How to Solve for Ray-Sphere Intersection Algebraically

[Bucky]

I'm trying to work through an explanation of how a ray-sphere intersection can be solved algebraically from here:
http://wiki.cgsociety.org/index.php/Ray_Sphere_Intersection

My problem is at this step:

we can find the t at which the ray intersects the sphere by setting ray(t) equal to p

(o + t d - c) . (o + t d - c) = r^2


To solve for t we first expand the above into a more recognisable quadratic equation form

(d.d)t^2 + 2 (o - c) . dt + (o - c) - r^2 = 0

I don't understand how they've expanded the formula.

I thought you just multiplied each term in the left bracket by each term in the right bracket...which gave me...

(o.o) + (c.c) - 2(c.o) - 2(d.c) + 2t(d.o) + t^2 (d.d) = r^2


Have I made a mistake or is there some trick I'm missing?

 

[badphysicist]

You're expanding it out too far. The idea is to get it in a quadratic form. You treated it as three terms being squared and they treated it as two terms. Treat (o-c) as one term in the initial brackets and td as the other term. Then expand keeping (o-c) together as one term.

 

[Defennder]

There isn't any trick. Just apply the distributive and commutative property of dot product to the vectors.