Thread: finding the min distance between two vectors

Hi there,
Im trying to find the minimum distance(sum of their radii) between two circles.
The circles have position vectors a radius and a velocity vector.

i can find the minimum(s) distance using Pythagoras and a loop. but this is costly i would much prefer to use some sort of differentiation.

a=sum of radii
b=(x1+Vx1*t)-(x2+Vx2*t)
c=(y1+Vy1*t)-(y2+Vy2*t)

Best regards,
Mark

a=x1
b=vx1
c=x2
d=vx2
e= sum of radii
t= time

e = (a+bt)-(c+dt)
e² = (a+bt)-(c+dt)* (a+bt)-(c+dt)

Multiply out

-2abt + 2ac + 2adt -a2 + 2bct + 2bdt2 -b2t2 -2cdt -c2 -d2t2 + e = 0

Rearrange

2bdt^2 - b2t^2 - d2t^2

-2abt + 2adt + 2bct – 2cdt

-a2 - c2 + 2ac+ e

= 0

Factorise

(2ab- b² - d²)t²

-2t(ab + ad + bc - cd)

-a² - c² + 2ac + e



This looks like a quadratic since if we remove all known variables we get

t² - 2t +c

Therefore we can solve using the x = (-b +- root(b² -4ac))/2a


Let x=t
Let a=A (a, b, d) = (2ab- b² - d²)
Let b=B (a, b, c, d) = (2(ab + ad + bc – cd))
Let c=C (a, c, e) = -a² - c² + 2ac + e

solve this for each plane we need to consider and find the closest positive time to zero.

iv done it!!

generalising the above forumula with 3d vectors

gives me the desired result

best regards
mark