Using Phase Shifts to Modify a Sine Curve

Date: 06/30/2005 at 11:03:53
From: John
Subject: Fattening a Sine wave.

Hi.  What modifiers can be used to fatten a sine wave while not 
affecting the zeros and peaks?  It seems the addition of a cos(angle) 
to the angle in the sin() can do this, but I have difficulty 
controlling the rest of the wave.

Date: 06/30/2005 at 12:47:25
From: Doctor Douglas
Subject: Re: Fattening a Sine wave.

Hi John.

You can think about this problem in the following way.  First, 
consider the graph of the sine wave:

  y(t)
   |
   |   cC                             This is crude ASCII art;
   | B     D                          view using a monospaced font.
   |        
  -A----+--e-E----+----I------>  t
   |             
   |           F     H
   |              Gg
   |

Curve ABCDEFGHI is your regular sine wave, which has zeroes at
t = 0, pi, 2*pi (points A,E,I respectively).  The peak is at t = pi/2 
(point C) and the valley is at t = 3*pi/2 (point G).

Now think about pulling the curve to get the shape you want.  For 
example, maybe you want to make the peak C occur earlier (say at point 
c) and G be later (at g).  Let us also advance point E by an amount 
twice the shift of c and g (point e).  Note that these movements are 
HORIZONTAL movements of the peak, valley, and zero-crossing.  You also 
can define horizontal movements of other points such as D to d and H 
to h (not shown), to keep a smooth curve.  Points A and I must remain 
fixed.  The new curve is AbcdefghI in the above diagram (although 
points b,d,f and h are not shown).

The function that goes through these points is the following:

    y = A*sin[2*pi*t + phi(t)]

where the phase phi(t) has the following properties:

    1.  If phi(t) = 0, the point is unshifted.
    2.  If phi(t) > 0, the point is advanced. (Wwhy?  If phi>0, then
        it takes a smaller value of t to arrive at that level.)
    3.  If phi(t) < 0, the point is retarded.

Now you can use your graph of y vs t to construct a set of points 
through which phi must pass.  You can make phi continuous, or 
piecewise linear, or use whatever interpolation method you want, as 
long as it goes through all of the control points {A,b,c,...} that you 
need.  It's usually wise to keep |phi(t)| small (certainly no more 
than 2*pi) so that your resulting sine wave is a gentle distortion, 
rather than being ripped apart into some unrecognizable shape.

For the above graph of y vs t, the graph of phi will look something
like this:

 phi(t)
   |         e                      phi(ABCDEFGHI) is identically 0.
   |       d 
   |    c                           phi(Abcdefghi) is some messy
   | b         f                    but continuous curve.  
  -A-B--C--D-E-F--G--H-I----        
   |                 h              
   |              g                 

Any graph that goes through AbcdefghI will work.  You can probably see 
how one could define phi as a piecewise linear function, as in the 
example in this archived answer:

  Irregular Sinusoidal Curves
    http://mathforum.org/library/drmath/view/65463.html 

But it doesn't have to be that way if you don't want.  For example, if 
the zero-crossing point E is *unshifted*, you can see that we get a 
graph of phi(t) that could look like this:

   |        
   |    c                           
   | b     d                         
  -A-B--C--D-E-F--G--H-I----        
   |           f     h              
   |              g                 

That is, phi(t) could have the form -K*sin(t), where the constant
K is probably not too large.  Now you can write y(t) in one line:
y = A*sin[2*pi*t - K*sin(t)].  This function will have an advanced
peak and a retarded valley.

You question above is how to "fatten a sine wave while not affecting
the zeros and peaks".  I'm not exactly sure what you mean, but perhaps 
it is a distortion of the form

  y(t)
   |
   |    C
   |bB     Dd
   |        
  -A---------E---------I------>  t
   |             
   |          fF     Hh
   |              G

where the zero-crossings A,E,I are unaffected, as are the peak C and 
the valley G.  The excursions BCD and FGH are "fattened" to bCd and 
fGh.  If you go through the graphing of phi in the same way as the 
example above, you get a graph that looks something like this:
 
   |                           
   |                            Points b and f are advanced.     
   | b         f                Points d and h are retarded.     
  -A-B--C--D-E-F--G--H-I----        
   |       d         h              
   |                               

Kind of looks like a sine-wave with twice the frequency, eh? Try 
y(t) = y = A*sin[2*pi*t - K*sin(2*t)] and see what happens.  Of course 
you could also make a piecewise linear function that goes through 
AbCdEfGhI in the last graph above.

I should remark that this is not the only way to generate distortions 
of a sine wave.  Another method is to write the distortion in terms of 
Fourier Series:

  y(t) = A*sin[2*pi*t] + B*sin[4*pi*t + b] + C*sin[4*pi*t + c] 
             + ...
If the various harmonics and their amplitudes {B,C,...} are somehow
important, then Fourier series is the way to go (this leads to
mathematics that involves lots of integral calculus).  But if you are 
drawing smooth computer graphics distortions, then maybe the phase 
modulation method is simpler to implement, requiring only a good 
working knowledge of the trigonometric functions.  It really depends 
on the nature of the problem and what you're trying to do.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/30/2005 at 13:25:24
From: John
Subject: Thank you (Fattening a Sine wave.)

You are horrifyingly brilliant.  Thank you for all the help.  I've
come up with:

m*sin(4*w * t/d - K * sin(2*4*w*t/d - k * sin(2*4*w*t/d)))

It's ugly, but embodies the needed skew and fattening and twisting to
nudge a sine wave onto the difference curve between angular and NURB
spaces representing a circle.  Thanks again!

John