Photo credit to Keith Johnston

In 2016, a friend asked me for help with a virtual reality problem. He was trying to find the collision in space between a baseball and a bat between frames of a physics simulator for a VR system. If you want to build VR software that can train professional athletes, accuracy is important.

The obvious approach, the one I thought of first, is a numerical solution but it is too slow. My desktop took 100-300ms to solve it numerically but we only have 30 ms between frames if we have a 30 hz frame rate. The code below solves the problem explicitly in roughly 10-20 µs. What starts as an 18 variable algebra problem can be reduced to the roots of a 4th order polynomial. It just takes several pages of algebra.

The code below is confusing. I can not understand why it works without seeing the derivation and I doubt anyone could. To understand why it works, see the documentation.

import matplotlib.pyplot as plt

import numpy as np

# Bat start and end positions are given as 2 tuples of x,y coordinates,

# the first tuple is the knob of the bat handle, the second is the end of the fat end of the bat

# the ball is defined by 1 tuple of x,y coordinates at the center

bat_start = ((0,0,0),(-1,5,0))

bat_end = ((1,0,0),(2,5,0))

ball_start = (30, 3.5, -10)

ball_end = (-10, 3.5, -10)

frame_length = 16.66 #ms 1/60*1000 frame length @ 60 Hz

ball_r,bat_r = [2.5,2.5]

# point = np.array(x,y,z)

# line = [np.array(x,y,z),np.array(x,y,z)]

def separation(point,line,inv =1):

lin_interp = lambda v1,v2,t: v1+(v2-v1)*t

v = (lambda t: lin_interp(i,j,t) for i,j in zip(line[0],line[1]))

tt = -np.dot((line[0]-point),(line[1]-line[0]))/np.dot((line[0]-line[1]),(line[0]-line[1]))

V = np.array([j(tt) for j in v])

d = np.sum((V-point)**2)**.5+inv*(bat_r+ball_r)

d_raw = np.sum(V-point)

return d,d_raw

def vec_interp(p1,p2):

return lambda t: (p2-p1)*t/frame_length+p1

def path_trace_min(bat_start,bat_end,ball_start,ball_end,show = False):

bat_start = [np.array(i) for i in bat_start]

bat_end = [np.array(i) for i in bat_end]

ball = [np.array(i) for i in [ball_start,ball_end]]

dE = vec_interp(bat_start[1],bat_end[1])

dS = vec_interp(bat_start[0],bat_end[0])

dB = vec_interp(ball[0],ball[1])

tt = np.linspace(0,frame_length,300)

inv = -1 if separation(dB(i),[dS(i),dE(i)])[1]<0 else 1

d = [separation(dB(i),[dS(i),dE(i)],inv)[0] for i in tt]

dr = [separation(dB(i),[dS(i),dE(i)])[1] for i in tt]

D = lambda i: separation(dB(i),[dS(i),dE(i)],inv)[1]

# find t, collision time

# print np.where(np.array(d)<0)[0]

if len(np.where(np.array(d)<0)[0])>1:

p2 = d[np.where(np.array(d)<0)[0][0]]

t = tt[np.where(np.array(d)<0)[0][0]]

print [t,p2]

else:

t=0;p2=10

if show:

plt.plot(tt,d); plt.legend();

plt.title('distance vs time'); plt.xlabel('time');

plt.ylabel('distance between ball and bat')

plt.plot(tt,dr);plt.show()

return t,dB,dS,dE,p2

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