How do you use Bézier curves for envelopes, waves, yadda yadda?

[Jafo]

Short version: How do you code a curve you've drawn into C++?

Long version: I love cubic Bézier curves; they're subtle, beautiful, and much easier to shape than stringing together a bunch of math functions. If you keep everything within ±1000, as is the rule in PostScript fonts, you just draw your curve, move the decimal point one place to the left, and you've got three very usable decimal digits that fit perfectly into the ±1.000 range. Unfortunately, transforming these lovely curves into functions is a bit of a mystery... straight lines are easy (just some if-statements), but how about the actual curves themselves?

The Wikipedia article gives the equation

B(t) = (1-t)^3*P0 + 3(1-t)^2t*P1 + 3(1-t)^2*P2 + t^3*P3,

wherein P0 is the starting point, P1 and P2 are controls, and P3 is the end; and t is in the set of 0 and 1. How do you know when to use 0 or 1? Using 0 would simply turn the whole equation into 0, so that's probably not normal. Also, all four points have two values each (x and y), but the equation seems to use only one; do you run this twice, then, once for x and once for y?

[duncanparsons]

t is now close to are to P0 or P3
B(0) = P0
B(1) = P3

anything else is how far along the curve you want to be. If used in animation, if your motion followed the curve for 10 seconds, and you want 25fps, then that's 250 frames/values of t, one for each frame.

Code: Select all

t=0;
LOOP 0 -> 250
  image.MOVETO(B(t));
  t = t + (1/250);
  WAIT (0.04 sec);

You could see t as a smoothness parameter almost.

So to get an algo, you expand the equation you quoted, such that you have
x = [(1-t)^3 * P0.x] + [3(1-t)^2t * P1.x] ..etc and
y = [(1-t)^3 * P0.y] + [3(1-t)^2t * P1.y] ..etc

that will give you co-ords for each value of t you give. The resolution of the spacing between successive values of t is your choice

If you want to iterate along the x axis plotting ys, then you have work to do: I'll let you think about it first

HTH
DSP

[Jafo]

If you want to iterate along the x axis plotting ys, then you have work to do: I'll let you think about it first

Yikes -- that much recursion at sampleRate just can't be a good thing. Nice job of filling in the blanks, btw.

Drat -- yet another idea for dynamically changing a transfer curve has led me nowhere. I really thought I had something workable, too: I was going to have the control points move in response to the last sample's output. Ah well, back to the drawing board...

[duncanparsons]

not that hard, actually. If you use Newton/Raphson, you can solve t for a given x in very few iterations. If done wisely, one could calc t for every nth x, get the y, then interpolate inbetween.

On a slightly related note, have you ever looked at the Douglas Peucker algorithm. I haven't quite worked out where I want to use in a VST yet, but the potential for sample reduction and FSU is awesome!

[nollock]

You can just use a 1D bezier. Basicly each dimension is a function in terms of 't', where t is the distance along the curve, ranging from [0..1]

So instead of

x = f(t)
y = f(t)

You just do

x = t
y = f(t)

which simplifies to

y = f(x), where x is in the same range as t, [0..1]

So..

y(x) = (1-x)^3*Y0 + 3(1-x)^2x*Y1 + 3(1-x)x^2*Y2 + x^3*Y3


It basicaly works out the same as a 2 dimensional bezier with the control points for X set at [0, 1/3, 2/3, 1], which makes x a linear function which directly maps to t.

[duncanparsons]

yeah that's one way around it, but wouldn't have worked for me with MIDIMadness. I like the idea, I hadn't considered before because it wouldn't have worked for me, but I can see contexts where it could be good

[Z1202]

Why in the world would one use Bezier curves and not exponential segments for envelopes?

Regards,
{Z}

[duncanparsons]

? I wouldn't use them for envelopes, unless pan envelopes or somesuch. There's a wealth of things they could be used for tho, you just have to find them

[Z1202]

? I wouldn't use them for envelopes, unless pan envelopes or somesuch. There's a wealth of things they could be used for tho, you just have to find them

What I meant to say, exponential envelopes would sound more natural. What can you do with Bezier (from the things that you want to do for the envelopes, that is), thank you cannot do with exponential (1st or 2nd order)?

Regards,
{Z}

[mdsp]

What I meant to say, exponential envelopes would sound more natural. What can you do with Bezier (from the things that you want to do for the envelopes, that is), thank you cannot do with exponential (1st or 2nd order)?

Regards,
{Z}

What do you mean by 2nd order exponential?

[Z1202]

What I meant to say, exponential envelopes would sound more natural. What can you do with Bezier (from the things that you want to do for the envelopes, that is), thank you cannot do with exponential (1st or 2nd order)?

Regards,
{Z}

What do you mean by 2nd order exponential?

Step response of a 2nd order nonresonating filter. Somewhat exotic for an envelope, I guess, but possible.

Regards,
{Z}

[mdsp]

Step response of a 2nd order nonresonating filter. Somewhat exotic for an envelope, I guess, but possible.

ah I thought I had missed something...

Here's what I found though googling for 2nd order exponentials.
http://demonstrations.wolfram.com/Conse ... tialDecay/

Nothing new, simple but effective. A good physically smooth and natural Attack/Decay envelope as an alternative to traditionnal synth enveloppes.

[Z1202]

Step response of a 2nd order nonresonating filter. Somewhat exotic for an envelope, I guess, but possible.

ah I thought I had missed something...

Here's what I found though googling for 2nd order exponentials.
http://demonstrations.wolfram.com/Conse ... tialDecay/

Nothing new, simple but effective. A good physically smooth and natural Attack/Decay envelope as an alternative to traditionnal synth enveloppes.

Looks more like impulse responses.

[Jafo]

Why in the world would one use Bezier curves and not exponential segments for envelopes?

Probably not for envelopes, no, but for other things. I'm looking for a way to create dynamic, tubelike saturation, which means transfer functions that change their shapes, and I'm working with what I know. I've seen differential equations from afar, but my math only went up to calculus, and that was back in the late '80s. I've tried writing compound functions, but I lack the math to eliminate discontinuities in every case. Cubic Béziers, however, which I use in designing fonts, are easy to visualize, easy to manipulate (with a mouse, at least), always continuous, and quite hip.

Basically, I need something with a slow rise from zero up to a certain point, then a rapid rise to a softish clip. The catch is that I want the rise and the hardness of the clipping to vary with what's come before. Béziers would work beautifully -- move the control points by a fraction of the previous sample's output.

For an example, here's what I was going to try as an experiment. In the positive phase, the curve goes from 0,0 to 700,700, with curves at 557,52 and 370,700 (a straight connection from the final point). Err, 0.700 and all that -- I'm thinking of drawing PostScript fonts. Multiply the previous sample's output by 0.1, and rotate the control points so that they're that many units (err, tenths) closer to the upper left-hand of the graph -- a faster, sharper curve.

*Shrug* Beyond my capabilities, alas. Maybe I'll play with manipulating sigmoids instead; their slopes are easy to vary.

[Music Engineer]

Step response of a 2nd order nonresonating filter. Somewhat exotic for an envelope, I guess, but possible.

ah I thought I had missed something...

Here's what I found though googling for 2nd order exponentials.
http://demonstrations.wolfram.com/Conse ... tialDecay/

Nothing new, simple but effective. A good physically smooth and natural Attack/Decay envelope as an alternative to traditionnal synth enveloppes.

i semi-recently derived some stuff that might be useful to put this 'consecutive exponential decay' to work:

www.rs-met.com/documents/notes/AttackDecayEnvelope.pdf

basically, this could also be implemented as one 2nd order filter with two real poles - and the (whole) envelope would then be the impulse-response of this filter. but i think Z was talking about a 2nd order filter with a complex pole pair and then we would use its step-response for segments? rather exotic indeed. did anyone actually try this? does it sound good?