Games101-Project-Triangle

绘制一个可旋转的三角形

model

Eigen::Matrix4f get_model_matrix(float rotation_angle)
{
    Eigen::Matrix4f model = Eigen::Matrix4f::Identity();
    model<<cos(rotation_angle/180*MY_PI),-sin(rotation_angle/180*MY_PI),0,0,
			sin(rotation_angle/180*MY_PI),cos(rotation_angle/180*MY_PI),0,0,
			0,0,1,0,
			0,0,0,1;
    return model;
}

view

Eigen::Matrix4f get_view_matrix(Eigen::Vector3f eye_pos)
{
    Eigen::Matrix4f view = Eigen::Matrix4f::Identity();
    Eigen::Matrix4f translate;
    translate << 1, 0, 0, -eye_pos[0], 
    			 0, 1, 0, -eye_pos[1], 
    			 0, 0, 1, -eye_pos[2], 
        		 0, 0, 0, 1;
    view = translate * view;
    return view;
}

projection

透视矩阵正交矩阵：Transformation

Eigen::Matrix4f get_projection_matrix(float eye_fov, float aspect_ratio, float zNear, float zFar)
{
    Eigen::Matrix4f projection;
    Eigen::Matrix4f orthotrans1;
    Eigen::Matrix4f orthotrans2;
    Eigen::Matrix4f persptrans;
    float f, n, l, r, b, t, fov;
    n = -zNear;
    f = -zFar;
    t = tan(eye_fov / 2) * abs(n);
    b = -t;
    r = t * aspect_ratio;
    l = -r;
    persptrans << n, 0, 0, 0,
        0, n, 0, 0,
        0, 0, n + f, -n * f,
        0, 0, 1, 0;
    orthotrans1 << 2 / (r - l), 0, 0, 0,
        0, 2 / (t - b), 0, 0,
        0, 0, 2 / (n - f), 0,
        0, 0, 0, 1;
    orthotrans2 << 1, 0, 0, -(r + l) / 2,
        0, 1, 0, -(t + b) / 2,
        0, 0, 1, -(n + f) / 2,
        0, 0, 0, 1;
    projection = orthotrans1 * orthotrans2 * persptrans;
    return projection;
}

draw

视口变换：Rasterization

void rst::rasterizer::draw(rst::pos_buf_id pos_buffer, rst::ind_buf_id ind_buffer, rst::Primitive type)
{
    if (type != rst::Primitive::Triangle)
    {
        throw std::runtime_error("Drawing primitives other than triangle is not implemented yet!");
    }
    auto& buf = pos_buf[pos_buffer.pos_id];
    auto& ind = ind_buf[ind_buffer.ind_id];

    float f1 = (100 - 0.1) / 2.0;
    float f2 = (100 + 0.1) / 2.0;

    Eigen::Matrix4f mvp = projection * view * model;
    for (auto& i : ind)
    {
        Triangle t;
		//三维增加一维w以实现运算 1表示向量 0表示点
        Eigen::Vector4f v[] = {
                mvp * to_vec4(buf[i[0]], 1.0f),
                mvp * to_vec4(buf[i[1]], 1.0f),
                mvp * to_vec4(buf[i[2]], 1.0f)
        };
		//w统一成1 x,y,z跟着变换
        for (auto& vec : v) {
            vec /= vec.w();
        }

        for (auto & vert : v)
        {
            //将x,y转换到屏幕上 从[-1,1]转到[0,windowsize]
            vert.x() = 0.5*width*(vert.x()+1.0);
            vert.y() = 0.5*height*(vert.y()+1.0);
            vert.z() = vert.z() * f1 + f2;
        }
		//取x,y,z舍弃w
        for (int i = 0; i < 3; ++i)
        {
            t.setVertex(i, v[i].head<3>());
            t.setVertex(i, v[i].head<3>());
            t.setVertex(i, v[i].head<3>());
        }

        t.setColor(0, 255.0,  0.0,  0.0);
        t.setColor(1, 0.0  ,255.0,  0.0);
        t.setColor(2, 0.0  ,  0.0,255.0);

        rasterize_wireframe(t);
    }
}

rasterize_wirefram

void rst::rasterizer::rasterize_wireframe(const Triangle& t)
{
    draw_line(t.c(), t.a());
    draw_line(t.c(), t.b());
    draw_line(t.b(), t.a());
}

绕任意过原点轴的旋转矩阵

Rodrigues’ rotation formula

Eigen::Matrix4f get_rotation(Vector3f axis, float angle)
{
    Matrix4f rotation = Matrix4f::Identity();
    float x = axis.x();
    float y = axis.y();
    float z = axis.z();
    float cosAngle = cos(angle / 180 * MY_PI);
    float sinAngle = sin(angle / 180 * MY_PI);
    rotation << cosAngle + (1 - cosAngle) * x * x, (1 - cosAngle)* x* y - sinAngle * z, (1 - cosAngle)* x* z + sinAngle * y, 0,
                (1 - cosAngle)* x* y + sinAngle * z, cosAngle + (1 - cosAngle) * y * y, (1 - cosAngle)* y* z - sinAngle * x, 0,
                (1 - cosAngle)* x* z - sinAngle * y, (1 - cosAngle)* y* z + sinAngle * x, cosAngle + (1 - cosAngle) * z * z, 0,
                0, 0, 0, 1;
    return rotation;
}

结果

绘制两个有颜色、有重叠的三角形

insideTriangle

判断点在三角形内：Shading

static bool insideTriangle(int x, int y, const Vector3f* _v)
{   
    //假设0，1，2对应A，B，C 判断点P
    //AB AC AP两两叉乘 看法向量是否是同一方向
    Vector3f v[3];
    for (int i = 0; i < 3; i++)
		v[i] = { _v[i].x(),_v[i].y(), 1.0 };
    Vector3f ab = v[1] - v[0];
    Vector3f bc = v[2] - v[1];
    Vector3f ca = v[0] - v[2];
    Vector3f ap = Vector3f(x, y, 1.0) - v[0];
    Vector3f bp = Vector3f(x, y, 1.0) - v[1];
    Vector3f cp = Vector3f(x, y, 1.0) - v[2];
    if (ab.cross(ap).z() > 0 && bc.cross(bp).z() > 0 && ca.cross(cp).z() > 0)
		return true;
    return false;
}

computeBarycentric2D

重心坐标：Shading

static std::tuple<float, float, float> computeBarycentric2D(float x, float y, const Vector3f* v)
{
    float c1 = (x*(v[1].y() - v[2].y()) + (v[2].x() - v[1].x())*y + v[1].x()*v[2].y() - v[2].x()*v[1].y()) / (v[0].x()*(v[1].y() - v[2].y()) + (v[2].x() - v[1].x())*v[0].y() + v[1].x()*v[2].y() - v[2].x()*v[1].y());
    
    float c2 = (x*(v[2].y() - v[0].y()) + (v[0].x() - v[2].x())*y + v[2].x()*v[0].y() - v[0].x()*v[2].y()) / (v[1].x()*(v[2].y() - v[0].y()) + (v[0].x() - v[2].x())*v[1].y() + v[2].x()*v[0].y() - v[0].x()*v[2].y());
    
    float c3 = (x*(v[0].y() - v[1].y()) + (v[1].x() - v[0].x())*y + v[0].x()*v[1].y() - v[1].x()*v[0].y()) / (v[2].x()*(v[0].y() - v[1].y()) + (v[1].x() - v[0].x())*v[2].y() + v[0].x()*v[1].y() - v[1].x()*v[0].y());
   
    return {c1,c2,c3};
}

rasterize_triangle

深度测试：Rasterization

void rst::rasterizer::rasterize_triangle(const Triangle& t) {
    auto v = t.toVector4();

    int x_min = std::min({ v[0].x(), v[1].x(), v[2].x() });
    int x_max = std::max({ v[0].x(), v[1].x(), v[2].x() });
    int y_min = std::min({ v[0].y(), v[1].y(), v[2].y() });
    int y_max = std::max({ v[0].y(), v[1].y(), v[2].y() });

    for (int x = x_min; x <= x_max; x++) {
        for (int y = y_min; y <= y_max; y++) {
            if (insideTriangle(x, y, t.v)) {
                auto [alpha, beta, gamma] = computeBarycentric2D(x, y, t.v);
                if (alpha >= 0 && beta >= 0 && gamma >= 0) {
                    float w_reciprocal = 1.0 / (alpha / v[0].w() + beta / v[1].w() + gamma / v[2].w());
                    float z_interpolated = alpha * v[0].z() / v[0].w() + beta * v[1].z() / v[1].w() + gamma * v[2].z() / v[2].w();
                    z_interpolated *= w_reciprocal;

                    int index = get_index(x, y);
                    if (z_interpolated < depth_buf[index]) {
                        depth_buf[index] = z_interpolated;
                        Eigen::Vector3f point(x, y, 0);
                        set_pixel(point, t.getColor());
                    }
                }
            }
        }
    }
}

MSAA

MSAA：Rasterization

int rst::rasterizer::MSAA(int x, int y, const Vector3f* _v, int n, int m, float z, Eigen::Vector3f color)
{
    //n*m超采样 每个大点细分成n*m小点 对小点深度测试 若可见 则记录小点的color
    //返回大点数量
    float size_x = 1.0 / n; // the size_x of every super sample pixel
    float size_y = 1.0 / m;

    int blocksinTriangle = 0;
    for (int i = 0; i < n; ++i)
        for (int j = 0; j < m; ++j) {
            if (z < msaa_depth_buf[get_index(x, y) * 4 + i * 2 + j] && insideTriangle_MSAA(x + i * size_x, y + j * size_y, _v, size_x, size_y)) {
                msaa_depth_buf[get_index(x, y) * 4 + i * 2 + j] = z;
                msaa_frame_buf[get_index(x, y) * 4 + i * 2 + j] = color;
                blocksinTriangle++;
            }
        }
    return blocksinTriangle;
}

void rst::rasterizer::rasterize_triangle_MSAA(const Triangle& t) {
    auto v = t.toVector4();

    int x_min = std::min({ v[0].x(), v[1].x(), v[2].x() });
    int x_max = std::max({ v[0].x(), v[1].x(), v[2].x() });
    int y_min = std::min({ v[0].y(), v[1].y(), v[2].y() });
    int y_max = std::max({ v[0].y(), v[1].y(), v[2].y() });

    int m = 2;
    int n = 2;

    for (int x = x_min; x <= x_max; x++) {
        for (int y = y_min; y <= y_max; y++) {
            auto [alpha, beta, gamma] = computeBarycentric2D(x, y, t.v);
            float w_reciprocal = 1.0 / (alpha / v[0].w() + beta / v[1].w() + gamma / v[2].w());
            float z_interpolated = alpha * v[0].z() / v[0].w() + beta * v[1].z() / v[1].w() + gamma * v[2].z() / v[2].w();
            z_interpolated *= w_reciprocal;
            int blocksinTriangle = MSAA(x, y, t.v, n, m, z_interpolated, t.getColor());
            //点在三角形内 
            if (blocksinTriangle > 0) {
                int idx = get_index(x, y);
                //颜色平均
                set_pixel(Eigen::Vector3f(x, y, z_interpolated), (msaa_frame_buf[idx * 4] + msaa_frame_buf[idx * 4 + 1] + msaa_frame_buf[idx * 4 + 2] + msaa_frame_buf[idx * 4 + 3]) / 4.0);
            }
        }
    }
}

static bool insideTriangle_MSAA(float x, float y, const Vector3f* _v, float size_x, float size_y)
{
    //x+0.25是小点中心 用这个判断是否在三角形内
    float x_mid = x + size_x / 2.0;
    float y_mid = y + size_y / 2.0;

    Vector3f v[3];
    for (int i = 0; i < 3; i++)
        v[i] = { _v[i].x(),_v[i].y(), 1.0 };
    Vector3f ab = v[1] - v[0];
    Vector3f bc = v[2] - v[1];
    Vector3f ca = v[0] - v[2];
    Vector3f ap = Vector3f(x_mid, y_mid, 1.0) - v[0];
    Vector3f bp = Vector3f(x_mid, y_mid, 1.0) - v[1];
    Vector3f cp = Vector3f(x_mid, y_mid, 1.0) - v[2];
    if (ab.cross(ap).z() > 0 && bc.cross(bp).z() > 0 && ca.cross(cp).z() > 0)
        return true;
    return false;
}

结果