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.

 

[baba89]

Hello,

I understand that you are having trouble with the problem presented. First, let's break down the problem into smaller parts to better understand it.

The first part defines a "pill" as a region within a distance R of a line segment between two points, B and B+H. This pill is essentially a cylinder with a radius R and two spheres at each of its endpoints. This can be visualized as a circular tube with two half-spheres at each end.

The second part introduces a particle with a starting point P and a vector V that defines its motion. The particle moves from P to P+V in one unit of time (t=1).

The task is to calculate the time t at which the particle first intersects the pill. This means finding the point at which the path of the particle crosses or touches the surface of the pill.

To solve this problem, you will need to use analytical geometry and equations that describe the motion of the particle and the geometry of the pill. I recommend researching equations for the intersection of a line and a cylinder, as well as equations for the intersection of a line and a sphere.

Once you have these equations, you can plug in the values for the starting point P, the vector V, and the geometry of the pill (B, H, R) to find the time t at which the particle intersects the pill.

I understand that this may seem overwhelming, but I encourage you to break down the problem further and research each component individually. You can also reach out to your professor or classmates for guidance and support.

I hope this helps you in your problem-solving process. Good luck!