Quiz 11: Continuous Collision Detection

In class we briefly brought up the question of how to detect collisions between two objects that are moving in time. Solving this problem exactly is already quite challenging for a pair of triangles (with new research on this seemingly basic question being published as recently as 2012!), so in this quiz we'll do what we know how to do best: start out simple, solve an easy problem, and build up additional complexity.

## Question 1: Point-Point Intersection

Checking whether two points intersect is easy: just see if they have identical coordinates! But what if these points are moving? In particular, consider two points p, q in 2D moving along linear trajectories

p(t) = (1-t)p_0 + t p_1


and

q(t) = (1-t)q_0 + t q_1


over a time interval 0 <= t <= 1. How would you test whether these points bump into each-other? Describe your strategy explicitly: what quantities would you evaluate, and what tests would you perform, in terms of the variables t, p0, p1, q0, and q1.

## Question 2: Ball-Ball Intersection

In reality, it's highly unlikely that two points will ever actually hit each other---imagine two particles of dust flying through deep space. However, as anyone who's ever played dodgeball knows, it is possible that two balls might hit each other mid-flight. Suppose in particular that our dodgeballs again have linear trajectories p(t) and q(t) in the interval 0 <= t <= 1 (exactly as above), but this time they have radii rp, rq > 0. How would you check if these two balls collide? Write out the equation(s) you would check, giving reasonably explicit expressions for how you'd compute the collision times t (if they exist). What are the possible interactions between the two balls, and how do they correspond to your equation(s)? [Hint: substitute complicated expressions for single variables when possible, in order to simplify your calculations.]

## Quiz 3: Point-Segment Intersection

It's also possible that for a moving point to collide with a moving line segment---this test starts to be useful for, say, checking if two triangles collide. In particular, consider a point following a linear trajectory

p(t) = (1-t)p_0 + t p_1


and an edge with endpoints $a,b \in \mathbb{R}^2$ both following linear trajectories

a(t) = (1-t)a_0 + t a_1,
b(t) = (1-t)b_0 + t b_1,


all in the time interval 0 <= t <= 1. Setup an equation or equations you must solve to determine whether the point and the edge collide. This time you do not have to solve for $t$ explicitly; just give the equation that needs to be solved. However, you should comment on what kind of equation it is, and whether you think it can/cannot easily be solved.