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Ray cast from INSIDE sphere or cylinder

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I've been using Christer Ericson's book for my standard ray-cast tests. Now in my particular case I need to cast a ray from a point that i know is contained by either a sphere or cylinder. I want to cast the ray outwards and get the appropriate t value and contact point on the boundary of the cylinder/sphere.

I figure that in the case of sphere or infinite cylinger i can simply solve for both t values and use those to figure out whats going on.

if discr < 0, theres no intersection,
if discr == 0, there is a simple contact tanger to surface, and
if discr > 0 there are 2 intersections. and 3 cases:

t1 and t2 are positive: the intersection is infront of the ray, i take the lowest t value for time of impact
t1 and t2 are negative: the intersection is behind the ray, return false
t1 is negative and t2 is positive: the ray origin is inside the object, the postive is its exit point in front of the ray. This is what i want?

Is this reasoning correct?

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