL, T>& a, T x) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) {
			out.m[j][i] = a.m[j][i] * x;
		}
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator / (const Matrix<ROW, COL, T>& a, T x) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) {
			out.m[j][i] = a.m[j][i] / x;
		}
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator * (T x, const Matrix<ROW, COL, T>& a) {
	return (a * x);
}

template<size_t ROW, size_t COL, typename T>
inline Matrix<ROW, COL, T> operator / (T x, const Matrix<ROW, COL, T>& a) {
	Matrix<ROW, COL, T> out;
	for (size_t j = 0; j < ROW; j++) {
		for (size_t i = 0; i < COL; i++) {
			out.m[j][i] = x / a.m[j][i];
		}
	}
	return out;
}

template<size_t ROW, size_t COL, typename T>
inline Vector<COL, T> operator * (const Vector<ROW, T>& a, const Matrix<ROW, COL, T>& m) {
	Vector<COL, T> b;
	for (size_t i = 0; i < COL; i++) 
		b[i] = vector_dot(a, m.Col(i));
	return b;
}

template<size_t ROW, size_t COL, typename T>
inline Vector<ROW, T> operator * (const Matrix<ROW, COL, T>& m, const Vector<COL, T>& a) {
	Vector<ROW, T> b;
	for (size_t i = 0; i < ROW; i++) 
		b[i] = vector_dot(a, m.Row(i));
	return b;
}


//---------------------------------------------------------------------
// 数学库：行列式和逆矩阵等，光照计算有用
//---------------------------------------------------------------------

// 行列式求值：一阶
template<typename T>
inline T matrix_det(const Matrix<1, 1, T> &m) {
	return m[0][0];
}

// 行列式求值：二阶
template<typename T>
inline T matrix_det(const Matrix<2, 2, T> &m) {
	return m[0][0] * m[1][1] - m[0][1] * m[1][0];
}

// 行列式求值：多阶行列式，即第一行同他们的余子式相乘求和
template<size_t N, typename T>
inline T matrix_det(const Matrix<N, N, T> &m) {
	T sum = 0;
	for (size_t i = 0; i < N; i++) sum += m[0][i] * matrix_cofactor(m, 0, i);
	return sum;
}

// 余子式：一阶
template<typename T>
inline T matrix_cofactor(const Matrix<1, 1, T> &m, size_t row, size_t col) {
	return 0;
}

// 多阶余子式：即删除特定行列的子式的行列式值
template<size_t N, typename T>
inline T matrix_cofactor(const Matrix<N, N, T> &m, size_t row, size_t col) {
	return matrix_det(m.GetMinor(row, col)) * (((row + col) % 2)? -1 : 1);
}

// 伴随矩阵：即余子式矩阵的转置
template<size_t N, typename T>
inline Matrix<N, N, T> matrix_adjoint(const Matrix<N, N, T> &m) {
	Matrix<N, N, T> ret;
	for (size_t j = 0; j < N; j++) {
		for (size_t i = 0; i < N; i++) ret[j][i] = matrix_cofactor(m, i, j);
	}
	return ret;
}

// 求逆矩阵：使用伴随矩阵除以行列式的值得到
template<size_t N, typename T>
inline Matrix<N, N, T> matrix_invert(const Matrix<N, N, T> &m) {
	Matrix<N, N, T> ret = matrix_adjoint(m);
	T det = vector_dot(m.Row(0), ret.Col(0));
	return ret / det;
}

// 输出到文本流
template<size_t ROW, size_t COL, typename T>
inline std::ostream& operator << (std::ostream& os, const Matrix<ROW, COL, T>& m) {
	for (size_t r = 0; r < ROW; r++) {
		Vector<COL, T> row = m.Row(r);
		os << row << std::endl;
	}
	return os;
}


//---------------------------------------------------------------------
// 工具函数
//---------------------------------------------------------------------
template<typename T> inline T Abs(T x) { return (x < 0)? (-x) : x; }
template<typename T> inline T Max(T x, T y) { return (x < y)? y : x; }
template<typename T> inline T Min(T x, T y) { return (x > y)? y : x; }

template<typename T> inline bool NearEqual(T x, T y, T error) { 
	return (Abs(x - y) < error); 
}

template<typename T> inline T Between(T xmin, T xmax, T x) { 
	return Min(Max(xmin, x), xmax); 
}

// 截取 [0, 1] 的范围
template<typename T> inline T Saturate(T x) {
	return Between<T>(0, 1, x);
}

// 类型别名
typedef Vector<2, float>  Vec2f;
typedef Vector<2, double> Vec2d;
typedef Vector<2, int>    Vec2i;
typedef Vector<3, float>  Vec3f;
typedef Vector<3, double> Vec3d;
typedef Vector<3, int>    Vec3i;
typedef Vector<4, float>  Vec4f;
typedef Vector<4, double> Vec4d;
typedef Vector<4, int>    Vec4i;

typedef Matrix<4, 4, float> Mat4x4f;
typedef Matrix<3, 3, float> Mat3x3f;
typedef Matrix<4, 3, float> Mat4x3f;
typedef Matrix<3, 4, float> Mat3x4f;


//---------------------------------------------------------------------
// 3D 数学运算
//---------------------------------------------------------------------

// 矢量转整数颜色
inline static uint32_t vector_to_color(const Vec4f& color) {
	uint32_t r = (uint32_t)Between(0, 255, (int)(color.r * 255.0f));
	uint32_t g = (uint32_t)Between(0, 255, (int)(color.g * 255.0f));
	uint32_t b = (uint32_t)Between(0, 255, (int)(color.b * 255.0f));
	uint32_t a = (uint32_t)Between(0, 255, (int)(color.a * 255.0f));
	return (r << 16) | (g << 8) | b | (a << 24);
}

// 矢量转换整数颜色
inline static uint32_t vector_to_color(const Vec3f& color) {
	return vector_to_color(color.xyz1());
}

// 整数颜色到矢量
inline static Vec4f vector_from_color(uint32_t rgba) {
	Vec4f out;
	out.r = ((rgba >> 16) & 0xff) / 255.0f;
	out.g = ((rgba >>  8) & 0xff) / 255.0f;
	out.b = ((rgba >>  0) & 0xff) / 255.0f;
	out.a = ((rgba >> 24) & 0xff) / 255.0f;
	return out;
}

// matrix set to zero
inline static Mat4x4f matrix_set_zero() {
	Mat4x4f m;
	m.m[0][0] = m.m[0][1] = m.m[0][2] = m.m[0][3] = 0.0f;
	m.m[1][0] = m.m[1][1] = m.m[1][2] = m.m[1][3] = 0.0f;
	m.m[2][0] = m.m[2][1] = m.m[2][2] = m.m[2][3] = 0.0f;
	m.m[3][0] = m.m[3][1] = m.m[3][2] = m.m[3][3] = 0.0f;
	return m;
}

// set to identity
inline static Mat4x4f matrix_set_identity() {
	Mat4x4f m;
	m.m[0][0] = m.m[1][1] = m.m[2][2] = m.m[3][3] = 1.0f; 
	m.m[0][1] = m.m[0][2] = m.m[0][3] = 0.0f;
	m.m[1][0] = m.m[1][2] = m.m[1][3] = 0.0f;
	m.m[2][0] = m.m[2][1] = m.m[2][3] = 0.0f;
	m.m[3][0] = m.m[3][1] = m.m[3][2] = 0.0f;
	return m;
}

// 平移变换
inline static Mat4x4f matrix_set_translate(float x, float y, float z) {
	Mat4x4f m = matrix_set_identity();
	m.m[3][0] = x;
	m.m[3][1] = y;
	m.m[3][2] = z;
	return m;
}

// 缩放变换
inline static Mat4x4f matrix_set_scale(float x, float y, float z) {
	Mat4x4f m = matrix_set_identity();
	m.m[0][0] = x;
	m.m[1][1] = y;
	m.m[2][2] = z;
	return m;
}

// 旋转变换，围绕 (x, y, z) 矢量旋转 theta 角度
inline static Mat4x4f matrix_set_rotate(float x, float y, float z, float theta) {
	float qsin = (float)sin(theta * 0.5f);
	float qcos = (float)cos(theta * 0.5f);
	float w = qcos;
	Vec3f vec = vector_normalize(Vec3f(x, y, z));
	x = vec.x * qsin;
	y = vec.y * qsin;
	z = vec.z * qsin;
	Mat4x4f m;
	m.m[0][0] = 1 - 2 * y * y - 2 * z * z;
	m.m[1][0] = 2 * x * y - 2 * w * z;
	m.m[2][0] = 2 * x * z + 2 * w * y;
	m.m[0][1] = 2 * x * y + 2 * w * z;
	m.m[1][1] = 1 - 2 * x * x - 2 * z * z;
	m.m[2][1] = 2 * y * z - 2 * w * x;
	m.m[0][2] = 2 * x * z - 2 * w * y;
	m.m[1][2] = 2 * y * z + 2 * w * x;
	m.m[2][2] = 1 - 2 * x * x - 2 * y * y;
	m.m[0][3] = m.m[1][3] = m.m[2][3] = 0.0f;
	m.m[3][0] = m.m[3][1] = m.m[3][2] = 0.0f;	
	m.m[3][3] = 1.0f;
	return m;
}

// 摄影机变换矩阵：eye/视点位置，at/看向哪里，up/指向上方的矢量
inline static Mat4x4f matrix_set_lookat(const Vec3f& eye, const Vec3f& at, const Vec3f& up) {
	Vec3f zaxis = vector_normalize(at - eye);
	Vec3f xaxis = vector_normalize(vector_cross(up, zaxis));
	Vec3f yaxis = vector_cross(zaxis, xaxis);
	Mat4x4f m;
	m.SetCol(0, Vec4f(xaxis.x, xaxis.y, xaxis.z, -vector_dot(eye, xaxis)));
	m.SetCol(1, Vec4f(yaxis.x, yaxis.y, yaxis.z, -vector_dot(eye, yaxis)));
	m.SetCol(2, Vec4f(zaxis.x, zaxis.y, zaxis.z, -vector_dot(eye, zaxis)));
	m.SetCol(3, Vec4f(0.0f, 0.0f, 0.0f, 1.0f));
	return m;
}


// D3DXMatrixPerspectiveFovLH
inline static Mat4x4f matrix_set_perspective(float fovy, float aspect, float zn, float zf) {
	float fax = 1.0f / (float)tan(fovy * 0.5f);
	Mat4x4f m = matrix_set_zero();
	m.m[0][0] = (float)(fax / aspect);
	m.m[1][1] = (float)(fax);
	m.m[2][2] = zf / (zf - zn);
	m.m[3][2] = - zn * zf / (zf - zn);
	m.m[2][3] = 1;
	return m;
}


//---------------------------------------------------------------------
// 位图库：用于加载/保存图片，画点，画线，颜色读取
//---------------------------------------------------------------------
class Bitmap
{
public:
	inline virtual ~Bitmap() { if (_bits) delete []_bits; _bits = NULL; }
	inline Bitmap(int width, int height): _w(width), _h(height) {
		_pitch = width * 4;
		_bits = new uint8_t[_pitch * _h];
		Fill(0);
	}

	inline Bitmap(const Bitmap& src): _w(src._w), _h(src._h), _pitch(src._pitch) {
		_bits = new uint8_t[_pitch * _h];
		memcpy(_bits, src._bits, _pitch * _h);
	}

	inline Bitmap(const char *filename) {
		Bitmap *tmp = LoadFile(filename);
		if (tmp == NULL) {
			std::string msg = "load failed: ";
			msg.append(filename);
			throw std::runtime_error(msg);
		}
		_w = tmp->_w; _h = tmp->_h; _pitch = tmp->_pitch; _bits = tmp->_bits;
		tmp->_bits = NULL;
		delete tmp;
	}

public:
	inline int GetW() const { return _w; }
	inline int GetH() const { return _h; }
	inline int GetPitch() const { return _pitch; }
	inline uint8_t *GetBits() { return _bits; }
	inline const uint8_t *GetBits() const { return _bits; }
	inline uint8_t *GetLine(int y) { return _bits + _pitch * y; }
	inline const uint8_t *GetLine(int y) const { return _bits + _pitch * y; }

public:

	inline void Fill(uint32_t color) {
		for (int j = 0; j < _h; j++) {
			uint32_t *row = (uint32_t*)(_bits + j * _pitch);
			for (int i = 0; i < _w; i++, row++) 
				memcpy(row, &color, sizeof(uint32_t));
		}
	}

	inline void SetPixel(int x, int y, uint32_t color) {
		if (x >= 0 && x < _w && y >= 0 && y < _h) {
			memcpy(_bits + y * _pitch + x * 4, &color, sizeof(uint32_t));
		}
	}

	inline uint32_t GetPixel(int x, int y) const {
		uint32_t color = 0;
		if (x >= 0 && x < _w && y >= 0 && y < _h) {
			memcpy(&color, _bits + y * _pitch + x * 4, sizeof(uint32_t));
		}
		return color;
	}

	inline void DrawLine(int x1, int y1, int x2, int y2, uint32_t color) {
		int x, y;
		if (x1 == x2 && y1 == y2) {
			SetPixel(x1, y1, color);
			return;
		}	else if (x1 == x2) {
			int inc = (y1 <= y2)? 1 : -1;
			for (y = y1; y != y2; y += inc) SetPixel(x1, y, color);
			SetPixel(x2, y2, color);
		}	else if (y1 == y2) {
			int inc = (x1 <= x2)? 1 : -1;
			for (x = x1; x != x2; x += inc) SetPixel(x, y1, color);
			SetPixel(x2, y2, color);
		}	else {
			int dx = (x1 < x2)? x2 - x1 : x1 - x2;
			int dy = (y1 < y2)? y2 - y1 : y1 - y2;
			int rem = 0;
			if (dx >= dy) {
				if (x2 < x1) x = x1, y = y1, x1 = x2, y1 = y2, x2 = x, y2 = y;
				for (x = x1, y = y1; x <= x2; x++) {
					SetPixel(x, y, color);
					rem += dy;
					if (rem >= dx) { rem -= dx; y += (y2 >= y1)? 1 : -1; SetPixel(x, y, color); }
				}
				SetPixel(x2, y2, color);
			}	else {
				if (y2 < y1) x = x1, y = y1, x1 = x2, y1 = y2, x2 = x, y2 = y;
				for (x = x1, y = y1; y <= y2; y++) {
					SetPixel(x, y, color);
					rem += dx;
					if (rem >= dy) { rem -= dy; x += (x2 >= x1)? 1 : -1; SetPixel(x, y, color); }
				}
				SetPixel(x2, y2, color);
			}
		}
	}

	struct BITMAPINFOHEADER { // bmih  
		uint32_t	biSize; 
		uint32_t	biWidth; 
		int32_t		biHeight; 
		uint16_t	biPlanes; 
		uint16_t	biBitCount;
		uint32_t	biCompression; 
		uint32_t	biSizeImage; 
		uint32_t	biXPelsPerMeter; 
		uint32_t	biYPelsPerMeter; 
		uint32_t	biClrUsed; 
		uint32_t	biClrImportant; 
	};

	// 读取 BMP 图片，支持 24/32 位两种格式
	inline static Bitmap* LoadFile(const char *filename) {
		FILE *fp = fopen(filename, "rb");
		if (fp == NULL) return NULL;
		BITMAPINFOHEADER info;
		uint8_t header[14];
		int hr = (int)fread(header, 1, 14, fp);
		if (hr != 14) { fclose(fp); return NULL; }
		if (header[0] != 0x42 || header[1] != 0x4d) { fclose(fp); return NULL; }
		hr = (int)fread(&info, 1, sizeof(info), fp);
		if (hr != 40) { fclose(fp); return NULL; }
		if (info.biBitCount != 24 && info.biBitCount != 32) { fclose(fp); return NULL; }
		Bitmap *bmp = new Bitmap(info.biWidth, info.biHeight);
		uint32_t offset;
		memcpy(&offset, header + 10, sizeof(uint32_t));
		fseek(fp, offset, SEEK_SET);
		uint32_t pixelsize = (info.biBitCount + 7) / 8;
		uint32_t pitch = (pixelsize * info.biWidth + 3) & (~3);
		for (int y = 0; y < (int)info.biHeight; y++) {
			uint8_t *line = bmp->GetLine(info.biHeight - 1 - y);
			for (int x = 0; x < (int)info.biWidth; x++, line += 4) {
				line[3] = 255;
				fread(line, pixelsize, 1, fp);
			}
			fseek(fp, pitch - info.biWidth * pixelsize, SEEK_CUR);
		}
		fclose(fp);
		return bmp;
	}

	// 保存 BMP 图片
	inline bool SaveFile(const char *filename, bool withAlpha = false) const {
		FILE *fp = fopen(filename, "wb");
		if (fp == NULL) return false;
		BITMAPINFOHEADER info;
		uint32_t pixelsize = (withAlpha)? 4 : 3;
		uint32_t pitch = (GetW() * pixelsize + 3) & (~3);
		info.biSizeImage = pitch * GetH();
		uint32_t bfSize = 54 + info.biSizeImage;
		uint32_t zero = 0, offset = 54;
		fputc(0x42, fp); 
		fputc(0x4d, fp);
		fwrite(&bfSize, 4, 1, fp);
		fwrite(&zero, 4, 1, fp);
		fwrite(&offset, 4, 1, fp);
		info.biSize = 40;
		info.biWidth = GetW();
		info.biHeight = GetH();
		info.biPlanes = 1;
		info.biBitCount = (withAlpha)? 32 : 24;
		info.biCompression = 0;
		info.biXPelsPerMeter = 0xb12;
		info.biYPelsPerMeter = 0xb12;
		info.biClrUsed = 0;
		info.biClrImportant = 0;
		fwrite(&info, sizeof(info), 1, fp);
		// printf("pitch=%d %d\n", (int)pitch, info.biSizeImage);
		for (int y = 0; y < GetH(); y++) {
			const uint8_t *line = GetLine(info.biHeight - 1 - y);
			uint32_t padding = pitch - GetW() * pixelsize;
			for (int x = 0; x < GetW(); x++, line += 4) {
				fwrite(line, pixelsize, 1, fp);
			}
			for (int i = 0; i < (int)padding; i++) fputc(0, fp);
		}
		fclose(fp);
		return true;
	}

	// 双线性插值
	inline uint32_t SampleBilinear(float x, float y) const {
		int32_t fx = (int32_t)(x * 0x10000);
		int32_t fy = (int32_t)(y * 0x10000);
		int32_t x1 = Between(0, _w - 1, fx >> 16);
		int32_t y1 = Between(0, _h - 1, fy >> 16);
		int32_t x2 = Between(0, _w - 1, x1 + 1);
		int32_t y2 = Between(0, _h - 1, y1 + 1);
		int32_t dx = (fx >> 8) & 0xff;
		int32_t dy = (fy >> 8) & 0xff;
		if (_w <= 0 || _h <= 0) return 0;
		uint32_t c00 = GetPixel(x1, y1);
		uint32_t c01 = GetPixel(x2, y1);
		uint32_t c10 = GetPixel(x1, y2);
		uint32_t c11 = GetPixel(x2, y2);
		return BilinearInterp(c00, c01, c10, c11, dx, dy);
	}

	// 纹理采样
	inline Vec4f Sample2D(float u, float v) const {
		uint32_t rgba = SampleBilinear(u * _w + 0.5f, v * _h + 0.5f);
		return vector_from_color(rgba);
	}

	// 纹理采样：直接传入 Vec2f
	inline Vec4f Sample2D(const Vec2f& uv) const {
		return Sample2D(uv.x, uv.y);
	}

	// 按照 Vec4f 画点
	inline void SetPixel(int x, int y, const Vec4f& color) {
		SetPixel(x, y, vector_to_color(color));
	}

	// 上下反转
	inline void FlipVertical() {
		uint8_t *buffer = new uint8_t[_pitch];
		for (int i = 0, j = _h - 1; i < j; i++, j--) {
			memcpy(buffer, GetLine(i), _pitch);
			memcpy(GetLine(i), GetLine(j), _pitch);
			memcpy(GetLine(j), buffer, _pitch);
		}
		delete []buffer;
	}

	// 水平反转
	inline void FlipHorizontal() {
		for (int y = 0; y < _h; y++) {
			for (int i = 0, j = _w - 1; i < j; i++, j--) {
				uint32_t c1 = GetPixel(i, y);
				uint32_t c2 = GetPixel(j, y);
				SetPixel(i, y, c2);
				SetPixel(j, y, c1);
			}
		}
	}

protected:

	// 双线性插值计算：给出四个点的颜色，以及坐标偏移，计算结果
	inline static uint32_t BilinearInterp(uint32_t tl, uint32_t tr, 
		uint32_t bl, uint32_t br, int32_t distx, int32_t disty) {
		uint32_t f, r;
		int32_t distxy = distx * disty;
		int32_t distxiy = (distx << 8) - distxy;  /* distx * (256 - disty) */
		int32_t distixy = (disty << 8) - distxy;  /* disty * (256 - distx) */
		int32_t distixiy = 256 * 256 - (disty << 8) - (distx << 8) + distxy;
		r = (tl & 0x000000ff) * distixiy + (tr & 0x000000ff) * distxiy
		  + (bl & 0x000000ff) * distixy  + (br & 0x000000ff) * distxy;
		f = (tl & 0x0000ff00) * distixiy + (tr & 0x0000ff00) * distxiy
		  + (bl & 0x0000ff00) * distixy  + (br & 0x0000ff00) * distxy;
		r |= f & 0xff000000;
		tl >>= 16; tr >>= 16; bl >>= 16; br >>= 16; r >>= 16;
		f = (tl & 0x000000ff) * distixiy + (tr & 0x000000ff) * distxiy
		  + (bl & 0x000000ff) * distixy  + (br & 0x000000ff) * distxy;
		r |= f & 0x00ff0000;
		f = (tl & 0x0000ff00) * distixiy + (tr & 0x0000ff00) * distxiy
		  + (bl & 0x0000ff00) * distixy  + (br & 0x0000ff00) * distxy;
		r |= f & 0xff000000;
		return r;
	}

protected:
	int32_t _w;
	int32_t _h;
	int32_t _pitch;
	uint8_t *_bits;
};


//---------------------------------------------------------------------
// 着色器定义
//---------------------------------------------------------------------

// 着色器上下文，由 VS 设置，再由渲染器按像素逐点插值后，供 PS 读取
struct ShaderContext {
	std::map<int, float> varying_float;    // 浮点数 varying 列表
	std::map<int, Vec2f> varying_vec2f;    // 二维矢量 varying 列表
	std::map<int, Vec3f> varying_vec3f;    // 三维矢量 varying 列表
	std::map<int, Vec4f> varying_vec4f;    // 四维矢量 varying 列表
};


// 顶点着色器：因为是 C++ 编写，无需传递 attribute，传个 0-2 的顶点序号
// 着色器函数直接在外层根据序号读取相应数据即可，最后需要返回一个坐标 pos
// 各项 varying 设置到 output 里，由渲染器插值后传递给 PS 
typedef std::function<Vec4f(int index, ShaderContext &output)> VertexShader;


// 像素着色器：输入 ShaderContext，需要返回 Vec4f 类型的颜色
// 三角形内每个点的 input 具体值会根据前面三个顶点的 output 插值得到
typedef std::function<Vec4f(ShaderContext &input)> PixelShader;


//---------------------------------------------------------------------
// RenderHelp
//---------------------------------------------------------------------
class RenderHelp
{
public:

	inline virtual ~RenderHelp() { Reset(); }

	inline RenderHelp() {
		_frame_buffer = NULL;
		_depth_buffer = NULL;
		_render_frame = false;
		_render_pixel = true;
	}

	inline RenderHelp(int width, int height) {
		_frame_buffer = NULL;
		_depth_buffer = NULL;
		_render_frame = false;
		_render_pixel = true;
		Init(width, height);
	}

public:

	// 复位状态
	inline void Reset() {
		_vertex_shader = NULL;
		_pixel_shader = NULL;
		if (_frame_buffer) delete _frame_buffer;
		_frame_buffer = NULL;
		if (_depth_buffer) {
			for (int j = 0; j < _fb_height; j++) {
				if (_depth_buffer[j]) delete []_depth_buffer[j];
				_depth_buffer[j] = NULL;
			}
			delete []_depth_buffer;
			_depth_buffer= NULL;
		}
		_color_fg = 0xffffffff;
		_color_bg = 0xff191970;
	}

	// 初始化 FrameBuffer，渲染前需要先调用
	inline void Init(int width, int height) {
		Reset();
		_frame_buffer = new Bitmap(width, height);
		_fb_width = width;
		_fb_height = height;
		_depth_buffer = new float*[height];
		for (int j = 0; j < height; j++) {
			_depth_buffer[j] = new float[width];
		}
		Clear();
	}

	// 清空 FrameBuffer 和深度缓存
	inline void Clear() {
		if (_frame_buffer) {
			_frame_buffer->Fill(_color_bg);
		}
		if (_depth_buffer) {
			for (int j = 0; j < _fb_height; j++) {
				for (int i = 0; i < _fb_width; i++) 
					_depth_buffer[j][i] = 0.0f;
			}
		}
	}

	// 设置 VS/PS 着色器函数
	inline void SetVertexShader(VertexShader vs) { _vertex_shader = vs; }
	inline void SetPixelShader(PixelShader ps) { _pixel_shader = ps; }

	// 保存 FrameBuffer 到 BMP 文件
	inline void SaveFile(const char *filename) { if (_frame_buffer) _frame_buffer->SaveFile(filename); }

	// 设置背景/前景色
	inline void SetBGColor(uint32_t color) { _color_bg = color; }
	inline void SetFGColor(uint32_t color) { _color_fg = color; }

	// FrameBuffer 里画点
	inline void SetPixel(int x, int y, uint32_t cc) { if (_frame_buffer) _frame_buffer->SetPixel(x, y, cc); }
	inline void SetPixel(int x, int y, const Vec4f& cc) { SetPixel(x, y, vector_to_color(cc)); }
	inline void SetPixel(int x, int y, const Vec3f& cc) { SetPixel(x, y, vector_to_color(cc)); }

	// FrameBuffer 里画线
	inline void DrawLine(int x1, int y1, int x2, int y2) {
		if (_frame_buffer) _frame_buffer->DrawLine(x1, y1, x2, y2, _color_fg);
	}

	// 设置渲染状态，是否显示线框图，是否填充三角形
	inline void SetRenderState(bool frame, bool pixel) {
		_render_frame = frame;
		_render_pixel = pixel;
	}

	// 判断一条边是不是三角形的左上边 (Top-Left Edge)
	inline bool IsTopLeft(const Vec2i& a, const Vec2i& b) {
		return ((a.y == b.y) && (a.x < b.x)) || (a.y > b.y);
	}

public:

	// 绘制一个三角形，必须先设定好着色器函数
	inline bool DrawPrimitive() {
		if (_frame_buffer == NULL || _vertex_shader == NULL) 
			return false;

		// 顶点初始化
		for (int k = 0; k < 3; k++) {
			Vertex& vertex = _vertex[k];

			// 清空上下文 varying 列表
			vertex.context.varying_float.clear();
			vertex.context.varying_vec2f.clear();
			vertex.context.varying_vec3f.clear();
			vertex.context.varying_vec4f.clear();

			// 运行顶点着色程序，返回顶点坐标
			vertex.pos = _vertex_shader(k, vertex.context);

			// 简单裁剪，任何一个顶点超过 CVV 就剔除
			float w = vertex.pos.w;
			
			// 这里图简单，当一个点越界，立马放弃整个三角形，更精细的做法是
			// 如果越界了就在齐次空间内进行裁剪，拆分为 0-2 个三角形然后继续
			if (w == 0.0f) return false;
			if (vertex.pos.z < 0.0f || vertex.pos.z > w) return false;
			if (vertex.pos.x < -w || vertex.pos.x > w) return false;
			if (vertex.pos.y < -w || vertex.pos.y > w) return false;

			// 计算 w 的倒数：Reciprocal of the Homogeneous W 
			vertex.rhw = 1.0f / w;

			// 齐次坐标空间 /w 归一化到单位体积 cvv
			vertex.pos *= vertex.rhw;

			// 计算屏幕坐标
			vertex.spf.x = (vertex.pos.x + 1.0f) * _fb_width * 0.5f;
			vertex.spf.y = (1.0f - vertex.pos.y) * _fb_height * 0.5f;

			// 整数屏幕坐标：加 0.5 的偏移取屏幕像素方格中心对齐
			vertex.spi.x = (int)(vertex.spf.x + 0.5f);
			vertex.spi.y = (int)(vertex.spf.y + 0.5f);

			// 更新外接矩形范围
			if (k == 0) {
				_min_x = _max_x = Between(0, _fb_width - 1, vertex.spi.x);
				_min_y = _max_y = Between(0, _fb_height - 1, vertex.spi.y);
			}
			else {
				_min_x = Between(0, _fb_width - 1, Min(_min_x, vertex.spi.x));
				_max_x = Between(0, _fb_width - 1, Max(_max_x, vertex.spi.x));
				_min_y = Between(0, _fb_height - 1, Min(_min_y, vertex.spi.y));
				_max_y = Between(0, _fb_height - 1, Max(_max_y, vertex.spi.y));
			}
		}

		// 绘制线框
		if (_render_frame) {
			DrawLine(_vertex[0].spi.x, _vertex[0].spi.y, _vertex[1].spi.x, _vertex[1].spi.y);
			DrawLine(_vertex[0].spi.x, _vertex[0].spi.y, _vertex[2].spi.x, _vertex[2].spi.y);
			DrawLine(_vertex[2].spi.x, _vertex[2].spi.y, _vertex[1].spi.x, _vertex[1].spi.y);
		}

		// 如果不填充像素就退出
		if (_render_pixel == false) return false;

		// 判断三角形朝向
		Vec4f v01 = _vertex[1].pos - _vertex[0].pos;
		Vec4f v02 = _vertex[2].pos - _vertex[0].pos;
		Vec4f normal = vector_cross(v01, v02);

		// 使用 vtx 访问三个顶点，而不直接用 _vertex 访问，因为可能会调整顺序
		Vertex *vtx[3] = { &_vertex[0], &_vertex[1], &_vertex[2] };

		// 如果背向视点，则交换顶点，保证 edge equation 判断的符号为正
		if (normal.z > 0.0f) {
			vtx[1] = &_vertex[2];
			vtx[2] = &_vertex[1];
		}
		else if (normal.z == 0.0f) {
			return false;
		}

		// 保存三个端点位置
		Vec2i p0 = vtx[0]->spi;
		Vec2i p1 = vtx[1]->spi;
		Vec2i p2 = vtx[2]->spi;

		// 计算面积，为零就退出
		float s = Abs(vector_cross(p1 - p0, p2 - p0));
		if (s <= 0) return false;

		// 三角形填充时，左面和上面的边上的点需要包括，右方和下方边上的点不包括
		// 先判断是否是 TopLeft，判断出来后会和下方 Edge Equation 一起决策
		bool TopLeft01 = IsTopLeft(p0, p1);
		bool TopLeft12 = IsTopLeft(p1, p2);
		bool TopLeft20 = IsTopLeft(p2, p0);

		// 迭代三角形外接矩形的所有点
		for (int cy = _min_y; cy <= _max_y; cy++) {
			for (int cx = _min_x; cx <= _max_x; cx++) {
				Vec2f px = { (float)cx + 0.5f, (float)cy + 0.5f };

				// Edge Equation
				// 使用整数避免浮点误差，同时因为是左手系，所以符号取反
				int E01 = -(cx - p0.x) * (p1.y - p0.y) + (cy - p0.y) * (p1.x - p0.x);
				int E12 = -(cx - p1.x) * (p2.y - p1.y) + (cy - p1.y) * (p2.x - p1.x);
				int E20 = -(cx - p2.x) * (p0.y - p2.y) + (cy - p2.y) * (p0.x - p2.x);


				// 如果是左上边，用 E >= 0 判断合法，如果右下边就用 E > 0 判断合法
				// 这里通过引入一个误差 1 ，来将 < 0 和 <= 0 用一个式子表达
				if (E01 < (TopLeft01? 0 : 1)) continue;   // 在第一条边后面
				if (E12 < (TopLeft12? 0 : 1)) continue;   // 在第二条边后面
				if (E20 < (TopLeft20? 0 : 1)) continue;   // 在第三条边后面

				// 三个端点到当前点的矢量
				Vec2f s0 = vtx[0]->spf - px;
				Vec2f s1 = vtx[1]->spf - px;
				Vec2f s2 = vtx[2]->spf - px;

				// 重心坐标系：计算内部子三角形面积 a / b / c
				float a = Abs(vector_cross(s1, s2));    // 子三角形 Px-P1-P2 面积
				float b = Abs(vector_cross(s2, s0));    // 子三角形 Px-P2-P0 面积
				float c = Abs(vector_cross(s0, s1));    // 子三角形 Px-P0-P1 面积
				float s = a + b + c;                    // 大三角形 P0-P1-P2 面积

				if (s == 0.0f) continue;

				// 除以总面积，以保证：a + b + c = 1，方便用作插值系数
				a = a * (1.0f / s);
				b = b * (1.0f / s);
				c = c * (1.0f / s);

				// 计算当前点的 1/w，因 1/w 和屏幕空间呈线性关系，故直接重心插值
				float rhw = vtx[0]->rhw * a + vtx[1]->rhw * b + vtx[2]->rhw * c;

				// 进行深度测试
				if (rhw < _depth_buffer[cy][cx]) continue;
				_depth_buffer[cy][cx] = rhw;   // 记录 1/w 到深度缓存

				// 还原当前像素的 w
				float w = 1.0f / ((rhw != 0.0f)? rhw : 1.0f);

				// 计算三个顶点插值 varying 的系数
				// 先除以各自顶点的 w 然后进行屏幕空间插值然后再乘以当前 w
				float c0 = vtx[0]->rhw * a * w;
				float c1 = vtx[1]->rhw * b * w;
				float c2 = vtx[2]->rhw * c * w;

				// 准备为当前像素的各项 varying 进行插值
				ShaderContext input;

				ShaderContext& i0 = vtx[0]->context;
				ShaderContext& i1 = vtx[1]->context;
				ShaderContext& i2 = vtx[2]->context;

				// 插值各项 varying
				for (auto const &it: i0.varying_float) {
					int key = it.first;
					float f0 = i0.varying_float[key];
					float f1 = i1.varying_float[key];
					float f2 = i2.varying_float[key];
					input.varying_float[key] = c0 * f0 + c1 * f1 + c2 * f2;
				}

				for (auto const &it: i0.varying_vec2f) {
					int key = it.first;
					const Vec2f& f0 = i0.varying_vec2f[key];
					const Vec2f& f1 = i1.varying_vec2f[key];
					const Vec2f& f2 = i2.varying_vec2f[key];
					input.varying_vec2f[key] = c0 * f0 + c1 * f1 + c2 * f2;
				}

				for (auto const &it: i0.varying_vec3f) {
					int key = it.first;
					const Vec3f& f0 = i0.varying_vec3f[key];
					const Vec3f& f1 = i1.varying_vec3f[key];
					const Vec3f& f2 = i2.varying_vec3f[key];
					input.varying_vec3f[key] = c0 * f0 + c1 * f1 + c2 * f2;
				}

				for (auto const &it: i0.varying_vec4f) {
					int key = it.first;
					const Vec4f& f0 = i0.varying_vec4f[key];
					const Vec4f& f1 = i1.varying_vec4f[key];
					const Vec4f& f2 = i2.varying_vec4f[key];
					input.varying_vec4f[key] = c0 * f0 + c1 * f1 + c2 * f2;
				}

				// 执行像素着色器
				Vec4f color = { 0.0f, 0.0f, 0.0f, 0.0f };

				if (_pixel_shader != NULL) {
					color = _pixel_shader(input);
				}

				// 绘制到 framebuffer 上，这里可以加判断，如果 PS 返回的颜色 alpha 分量
				// 小于等于零则放弃绘制，不过这样的话要把前面的更新深度缓存的代码挪下来，
				// 只有需要渲染的时候才更新深度。
				_frame_buffer->SetPixel(cx, cy, color);
			}
		}

		// 绘制线框，再画一次避免覆盖
		if (_render_frame) {
			DrawLine(_vertex[0].spi.x, _vertex[0].spi.y, _vertex[1].spi.x, _vertex[1].spi.y);
			DrawLine(_vertex[0].spi.x, _vertex[0].spi.y, _vertex[2].spi.x, _vertex[2].spi.y);
			DrawLine(_vertex[2].spi.x, _vertex[2].spi.y, _vertex[1].spi.x, _vertex[1].spi.y);
		}

		return true;
	}

protected:

	// 顶点结构体
	struct Vertex {
		ShaderContext context;    // 上下文
		float rhw;                // w 的倒数
		Vec4f pos;                // 坐标
		Vec2f spf;                // 浮点数屏幕坐标
		Vec2i spi;                // 整数屏幕坐标
	};

protected:
	Bitmap *_frame_buffer;    // 像素缓存
	float **_depth_buffer;    // 深度缓存

	int _fb_width;            // frame buffer 宽度
	int _fb_height;           // frame buffer 高度
	uint32_t _color_fg;       // 前景色：画线时候用
	uint32_t _color_bg;       // 背景色：Clear 时候用

	Vertex _vertex[3];        // 三角形的三个顶点

	int _min_x;               // 三角形外接矩形
	int _min_y;
	int _max_x;
	int _max_y;

	bool _render_frame;       // 是否绘制线框
	bool _render_pixel;       // 是否填充像素

	VertexShader _vertex_shader;
	PixelShader _pixel_shader;
};


#endif