Spiralling String Curve

by Jakob Schmid 2004-08-14

Task:  We want to make a S-shaped parameter curve

First we make a simple circular curve:

>	plot([ cos(t), sin(t), t=0..2*Pi], axes=none);

We can easily make it into a spiral:

>	plot([ cos(t)*t, sin(t)*t, t=0..2*Pi], axes=none);

We can also translate the circular curve diagonally by adding t  to x  and y :

>	plot([ cos(t*3*Pi)+t, sin(t*3*Pi)+t, t=0..Pi], axes=none);

We want to make the rotation change direction in the middle of the curve, so we use sin(t)  as the running variable instead of t:

>	plot([ cos(sin(t)*3*Pi)+t, sin(sin(t)*3*Pi)+t, t=0..Pi], axes=none);

We make it spiral by multiplying with t:

>	plot([ cos(sin(t)*3*Pi)*t+t, sin(sin(t)*3*Pi)*t+t, t=0..Pi], axes=none);

The spiralling radius should be low at the beginning, high in the middle and low in the end. We do this by multiplying with sin(t)  instead of t (actually, 2sin(t) , but that's just an adjustment)

>	plot([ cos(sin(t)*6*Pi)*2*sin(t)+t, sin(sin(t)*6*Pi)*2*sin(t)+t, t=0..Pi], axes=none);

If we parameterize the above formula, we get:

>	dc:=1:          # direction changes sr:=6:          # spiralling rounds r:=2:           # spiral relative radius ss:=1:          # spiral radius change speed tx:=1: ty:=1: # translation of x and y C:=[cos(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*tx, sin(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*ty, t=0..Pi];

Changing the parameters lead to many wonderous shapes:

>

dc:=1:          # direction changes sr:=4:          # spiralling rounds r:=2:           # spiral relative radius ss:=3:          # spiral radius change speed tx:=1: ty:=-1:  # translation of x and y C:=[cos(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*tx, sin(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*ty, t=0..Pi]: plot(C, axes=none);

>

dc:=1:         # direction changes sr:=10:        # spiralling rounds r:=0.9:        # spiral relative radius ss:=11:        # spiral radius change speed tx:=1: ty:=1:  # translation of x and y C:=[cos(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*tx, sin(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*ty, t=0..Pi]: plot(C, axes=none);

The snake shape is made with these parameters (also note the change in the interval t runs):

>

dc:=1:         # direction changes sr:=2:         # spiralling rounds r:=1.3:        # spiral relative radius ss:=1.9:       # spiral radius change speed tx:=.8: ty:=1: # translation of x and y C:=[cos(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*tx, sin(sin(t*dc)*sr*Pi)*r*sin(ss*t)+t*ty, t=.7..2.3]: plot(C, axes=none, scaling=CONSTRAINED);

We'll differentiate x(t)  and y(t)  to be able to calculate the tangent line to any point:

>

Dc:=1:         # direction changes Sr:=2:         # spiralling rounds R:=1.3:        # spiral relative radius Ss:=1.9:       # spiral radius change speed Tx:=.8: Ty:=1: # translation of x and y f  := t -> Pi*Sr*sin(Dc*t): x  := t -> cos(f(t))*R*sin(Ss*t)+t*Tx: y  := t -> sin(f(t))*R*sin(Ss*t)+t*Ty: The derivatives were done by hand (didn't seem like Maple was up to the task of differentiating x(t)  and y(t) ): Dx := t -> R*(cos(f(t))*cos(Ss*t)*Ss            - sin(f(t))*Pi*Sr*cos(Dc*t)*Dc*sin(Ss*t))+Tx: Dy := t -> R*(sin(f(t))*cos(Ss*t)*Ss            + cos(f(t))*Pi*Sr*cos(Dc*t)*Dc*sin(Ss*t))+Ty: C:=[x(t), y(t), t=.7..2.3]:

The derivates look like this:

>	plot([Dx(t), Dy(t)], t=.7..2.3, axes=none);