Question

I’m in the process of building a free open source OpenGL3-based 3D game engine (it’s not a school assignment, rather it’s for personal skill development and to give something back to the open source community). I’ve reached the stage where I need to learn lots of related math, so I’m reading a great textbook called “Mathematics for 3D Game Programming and Computer Graphics, 3rd Edition”.

I’ve hit a snag early on trying to do the book’s exercises though, as my attempt at implementing the “Gram-Schmidt Orthogonalization algorithm” in C++ is outputting a wrong answer. I’m no math expert (although I’m trying to get better), and I have very limited experience looking at a math algorithm and translating it into code (limited to some stuff I learned from Udacity.com). Anyway, it would really help if someone could look at my incorrect code and give me a hint or a solution.

Here it is:

/*
The Gram-Schmidt Orthogonalization algorithm is as follows:

    Given a set of n linearly independent vectors Beta = {e_1, e_2, ..., e_n},
    the algorithm produces a set Beta' = {e_1', e_2', ..., e_n'} such that
    dot(e_i', e_j') = 0 whenever i != j.

    A. Set e_1' = e_1
    B. Begin with the index i = 2 and k = 1
    C. Subtract the projection of e, onto the vectors e_1', e_2', ..., e_(i-1)'
       from e_i, and store the result in e_i', That is,

                             dot(e_i, e_k')
       e_i' = e_i - sum_over(-------------- e_k')
                                e_k'^2

    D. If i < n, increment i and loop back to step C.
*/

#include <iostream>
#include <glm/glm.hpp>

glm::vec3 sum_over_e(glm::vec3* e, glm::vec3* e_prime, int& i)
{
    int k = 0;
    glm::vec3 result;

    while (k < i-2)
    {
        glm::vec3 e_prime_k_squared(pow(e_prime[k].x, 2), pow(e_prime[k].y, 2), pow(e_prime[k].z, 2));
        result += (glm::dot(e[i], e_prime[k]) / e_prime_k_squared) * e_prime[k];
        k++;
    }

    return result;
}

int main(int argc, char** argv)
{
    int n = 2;  // number of vectors we're working with
    glm::vec3 e[] = {
        glm::vec3(sqrt(2)/2, sqrt(2)/2, 0),
        glm::vec3(-1, 1, -1),
        glm::vec3(0, -2, -2)
    };

    glm::vec3 e_prime[n];
    e_prime[0] = e[0];  // step A

    int i = 0;  // step B

    do  // step C
    {
        e_prime[i] = e[i] - sum_over_e(e, e_prime, i);

        i++;    // step D
    } while (i-1 < n);

    for (int loop_count = 0; loop_count <= n; loop_count++)
    {
        std::cout << "Vector e_prime_" << loop_count+1 << ": < " 
                  << e_prime[loop_count].x << ", " 
                  << e_prime[loop_count].y << ", " 
                  << e_prime[loop_count].z << " >" << std::endl;
    }

    return 0;
}

This code outputs:

Vector e_prime_1: < 0.707107, 0.707107, 0 >
Vector e_prime_2: < -1, 1, -1 >
Vector e_prime_3: < 0, -2, -2 >

but the correct answer is supposed to be:

Vector e_prime_1: < 0.707107, 0.707107, 0 >
Vector e_prime_2: < -1, 1, -1 >
Vector e_prime_3: < 1, -1, -2 >

Edit: Here’s the code that produces the correct answer:

#include <iostream>
#include <glm/glm.hpp>

glm::vec3 sum_over_e(glm::vec3* e, glm::vec3* e_prime, int& i)
{
    int k = 0;
    glm::vec3 result;

    while (k < i-1)
    {
        float e_prime_k_squared = glm::dot(e_prime[k], e_prime[k]);
        result += ((glm::dot(e[i], e_prime[k]) / e_prime_k_squared) * e_prime[k]);
        k++;
    }

    return result;
}

int main(int argc, char** argv)
{
    int n = 3;  // number of vectors we're working with
    glm::vec3 e[] = {
        glm::vec3(sqrt(2)/2, sqrt(2)/2, 0),
        glm::vec3(-1, 1, -1),
        glm::vec3(0, -2, -2)
    };

    glm::vec3 e_prime[n];
    e_prime[0] = e[0];  // step A

    int i = 0;  // step B

    do  // step C
    {
        e_prime[i] = e[i] - sum_over_e(e, e_prime, i);

        i++;    // step D
    } while (i < n);

    for (int loop_count = 0; loop_count < n; loop_count++)
    {
        std::cout << "Vector e_prime_" << loop_count+1 << ": < " 
                  << e_prime[loop_count].x << ", " 
                  << e_prime[loop_count].y << ", " 
                  << e_prime[loop_count].z << " >" << std::endl;
    }

    return 0;
}

Answer 1

The problem is probably in the way you define e_k'^2. As far as vector math goes, the square of a vector is usually taken to be the square of its norm. Therefore,

double e_prime_k_squared = glm::dot(e_prime_k, e_prime_k);

Moreover, dividing by a vector is undefined (I wonder why GLM allows it?), so if e_k'^2 is a vector, the whole thing is undefined.