helix-shaped movement of an object in Matlab - matlab

Question

I have found this equation in a paper which represents a helix-shaped movement of an object:
When I plotted the S vector in Matlab I got a different result, not helix shape.
Where is the problem, is it in the equation or in the code?
l is a random number in [-1,1]
r is a random vector in [0,1]
b is a constant for defining the shape of the logarithmic spiral.
Matlab code:
dim =3;
Max_iter =10;
X_star=zeros(1,dim);
ub = 100;
lb = -100;
X=rand(1,dim).*(ub-lb)+lb;
S = [];
t=0;
while t<Max_iter
a=-1+t*((-1)/Max_iter);
r=rand();
b=1;
l=(a-1)*rand + 1;
for j=1:size(X,2)
D= abs(X_star(j) - X(1,j));
X(1,j)= D * exp(b.*l).* cos(l.*2*pi) + X_star(j);
end
X_star=X(1,:);
S = [S X];
plot(S);
t = t+1;
end

Answer 1

that equations does not look like helix at all. Lets do this for 3D. First some definitions:
p0,p1 - centers of start and end of the helix 3D points
r - scalar radius of the helix
m - scalar number of screws (if not integer than the screws will not start and end at the same angular position)
We first need TBN vectors:
n = p1-p0
t = (1,0,0)
if (|dot(n/|n|,t)|>0.9) t = (0,1,0)
b = cross(n,t)
b = r*b / |b|
t = cross(b,n)
t = r*t / |t|
And now we can do the helix using TBN as basis vectors:
// x' y' z'
p(a) = p0 + t*cos(2*Pi*m*a) + b*sin(2*Pi*m*a) + n*a
Where a = <0.0,1.0> is the parameter which point on helix you want. If you move out of this interval you will extrapolate the helix before p0 and after p1. You can consider a is the time...
As you can see the equation is just 2D parametric circle (x',y') + linear movement (z') and transformed by basis vectors to p0,p1 direction and position.
If you need equations/code for the dot,cross and abs value of 3D vectors see:
Understanding 4x4 homogenous transform matrices
They are at the bottom of that answer of mine.