Creating a path for objects to follow

Here's a function for plotting bezier curves. cp0-cp3 are your control points and t is a value from 0 to 1. The idea is to decide how many steps will be needed (the more steps, the smoother the line) then call BezierPoint to get the x-y point at that step.

Type TPoint
 Field X:Int
 Field Y:Int
End Type

Function BezierPoint:TPoint(t:Float, cp0:TPoint, cp1:TPoint, cp2:TPoint, cp3:TPoint)
Local ax:Float, bx:Float, cx:Float
Local ay:Float, by:Float, cy:Float

Local TSquared:Float, TCubed:Float
Local PointResult:TPoint = New TPoint

cx = 3.0 * (cp1.x - cp0.x)
bx = 3.0 * (cp2.x - cp1.x) - cx
ax = cp3.x - cp0.x - cx - bx
        
cy = 3.0 * (cp1.y - cp0.y)
by = 3.0 * (cp2.y - cp1.y) - cy
ay = cp3.y - cp0.y - cy - by         
tSquared = t * t;
tCubed = tSquared * t;
        
Pointresult.x = (ax * tCubed) + (bx * tSquared) + (cx * t) + cp0.x;
Pointresult.y = (ay * tCubed) + (by * tSquared) + (cy * t) + cp0.y;
        
Return Pointresult
End Function

Here is an example program using it.

Const NumPoints = 100
Const td:Float = 1.0/NumPoints
Include "Bezier.bmx"

Graphics 640,480,32

Local cp0:TPoint = New TPoint
Local cp1:TPoint = New TPoint
Local cp2:TPoint = New TPoint
Local cp3:TPoint = New TPoint
Local result:TPoint

cp0.x = 0
cp0.y = 300
cp1.x = 100
cp1.y = 200
cp2.x = 500
cp2.y = 200
cp3.x = 600
cp3.y = 300

While Not KeyHit(KEY_ESCAPE)
 If KeyDown(KEY_LEFT)
  cp1.x :- 1
 End If
 If KeyDown(KEY_RIGHT)
  cp1.x :+ 1
 End If
 If KeyDown(KEY_K)
  cp2.x :- 1
 End If
 If KeyDown(KEY_L)
  cp2.x :+ 1
 End If
 If KeyDown(KEY_UP)
  cp1.y :- 1
 End If
 If KeyDown(KEY_DOWN)
  cp1.y :+ 1
 End If
 If KeyDown(KEY_O)
  cp2.y :- 1
 End If
 If KeyDown(KEY_COMMA)
  cp2.y :+ 1
 End If


 Cls
 SetColor 255,255,255
 For i = 0 Until numpoints
  t:Float = i*td
  result = BezierPoint(t,cp0,cp1,cp2,cp3)
  Plot(result.x,result.y)
 Next
 SetColor 255,0,0
 DrawRect(cp0.x-2,cp0.y-2,4,4)
 DrawRect(cp1.x-2,cp1.y-2,4,4)
 DrawRect(cp2.x-2,cp2.y-2,4,4)
 DrawRect(cp3.x-2,cp3.y-2,4,4)
 FlushMem(); Flip
Wend

Just use the arrow keys and KLO, to move the control points around