[Closed] arc between two points?

Gravey: Gosh, I miss the days of college maths… wait, maybe not all that much Anyway, the functions are just a few substitutions away. For a start, let’s say we switch the distance function for a lenght of vector.

fn circumcenter p1 p2 p3 = 
  (
  	local BC = length (p3 - p2)
  	local CA = length (p3 - p1)
  	local AB = length (p2 - p1)
  	
  	local u = BC^2*(CA^2+AB^2-BC^2)
  	local v = CA^2*(AB^2+BC^2-CA^2)
  	local w = AB^2*(BC^2+CA^2-AB^2)
  
  	barycentricToWorld p1 p2 p3 u v w
  )

This is just to make the transtion easier to follow, as we now use each of these vectors as a variable and the dot product of the vector with itself substitutes the squared distance (getting rid of squaring the square root):

fn circumcenter p1 p2 p3 = 
  (
  	local a = p3 - p2
  	local b = p3 - p1
  	local c = p2 - p1
  	
  	local u = dot a a * (dot b b + dot c c - dot a a)
  	local v = dot b b * (dot c c + dot a a - dot b b)
  	local w = dot c c * (dot a a + dot b b - dot c c)
  
  	barycentricToWorld p1 p2 p3 u v w
  )

Now, all the negative dot product terms can also be subsituted,

	dot a a == dot (b - c) (b - c)
  	dot a a == dot b b - 2 * dot b c + dot c c
  	
  	dot b b == dot (c - -a) (c - -a)
  	dot b b == dot c c - 2 * dot c -a + dot a a --> dot -a -a == dot a a
  
  	dot c c == dot (a - b) (a - b)
  	dot c c == dot a a - 2 * dot a b + dot b b

So the three terms are in the end:

	dot a a * (dot b b + dot c c - dot b b + 2 * dot b c - dot c c)
  	--> 2 * dot a a * dot b c
  
  	dot b b * (dot c c + dot a a - dot c c + 2 * dot c -a - dot a a)
  	--> 2 * dot b b * dot c -a
  
  	dot c c *  (dot a a + dot b b - dot a a + 2 * dot a b - dot b b)
  	--> 2 * dot c c * dot a b

No square root anymore, just basic multiplication and stuff – of course here it’s no big deal

what’s wrong with this one:

fn circumcenter p1 p2 p3 =
(
	fn barycentricToWorld p1 p2 p3 u v w = (u*p1 + v*p2 + w*p3) / (u + v + w)

	local a = p3 - p2
	local b = p1 - p3
	local c = p2 - p1

	local u = (dot a a) * (dot c b)
	local v = (dot b b) * (dot c a)
	local w = (dot c c) * (dot b a)

	barycentricToWorld p1 p2 p3 u v w
)

fastest, cleanest…