Re: Difficulty with a Spiral Equation !

On Nov 9, 8:28 pm, monir <mon...@xxxxxxxxxxxx> wrote:
Hello;

1) I've 6 analytically derived data points (T*=0.09)

What does "(T*=0.09)" mean in this context?

and the 1st
derivative at the last point:
i x y
1 -0.236435 0.937134
2 -0.232600 0.951276
3 -0.233333 0.926882
4 -0.242256 0.955982
5 -0.228409 0.974592
6 -0.211085 0.949008 & (dy/dx)= - 7.2113388

2) My analytical model postulates that the above 6 points lies on a
smooth CLOCKWISE spiral (with no intersecting turns) joining pnt# 1,
pnt# 2, ..., pnt# 6 in the same order.

Does your model not tell you anything about the functional form of the
spiral then? There will be infinitely many ways to do this, so it's
hard to know where to start, and hard to see how "absolutely any
spiral that fits" could make sense in the context of whatever it is
you're doing.

Looking at the points, it doesn't seem to me as if you're going to get
a "nice" spiral -- it'll be squashed and mis-shapen. Is that what
you're expecting?

Couple of other things: as far as I can see, your example below *does*
have intersecting turns after it leaves point 6. Or do you only
require the spiral to be non-intersecting on its path from point 1 to
point 6? Do you have any requirements on the number of turns between
points? For example, your curve below makes approximately a quarter-
turn between points 1 and 2, but it could in theory make 10 and a
quarter, or a hundred and a quarter, or whatever... And finally,
presumably you don't care where the centre is? Is that right?


3) I've tried a number of possible spiral formulations with no
success. The most promising attempt was to represent the spiral by
the equation:
r = (a + b.th + c.th^2 + d.th^3).Exp(m.th)
with its centre O at (f,g)
"r" is the distance from the centre O to pnt # i, i=1, 6
"th" is the angle measured clockwise from the vector: O(f,g) ----> pnt
# 1
(obviously, "th" for pnt # 1 is zero)

4) So we have 7 unknowns: a, b, c, d, m, f, g
and 7 conditions: 6 points i=1, 6 and the slope (dy/dx) at the
last point i=6

5) I couldn't analytically solve the problem !!! The best I could
get:
a = 0.0094859
b = - 0.0020353
c = 0.0002411
d = - 0.0000122
m = 0.2693740
f = - 0.2281596
g = 0.9417704
The spiral looks good but has one critical problem! It refuses to
pass through pnt # 5 !!!! It passes through pnt 1, 2, 3, 4, 6 and
satisfies the slope condition at pnt 6.

6) The beauty of the above spiral formula (item 3.) is its flexibility
in accommodating 5-pnt, 4-pnt, and 3-pnt spirals:
> for a 5-point + {(dy/dx) at i=5}, one drops the "d" term;
> for a 4-point + {(dy/dx) at i=4}, then one drops the "c" & "d"
terms ; and
> for a (min) 3-point + {(dy/dx) at i=3}, one drops the terms "b",
"c" & "d"

7) There might be a robust analytical way to derive the 7 unknowns a,
b, c, ..., etc., and/or a better spiral representation, or even having
pnt # 1 as its centre !!

Your expert help would be greatly appreciated. Thank you.
Monir


.

Relevant Pages

  • Re: Difficulty with a Spiral Equation !
    ... smooth CLOCKWISE spiral joining pnt# 1, ... have intersecting turns after it leaves point 6. ... I think this is made easier by stretching in the x-direction, ... can get some r-theta plots that look nice and smooth and amenable to ...
    (sci.math)
  • Re: Difficulty with a Spiral Equation !
    ... smooth CLOCKWISE spiral joining pnt# 1, ... have intersecting turns after it leaves point 6. ... I think this is made easier by stretching in the x-direction, ...
    (sci.math)
  • Re: Difficulty with a Spiral Equation !
    ... smooth CLOCKWISE spiral joining pnt# 1, ... have intersecting turns after it leaves point 6. ... presumably you don't care where the centre is? ... the sample 6-point data set. ...
    (sci.math)