Data Points Joined by Spiral

[monirg]

Hello;

I've non-monotonic analytically derived data points: (x,y)i, i=1, n
I would like to join the points by a smooth curve with the following constraints:
a. points are joined in the same order, i.e.; pnt# 1, pnt# 2, ..., pnt# N
b. the curve progresses in clockwise direction ONLY starting at pnt # 1
c. the curve does not cross itself
d. the curve has a minimum possible turns
e. the curve has a known (dy/dx) at pnt # n

By examining the data, the points appear to lie on a spiral of some kind.

Here're two sets of data points for illustration purposes
(Ref. HTK006,T*=0.08,T*=0.05):
SET # 1:
n = 5, (dy/dx) at pnt# 5 = - 4.1343
pnt#1 x = - 0.2062 , y = 0.9369
pnt#2 x = - 0.1937 , y = 0.9454
pnt#3 x = - 0.2043 , y = 0.9263
pnt#4 x = - 0.1797 , y = 0.9640
pnt#5 x = - 0.1542 , y = 0.9309

SET # 2:
n = 4, (dy/dx) at pnt# 4 = - 8.6897
pnt#1 x = - 0.1206 , y = 0.9698
pnt#2 x = - 0.1292 , y = 0.9470
pnt#3 x = - 0.1282 , y = 0.9785
pnt#4 x = - 0.0974 , y = 0.9508

(n=3 and n=10 are generally the min and max data points per set)

How can one develop such spiral ???

Your expert opinion would be greatly appreciated.

 

[Damon Ostrander]

Hi Monir,

I don't know if this is what you are after, but perhaps you will at least find it interesting and can perhaps glean some ideas:

Sub DrawSpiral()
'
' Written 1/14/99 by Damon Ostrander
'
    Const A = 5
    Const B = 1#
    Const TwoPi = 2 * 3.1415926535
    Const nRevs = 4
    
    Dim x0      As Single
    Dim y0      As Single
    Dim x       As Single
    Dim y       As Single
    Dim R       As Double
    Dim Theta   As Double
    
        x0 = 250
        y0 = 350
    
    With ActiveSheet
        
        'Uh oh.  Here comes some math!  Have to calculate x and y
         
        For Theta = TwoPi / 30 To nRevs * TwoPi Step TwoPi / 30
        
            R = A + B * Theta
            
            x = x0 + R * Sin(Theta)
            y = y0 - R * Cos(Theta)
            
            .Shapes.AddLine(x0, y0, x, y).Select (False)
            
            x0 = x
            y0 = y
            
        Next Theta
     
   End With
   
   ActiveWindow.Selection.ShapeRange.Group
    
End Sub

Of course, it is quite easy to plot a spiral from the data you have using a Scatter type plot with data smoothing (perhaps you have already done this) and to use a similiar algorithm to the above directly on a worksheet to calculate the spiral point coordinates.

Keep Excelling.

Damon