How Do You Calculate the Intersection Time of a Particle and a 3D Pill Shape?

[21omega12]

I am very confused about the problem presented below and would appreciate any help in visualization or pointing me in the direction of an article or equation that will set me on the right path. I can't seem to get past the initial set up of the problem at this point and am quite frustrated at the moment.

Homework Statement

Using a point B, a vector H, and a radius R in 3D space, define a
"pill" as the region within distance R of the line segment between B
and B+H. The pill will be a cylinder of radius R joined with spheres
centered at each of its endpoints. A second point P and vector V define
the motion of a particle: it begins at P at time t=0 and reaches P+V
by t=1.

Calculate the time t (if it is defined and between zero and one) at
which the particle first intersects the pill. This should be an
analytic solution, not a high-level algorithm.

 

[tiny-tim]

Welcome to PF!

Using a point B, a vector H, and a radius R in 3D space, define a
"pill" as the region within distance R of the line segment between B
and B+H. The pill will be a cylinder of radius R joined with spheres
centered at each of its endpoints. A second point P and vector V define
the motion of a particle: it begins at P at time t=0 and reaches P+V
by t=1.

Calculate the time t (if it is defined and between zero and one) at
which the particle first intersects the pill. This should be an
analytic solution, not a high-level algorithm.

Hi 21omega12! Welcome to PF!

B to B+H is a line.

It has a cylinder round it, of radius R, and two hemispheres at the ends.

Hint: If the particle intersects the pill, then it must intersect either the cylinder or one of the hemispheres first.

So just treat it as three separate problems, find t for each problem (if it exists), and then take the minimum of the solutions.