exu | Universal Executables are an architecture

I am working on a 2D collider system that breaks up shapes into one possible primitive: impenetrable segments that are defined by two points. To provide collision detection for this system, I am using a static collision detection approach that calculates the distance between the edge of one segment and the currently handled segment (point/line distance) once every frame. If the distance is too small, a collision is triggered during that frame. This works fine but has the known problem of tunneling if one or more bodies exhibit high speeds. So I am tinkering with alternatives.

Now I want to introduce continuous collision detection (CCD) that operates on dynamic points / dynamic segments. My problem is: I don't exactly know how. I do know how to do continuous collision between two moving points, a moving point and a static segment but not how to do CCD between a moving point (defined by point P) and a moving segment (defined by points U and V, both can move completely freely).

illustration of problem

I have seen similar questions beeing asked on SO and other platforms, but not with these exact requirements:

• both point and segment are moving
• segment can be rotating and stretching (because U and V are moving freely)
• collision time and collision point need to be found accurately between two frames (CCD, no static collision test)
• I prefer a mathematically perfect solutiuon, if possible (no iterative approximation algorithms, swept volumes)
• note: the swept line shape will not always be a convex polygon, because of the freedom of the U,V points (see image)
• note: testing for a collision with the swept volume test is inaccurate because a collision point with the polygon does not mean a collision point in the actual movement (see image, the point will have left the polygon once the actual segment has crossed the trajectory of the point)

So far I came up with the following approach, given:

• sP (P at start of frame),
• eP (P at end of frame),
• sU (U at start of frame),
• eU (U at end of frame),
• sV (V at start of frame),
• eV (V at end of frame)

Question: Will they collide? If yes, when and where?

To answer the question of "if", I found this paper to be useful: https://www.cs.ubc.ca/~rbridson/docs/brochu-siggraph2012-ccd.pdf (section 3.1) but I could not derive the answers to "when" and "where". I also found an alternative explanation of the problem here: http://15462.courses.cs.cmu.edu/fall2018/article/13 (3rd Question)

Solution:

Model temporal trajectory of each point during a frame as linear movement (line trajectory for 0 <= t <= 1)

• P(t) = sP * (1 - t) + eP * t
• U(t) = sU * (1 - t) + eU * t
• V(t) = sV * (1 - t) + eV * t

(0 <= a <= 1 represents a location on the segment defined by U and V):

• UV(a, t) = U(t) * (1 - a) + V(t) * a

Model collision by equating point and segment equations:

• P(t) = UV(a, t)
• P(t) = U(t) * (1 - a) + V(t) * a

Derive a function for the vector from point P to a point on the segment (see picture of F):

• F(a, t) = P(t) - (1 - a) * U(t) - a * V(t)

To now find a collision, one needs to find a and t, so that F(a, t) = (0, 0) and a,t in [0, 1]. This can be modeled as a root finding problem with 2 variables.

Insert the temporal trajectory equations into F(a, t):

• F(a, t) = (sP * (1 - t) + eP * t) - (1 - a) * (sU * (1 - t) + eU * t) - a * (sV * (1 - t) + eV * t)

Separate the temporal trajectory equations by dimension (x and y):

• Fx(a, t) = (sP.x * (1 - t) + eP.x * t) - (1 - a) * (sU.x * (1 - t) + eU.x * t) - a * (sV.x * (1 - t) + eV.x * t)

• Fy(a, t) = (sP.y * (1 - t) + eP.y * t) - (1 - a) * (sU.y * (1 - t) + eU.y * t) - a * (sV.y * (1 - t) + eV.y * t)

Now we have two equations and two variables that we want to solve for (Fx, Fy and a, t respectively), so we should be able to use a solver to get a and t to only then check if they lie within [0, 1].. right?

When I plug this into Python sympy to solve: