Equations for projecting 3d on 2d screen code example

Example: 3d projection onto 2d plane algorithm

#include <vector>
#include <cmath>
#include <stdexcept>
#include <algorithm>

struct Vector
{
    Vector() : x(0),y(0),z(0),w(1){}
    Vector(float a, float b, float c) : x(a),y(b),z(c),w(1){}

    /* Assume proper operator overloads here, with vectors and scalars */
    float Length() const
    {
        return std::sqrt(x*x + y*y + z*z);
    }
    
    Vector Unit() const
    {
        const float epsilon = 1e-6;
        float mag = Length();
        if(mag < epsilon){
            std::out_of_range e("");
            throw e;
        }
        return *this / mag;
    }
};

inline float Dot(const Vector& v1, const Vector& v2)
{
    return v1.x*v2.x + v1.y*v2.y + v1.z*v2.z;
}

class Matrix
{
    public:
    Matrix() : data(16)
    {
        Identity();
    }
    void Identity()
    {
        std::fill(data.begin(), data.end(), float(0));
        data[0] = data[5] = data[10] = data[15] = 1.0f;
    }
    float& operator[](size_t index)
    {
        if(index >= 16){
            std::out_of_range e("");
            throw e;
        }
        return data[index];
    }
    Matrix operator*(const Matrix& m) const
    {
        Matrix dst;
        int col;
        for(int y=0; y<4; ++y){
            col = y*4;
            for(int x=0; x<4; ++x){
                for(int i=0; i<4; ++i){
                    dst[x+col] += m[i+col]*data[x+i*4];
                }
            }
        }
        return dst;
    }
    Matrix& operator*=(const Matrix& m)
    {
        *this = (*this) * m;
        return *this;
    }

    /* The interesting stuff */
    void SetupClipMatrix(float fov, float aspectRatio, float near, float far)
    {
        Identity();
        float f = 1.0f / std::tan(fov * 0.5f);
        data[0] = f*aspectRatio;
        data[5] = f;
        data[10] = (far+near) / (far-near);
        data[11] = 1.0f; /* this 'plugs' the old z into w */
        data[14] = (2.0f*near*far) / (near-far);
        data[15] = 0.0f;
    }

    std::vector<float> data;
};

inline Vector operator*(const Vector& v, const Matrix& m)
{
    Vector dst;
    dst.x = v.x*m[0] + v.y*m[4] + v.z*m[8 ] + v.w*m[12];
    dst.y = v.x*m[1] + v.y*m[5] + v.z*m[9 ] + v.w*m[13];
    dst.z = v.x*m[2] + v.y*m[6] + v.z*m[10] + v.w*m[14];
    dst.w = v.x*m[3] + v.y*m[7] + v.z*m[11] + v.w*m[15];
    return dst;
}

typedef std::vector<Vector> VecArr;
VecArr ProjectAndClip(int width, int height, float near, float far, const VecArr& vertex)
{
    float halfWidth = (float)width * 0.5f;
    float halfHeight = (float)height * 0.5f;
    float aspect = (float)width / (float)height;
    Vector v;
    Matrix clipMatrix;
    VecArr dst;
    clipMatrix.SetupClipMatrix(60.0f * (M_PI / 180.0f), aspect, near, far);
    /*  Here, after the perspective divide, you perform Sutherland-Hodgeman clipping 
        by checking if the x, y and z components are inside the range of [-w, w].
        One checks each vector component seperately against each plane. Per-vertex
        data like colours, normals and texture coordinates need to be linearly
        interpolated for clipped edges to reflect the change. If the edge (v0,v1)
        is tested against the positive x plane, and v1 is outside, the interpolant
        becomes: (v1.x - w) / (v1.x - v0.x)
        I skip this stage all together to be brief.
    */
    for(VecArr::iterator i=vertex.begin(); i!=vertex.end(); ++i){
        v = (*i) * clipMatrix;
        v /= v.w; /* Don't get confused here. I assume the divide leaves v.w alone.*/
        dst.push_back(v);
    }

    /* TODO: Clipping here */

    for(VecArr::iterator i=dst.begin(); i!=dst.end(); ++i){
        i->x = (i->x * (float)width) / (2.0f * i->w) + halfWidth;
        i->y = (i->y * (float)height) / (2.0f * i->w) + halfHeight;
    }
    return dst;
}