OpenGL : how to get the matrix that mirror the Scene? - c++

Question

Context:
I try to make a mirror in OpenGL.
So what I need to define a mirror is a plane, defined by a point p and two normal vectors n1 and n2. n3=n1^n2 defines a vector oriented to the viewer of the mirror (the sign is important to discard fragments that are behind the mirror and also would be reflected in front of the mirror). n1, n2 and n3 define a basis in the coordinates of the mirror (with origin p an arbitrary point located on the mirror plane).
Now, to draw the reflection of the scene I need the matrix reflectionMatrix that transform any point into its mirrored. I don't even know if it's possible, but what I tried looks ok but there is problems when the mirror is very close to the point.
I tried to use transition matrix used in linear algebra, but I don't even know if I can use them with transformation matrix. I am also a bit confused about the transformation order.
The code is that :
/// #brief Compute change-of-basis matrix.
/// From basis {e1, e2, e3} to canonical basis of R^3.
mat4 transition(const glm::vec3& e1, const glm::vec3& e2, const glm::vec3& e3)
{
// GLM is column major: mat4(col1, col2, col3, col4)
return {
vec4(e1, 0),
vec4(e2, 0),
vec4(e3, 0),
vec4(vec3(0), 1)
};
}
glm::mat4 getReflectionMatrix() {
const glm::vec3 mirrorPos{scene.mirrorPos};
const glm::vec3 n1{1, 0, 0};
const glm::vec3 n2{0, 1, 0};
const glm::vec3 n3{glm::cross(n1, n2)};
// n1, n2 are orthogonal vectors of the mirror
// n3 is to complete the basis
// the vertices should be in trigonometric order to work
glm::mat4 reflectionMatrix{1};
// Let p be the point we mirror
// Compute p1 = p relative to the mirror [Translation in global coordinate system ~= premultiply]
reflectionMatrix = glm::translate(reflectionMatrix, -mirrorPos);
// Compute p2 = same as p1 but in coordinate system {n1, n2, n3}
reflectionMatrix *= glm::inverse(transition(n1, n2, n3));
// Compute p3 = (p2.x, p2.y, -p2.z) to go into the mirror (z is orthogonal distance from mirror)
reflectionMatrix = glm::scale(reflectionMatrix, {1, 1, -1});
// Compute p4 = same as p3 but in canonical coordinate system (again relative to the mirror)
reflectionMatrix *= transition(n1, n2, n3);
// Compute p5 = p4 from relative to the mirror to global space [Translation in global coordinate system]
reflectionMatrix = glm::translate(reflectionMatrix, mirrorPos);
return reflectionMatrix;
}
What is wrong with the reflection matrix? Or is there a simpler way