What's the equation for calculating normal vector at a point?

Question

What's the equation for calculating normal vector at a point?

Particularly I'm trying understand the function given here:

Vec3f Sphere::GetNormalAt(const Vec3f &pos) const {   
    return cv::normalize(pos - points[o]); 
}

So what exactly is this as an equation?

It's some kind of

$$\frac{pos-p}{||pos-p||}$$

but what point is $p$?

Also, I'm particularly trying to get this function in order to use here: https://bobobobo.wordpress.com/tag/ray/
where you see the function getNormalAt() being used for something in Ray-Sphere interaction.

Additionally here one can find the function with some algebra:

private Vector GetNormalAt(Vector v)
{
    double x = (2 / (_a * _a)) * v.GetX();
    double y = (2 / (_b * _b)) * v.GetY();
    double z = (2 / (_c * _c)) * v.GetZ();
    Vector normal = new Vector(x, y, z);
    normal.NornVector();
    return normal * -_normalSign;
}

Answer 1

Well, looking at the function, it is a method of a sphere class.

Consider a sphere and a point $p$ on the sphere, of radius $r$, with center $c$.

The normal at $p$ is pointing in the same direction as $n = p - c$ (i.e. pointing outward from the center of the sphere). Thus, the unit normal is just: $$ n(p) = \frac{p-c}{||p-c||_2} $$ So I can only assume $c$ is the point to which you are referring.

The second class you link to is an ellipsoid class. Again, the calculation is specific to that surface.

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For reference, in general, one computes surface normals by finding two linearly independent tangent vectors to the surface $T_1(p)$ and $T_2(p)$ at point $p$, and computing the normal as: $$ n(p) = \frac{T_1(p)\times T_2(p)}{||T_1(p)\times T_2(p)||_2} $$

(The cases you are looking at have special geometry such that this is not required.)