Angle of reflection relative to coordinate system

Question

I have a two 2D points u = (ux, uy) and v = (vx, vy) that define a line segment. Additionally I have an angle θ that is defined relative to the coordinate system (angle to x-axis), indicating the directing of a moving particle. Is there a simple way to find the angle of reflection resulting (again, relative to the coordinate system) from that particle bouncing off the line segment?

So far I have found the angle of the line segment θuv = numpy.arctan2(vy-uy, vx-ux), taken the difference Δθ = θuv - θ and set the resulting angle to θ_reflected = θ - 2*Δθ. Sometimes, this seems to work, but other times it's completely off.

Answer 1

Segment has length

leng = hypot(vy-uy, vx-ux)

and unit direction vector (perhaps in numpy there is ready function like normalized)

dx = (vx-ux) / leng
dy = (vy-uy) / leng

Unit normal to segment

nx = - dy
ny = dx

particle direction vector is

px = cos(θ)
py = sin(θ)

Reflected vector

dott = dot(p, n) = px * nx + py * ny
rx = px - 2 * dott * nx
ry = py - 2 * dott * ny

If you need angle

θ_reflected = atan2(ry, rx)

but sometimes particle direction vector components are more useful