Function is circle within circle and correction? - math

Question

I am trying to implement a mathematical procedure to ensure that a circle c1 is completely inside another circle c2.
It should work the following way:
Given c1(x, y, r) and c2(x, y, r) and c2.r>c1.r
return true if c1 is inside c2
return a vector V(x,y) being the minimum correction to apply to c1 so it is inside c2.
How does it look to you? should be easy for a mathematician or a physicist but it's quite hard for me.
I already tried an implementation in lua, but there's definitely something wrong on it.
local function newVector(P1, P2)
local w, h=(P2.x-P1.x), (P2.y-P1.y)
local M=math.sqrt(w^2 + h^2)
local alpha=math.atan(h/w)
return {m=M, alpha=alpha}
end
local function isWithin(C1, C2)
local V12=newVector(C1, C2)
local R12=C2.r-C1.r
local deltaR=R12-V12.m
if deltaR>=0 then
return true
else
local correctionM=(V12.m+deltaR) --module value to correct
local a=V12.alpha
print("correction angle: "..math.deg(a))
local correctionX=correctionM*math.cos(a)
local correctionY=correctionM*math.sin(a)
return {x=correctionX, y=correctionY}
end
end
Thanks!

Answer 1

Isn't it enough to check that distance(Center1, Center2) + Radius1 <= Radius2 ?
local function isWithin(C1, C2)
local distance = math.sqrt((C1.x-C2.x)^2+(C1.y-C2.y)^2)
return distance + C1.r <= C2.r + Epsilon
Epsilon is used in order to avoid numerical errors. (e.g. Epsilon = 1e-9)
Sounds easy this way.

Answer 2

Ok, finally I have it working fine.
The key was to use math.atan2 as lhf suggested. And I had some numerical mistake in the correction value.
Here's the final code. I have also included the circleFromBounds function that I use to create the circles from any corona display object.
local function newVector(P1, P2)
local w, h=(P2.x-P1.x), (P2.y-P1.y)
local M=math.sqrt(w^2 + h^2)
local alpha=math.atan2(h, w)
return {m=M, alpha=alpha}
end
local function isWithin(C1, C2)
local V12=newVector(C1, C2)
local epsilon = 10^(-9)
if (V12.m + C1.r <= C2.r + epsilon) then
return true
else
local correctionM=(C2.r-(C1.r+V12.m)) --module value to correct
local A=V12.alpha
local correctionX=correctionM*math.cos(A)
local correctionY=correctionM*math.sin(A)
return {x=correctionX, y=correctionY}
end
end
local function circleFromBounds(bounds, type)
local x, y, r
local width, height = (bounds.xMax-bounds.xMin), (bounds.yMax-bounds.yMin)
local ratio=width/height
x=bounds.xMin+width*.5
y=bounds.yMin+height*.5
if "inner"==type then
if ratio>1 then
r=height*.5
else
r=width*.5
end
elseif "outer"==type then
local c1, c2 = width*.5, height*.5
r = math.sqrt(c1^2 + c2^2)
end
return {x=x, y=y, r=r}
end
Thanks for your help!