How does a point squared equal the radius squared?

Question

I am working on a simple ray tracer but I don't understand some of the formulas.

One that is bugging me at the moment is this:

If a sphere is centred at origin, a point p lies on a sphere of radius r² if
p*p = r²

Now, as I understand it, 'p' is a point in 3D space, (x,y,z), so how is it possible to square the point and have it equal the radius squared?

For what it's worth, I'm reading this article on calculating the intersection of a ray and a sphere.

Answer 1

There are two ways to "multiply" vectors: the dot product and the cross product.

In this case, they're using the dot product, and it is very easy to see that, if p is centered at the origin, then:

  dot(p, p)
= (p.x * p.x) + (p.y * p.y) + (p.z * p.z)
= mag(p) ^ 2 // per the pythagorean theorem

Now, if the magnitude of your vector happens to be the radius of the sphere, then your point will lie exactly in the sphere.

Answer 2

You seem to have paraphrased the formula incorrectly. In vector algebra, there are several types of multiplication and they use different symbols. The original article uses the dot product (·) which is not equivalent to the generic multiplication symbol *. It also uses bold face to signify vectors as opposed to scalar quantities, i.e. r² is not the same as r². In fact, as you rightfully noted, you cannot really 'square' a vector.

The formula as it was originally written is:

p · p = r²

If two vectors point in the same direction, as p and p obviously do, their dot product equals the square of their absolute values or 'lengths'. By definition, any point at a distance r from the origin lies on the sphere with radius r centered at the origin. Ergo, if the square of a vector's length equals the square of this sphere's radius (p · p = r²), then the point lies on a sphere with radius r. The article mentions the sphere's radius is r²; this is incorrect.