do
	local pi			=math.pi
	local halfpi		=pi/2
	local cos			=math.cos
	local sin			=math.sin
	local acos			=math.acos
	local v3			=Vector3.new
	local nv			=v3()
	local cf			=CFrame.new
	local nc			=cf()
	local components	=nc.components
	local tos			=nc.toObjectSpace
	local vtos			=nc.vectorToObjectSpace
	local ptos			=nc.pointToObjectSpace
	local backcf		=cf(0,0,0,-1,0,0,0,1,0,0,0,-1)
	local lerp			=nc.lerp

	cframe.identity		=nc
	cframe.new			=cf
	cframe.vtws			=nc.vectorToWorldSpace
	cframe.tos			=nc.toObjectSpace
	cframe.ptos			=nc.pointToObjectSpace
	cframe.vtos			=nc.vectorToObjectSpace


	function cframe.fromaxisangle(x,y,z)
		if not y then
			x,y,z=x.x,x.y,x.z
		end
		local m=(x*x+y*y+z*z)^0.5
		if m>1e-5 then
			local si=sin(m/2)/m
			return cf(0,0,0,si*x,si*y,si*z,cos(m/2))
		else
			return nc
		end
	end

	function cframe.toaxisangle(c)
		local _,_,_,
			xx,yx,zx,
			xy,yy,zy,
			xz,yz,zz=components(c)
		local co=(xx+yy+zz-1)/2
		if co<-0.99999 then
			local x=xx+yx+zx+1
			local y=xy+yy+zy+1
			local z=xz+yz+zz+1
			local m=pi*(x*x+y*y+z*z)^-0.5
			return v3(m*x,m*y,m*z)
		elseif co<0.99999 then
			local x=yz-zy
			local y=zx-xz
			local z=xy-yx
			local m=acos(co)*(x*x+y*y+z*z)^-0.5
			return v3(m*x,m*y,m*z)
		else
			return nv
		end
	end

	function cframe.direct(c,look,newdir,t)
		local lx,ly,lz		=look.x,look.y,look.z
		local rv			=vtos(c,newdir)
		local rx,ry,rz		=rv.x,rv.y,rv.z
		local rl			=((rx*rx+ry*ry+rz*rz)*(lx*lx+ly*ly+lz*lz))^0.5
		local d				=(lx*rx+ly*ry+lz*rz)/rl
		if d<-0.99999 then
			return c*backcf
		elseif d<0.99999 then
			if t then
				local th	=t*acos(d)/2
				local qw	=cos(th)
				local m		=rl*((1-d*d)/(1-qw*qw))^0.5
				return		c*cf(
							0,0,0,
							(ly*rz-lz*ry)/m,
							(lz*rx-lx*rz)/m,
							(lx*ry-ly*rx)/m,
							qw
							)
			else
				local qw	=((d+1)/2)^0.5
				local m		=2*qw*rl
				return		c*cf(
							0,0,0,
							(ly*rz-lz*ry)/m,
							(lz*rx-lx*rz)/m,
							(lx*ry-ly*rx)/m,
							qw
							)
			end
		else
			return			c
		end
	end

	function cframe.toquaternion(c)
		local x,y,z,
			xx,yx,zx,
			xy,yy,zy,
			xz,yz,zz	=components(c)
		local tr		=xx+yy+zz
		if tr>2.99999 then
			return		x,y,z,0,0,0,1
		elseif tr>-0.99999 then
			local m		=2*(tr+1)^0.5
			return		x,y,z,
						(yz-zy)/m,
						(zx-xz)/m,
						(xy-yx)/m,
						m/4
		else
			local qx	=xx+yx+zx+1
			local qy	=xy+yy+zy+1
			local qz	=xz+yz+zz+1
			local m		=(qx*qx+qy*qy+qz*qz)^0.5
			return		x,y,z,qx/m,qy/m,qz/m,0
		end
	end

	function cframe.power(c,t)
--[[		local x,y,z,
			xx,yx,zx,
			xy,yy,zy,
			xz,yz,zz	=components(c)
		local tr		=xx+yy+zz
		if tr>2.99999 then
			return		cf(t*x,t*y,t*z)
		elseif tr>-0.99999 then
			local m		=2*(tr+1)^0.5
			local qw	=m/4
			local th	=acos(qw)
			local s		=(1-qw*qw)^0.5
			local c		=sin(th*t)/s
			return		cf(
						t*x,t*y,t*z,
						c*(yz-zy)/m,
						c*(zx-xz)/m,
						c*(xy-yx)/m,
						c*qw+sin(th*(1-t))/s
						)
		else
			local qx	=xx+yx+zx+1
			local qy	=xy+yy+zy+1
			local qz	=xz+yz+zz+1
			local c		=sin(halfpi*t)/(qx*qx+qy*qy+qz*qz)^0.5
			return		cf(
						t*x,t*y,t*z,
						c*qx,
						c*qy,
						c*qz,
						sin(halfpi*(1-t))
						)
end]]
		return lerp(nc,c,t)
	end

	--local power=cframe.power
	--[[function cframe.interpolate(c0,c1,t)
		--return c0*power(tos(c0,c1),t)
	end]]
	cframe.interpolate=lerp

	--local toquaternion=cframe.toquaternion
	function cframe.interpolator(c0,c1,c2)
		--[[if c1 then
			local x0,y0,z0,qx0,qy0,qz0,qw0=toquaternion(c0)
			local x1,y1,z1,qx1,qy1,qz1,qw1=toquaternion(c1)
			local x,y,z=x1-x0,y1-y0,z1-z0
			local c=qx0*qx1+qy0*qy1+qz0*qz1+qw0*qw1
			if c<0 then
				qx0,qy0,qz0,qw0=-qx0,-qy0,-qz0,-qw0
			end
			if c<0.9999 then
				local s=(1-c*c)^0.5
				local th=acos(c)
				return function(t)
					local s0=sin(th*(1-t))/s
					local s1=sin(th*t)/s
					return cf(
						x0+t*x,
						y0+t*y,
						z0+t*z,
						s0*qx0+s1*qx1,
						s0*qy0+s1*qy1,
						s0*qz0+s1*qz1,
						s0*qw0+s1*qw1
					)
				end
			else
				return function(t)
					return cf(x0+t*x,y0+t*y,z0+t*z,qx1,qy1,qz1,qw1)
				end
			end
		else
			local x,y,z,qx,qy,qz,qw=cframe.toquaternion(c0)
			if qw<0.9999 then
				local s=(1-qw*qw)^0.5
				local th=acos(qw)
				return function(t)
					local s1=sin(th*t)/s
					return cf(
						t*x,
						t*y,
						t*z,
						s1*qx,
						s1*qy,
						s1*qz,
						sin(th*(1-t))/s+s1*qw
					)
				end
			else
				return function(t)
					return cf(t*x,t*y,t*z,qx,qy,qz,qw)
				end
			end
		end]]
		if c2 then
			return function(t)
				return lerp(lerp(c0,c1,t),lerp(c1,c2,t),t)
			end
		elseif c1 then
			return function(t)
				return lerp(c0,c1,t)
			end
		else
			return function(t)
				return lerp(nc,c0,t)
			end
		end
	end

	function cframe.jointleg(r0,r1,c,p,a)
		local t=ptos(c,p)
		local tx,ty,tz=t.x,t.y,t.z

		--Calculate inverse kinemetics equation
		local d=(tx*tx+ty*ty+tz*tz)^0.5
		local nx,ny,nz=tx/d,ty/d,tz/d
		d=r0+r1<d and r0+r1 or d
		local l=(r1*r1-r0*r0-d*d)/(2*r0*d)
		local h=-(1-l*l)^0.5

		--Generate the natural quaternion for the shoulder.
		local m=(2*(1+h*ny+l*nz))^0.5
		local qw,qx,qy,qz=m/2,(h*nz-l*ny)/m,l*nx/m,-h*nx/m

		--If a, then rotate the natural quaternion by a.
		if a then
			local co,si=cos(a/2),sin(a/2)
			qw,qx,qy,qz=co*qw-si*(nx*qx+ny*qy+nz*qz),
				co*qx+si*(nx*qw-nz*qy+ny*qz),
				co*qy+si*(ny*qw+nz*qx-nx*qz),
				co*qz+si*(nz*qw-ny*qx+nx*qy)
		end

		--Generate the quaternion for the lower arm and return.
		local g=(d*l+r0)/(d*d+2*d*l*r0+r0*r0)^0.5
		local co=((1-g)/2)^0.5
		local si=-((1+g)/2)^0.5
		return c*cf(-r0*2*(qx*qz+qy*qw),
				r0*2*(qx*qw-qy*qz),
				r0*(2*(qx*qx+qy*qy)-1),
				co*qx+si*qw,
				co*qy+si*qz,
				co*qz-si*qy,
				co*qw-si*qx),
			c*cf(0,0,0,qx,qy,qz,qw)
	end

	function cframe.jointarm(r0,r1,c,p,a)
		local t=ptos(c,p)
		local tx,ty,tz=t.x,t.y,t.z

		--Calculate inverse kinemetics equation
		local d=(tx*tx+ty*ty+tz*tz)^0.5
		local nx,ny,nz=tx/d,ty/d,tz/d
		d=r0+r1<d and r0+r1 or d
		local l=(r1*r1-r0*r0-d*d)/(2*r0*d)
		local h=(1-l*l)^0.5

		--Generate the natural quaternion for the shoulder.
		local m=(2*(1+h*ny+l*nz))^0.5
		local qw,qx,qy,qz=m/2,(h*nz-l*ny)/m,l*nx/m,-h*nx/m

		--If a, then rotate the natural quaternion by a.
		if a then
			local co,si=cos(a/2),sin(a/2)
			qw,qx,qy,qz=co*qw-si*(nx*qx+ny*qy+nz*qz),
				co*qx+si*(nx*qw-nz*qy+ny*qz),
				co*qy+si*(ny*qw+nz*qx-nx*qz),
				co*qz+si*(nz*qw-ny*qx+nx*qy)
		end

		--Generate the quaternion for the lower arm and return.
		local g=(d*l+r0)/(d*d+2*d*l*r0+r0*r0)^0.5
		local co=((1-g)/2)^0.5
		local si=((1+g)/2)^0.5
		return c*cf(-r0*2*(qx*qz+qy*qw),
				r0*2*(qx*qw-qy*qz),
				r0*(2*(qx*qx+qy*qy)-1),
				co*qx+si*qw,
				co*qy+si*qz,
				co*qz-si*qy,
				co*qw-si*qx),
			c*cf(0,0,0,qx,qy,qz,qw)
	end
end