How change the spiral movement in 2D space?

I have two points in 2D space as we can see in the figure to move from the blue point to the red points we can use the equation (1). Where b is a constant used to limit the shape of the logarithmic spiral, l is a random number in [−1,1], which is used to control the indentation effect of the movement, D indicates the distance between blue points and the current point

I need another movement that can move from blue points to the red points like in the figure

You can use sinusoidal model.

For start point (X0, Y0) and end point (X1,Y1) we have vector end-start, determine it's length - distance between points L, and angle of vector Fi (using atan2).

Then generate sinusoidal curve for some standard situation - for example, along OX axis, with magnitude A, N periods for distance 2 * Pi * N:

Scaled sinusoid in intermediate point with parameter t, where t is in range 0..1 (t=0 corresponds to start point (X0,Y0))

X(t) = t * L
Y(t) =  A * Sin(2 * N * Pi * t)

Then shift and rotate sinusoid using X and Y calculated above

X_result = X0 + X * Cos(Fi) - Y * Sin(Fi)
Y_result = Y0 + X * Sin(Fi) + Y * Cos(Fi)

Example Python code:

import math
x0 = 100
y0 = 100
x1 = 400
y1 = 200
nperiods = 4
amp = 120
nsteps = 20
leng = math.hypot(x1 - x0, y1 - y0)
an = math.atan2(y1 - y0, x1 - x0)
arg =  2 * nperiods* math.pi
points = []
for t in range(1, nsteps + 1):
    r = t / nsteps
    xx = r * leng
    yy =  amp * math.sin(arg * r)
    rx = x0 + xx * math.cos(an) - yy * math.sin(an)
    ry = y0 + xx * math.sin(an) + yy * math.cos(an)
    points.append([rx, ry])
print(points)

Draw points: