03-18-08 - 1

On Unit Interval Remappers

Some people have looked at the "controller/lerper" problem and mentioned interpolators. Interpolators are great things but they are not useful as controllers, because the target is not necessarilly constant, and even the "current" object is not under 100% control of the controller. The target can be jumping around, and hell the object may have other forces being applied to it as well. If you try to use an interpolator and just keep resetting it, you will have discontinuities and very ugly responses.

Anyway, it reminded me all the 0->1 : 0->1 float functions. Everybody knows so-called "smooth step" which IMO is more correctly called a hermite lerp because I'm sloppy and say "lerp" for any interpolation :

hermitelerp(t) = (3 - 2*t)*t*t
So, like you have your t in [0,1] which is your lerp parameter, and you want to get between two values A and B, instead of A + (B-A)*t , you use A + (B-A)*hermitelerp(t).

BTW amateurs can easily get into over using this. It's not necessarilly "better" than a lerp. Yes it is C1 (velocity continuous) if you require that the end points are zero velocity. It makes the curve slow at the ends and fast in the middle. Sometimes that's bad. I've seen people just tossing hermitelerps everywhere because it's "smoother" and of course that's retarded.

Of course very similar is

coslerp(t) = 0.5 - 0.5 * cos(t * pi)
which is C-inf if the end points are non-moving. Another problem with these of course is if you use them repeatedly to chase a target, you get really stair-steppy motion, you stop, speed up, slow to a stop, speed up, slow to a stop, it's dumb.

Anyway, we all know that, but if we think a bit more about our lerpers, it's obvious that there's this whole class of functions that remap the unit interval :

f : [0,1] -> [0,1]
f(0) = 0
f(1) = 1
A true remapper is also always in bounds and monotonically increasing in [0,1] :
0 <= f(t) <= 1 , for t in [0,1]
(d/dt) f(t) >= 0 , for t in [0,1]
though these last two strict requirements can often be relaxed depending on the application.

Of course the identity f(t) = t is your basic option.

I think of these as "response shapers". There are lots of times when you are really doing some kind of arbitrary response curve and you don't realize it. For example, say you want to show the user's health through color. You might do something like :

color = red + (green - red) * (health / health_max)
But really what you should have written was :
t = health / health_max;
color = red + (green - red) * response(t);
where response() is a unit interval reshaper. Of course the identity is one option, but it's not necessarily the best one.

It's particularly obvious that a response() curve belongs anywhere that you change units, because the meaning of values in one set of units is not necessarily linear in the other units. I don't really like the notion of sticking response() curves arbitrarily all over to fudge things within the same units - as long as you're in one set of units you should be able to do math to tell you what the curves should be like, but when you change units all bets are off.

One example is the one we used already - any time you convert a variable into a graphical indicator, such as color, brightness, the size of an explosion, etc. - those could easily get a response() curve. The other classic example is converting a mouse or gamepad stick to physical units. Instead of

torque = k * (stick deflection)
it should be
torque = k * response( stick deflection )

Now of course if your unit conversion is meters -> feet that's not what I mean by a unit change, and if it's something where you have a natural equation to convert units, that doesn't call for a response curve, but pretty much any time you change units using some non-physical unitful constant multiplier, it's suspicious. What makes you think those units are on the same scale that you can just use a linear conversion? You should assume you need a response curve, then you just might use the identity response.

Let's look at a few remappers. You can build a bag of tricks and get an intuition for these things and then start tossing them in the right places and understand how they affect control response.

First of all obviously you need to first map your values into 0->1 using something like :

float fmakelerper(float val,float lo,float hi) { return fclampunit( (val - lo) / (hi - lo) ); };
I wind up writing a lot of
flerp(response( fmakerlerper(val,lo,hi) ) ,lo,hi)

Now, given a few basic remappers, there are various transforms you can do which create new valid remappers.

Any time you have a remapper(t), you also have

remap2(t) = 1 - remap1( 1-t )
which is the reflection in X and Y (or the 180 rotation if you prefer). So for example if you have a curve that's always above the straight line, you can change that to a curve that's always below using 1-curve(1-t). And of course if you have any two remappers, then any lerp between remappers is also a remapper :
remap3(t,k) = k * remap1(t) + (1-k) * remap2(t)
This is a way you can expose shape control to the user or artist. Lerping a shaped curve with the identity curve is a good way to tweak the nonlinearity. Finally you can also remap a remap :
remap3(t) = remap1( remap2(t) )
Or any combination of these ways to make a new one.

The first ones that are obvious are the polynomials. Any (t ^ n) works, but we usually work with

square(t) = t*t
cube(t) = t*t*t
Intuition : these push the curve downward. That makes the response stay low longer and then shoot up at the end. This is useful for giving a linear ramp more "impact" and is used a lot in graphics. Linear ramps look boring, if you want a light to turn on over time t, something like square(t) is better. Obviously cube is the same thing but more extreme. These are also useful for providing fine control in UI elements like gamepad sticks or microscope zooms or camera focus; it makes small deflections even smaller, so that you can do very fine work, but it still lets you ramp up to a full speed change when you crank it.

Of course you can do powers of polynomials < 1 too , the basic gamma correction approximation is sqrt :

sqrt(t) = t^.5
intuitively this makes low values jump up fast, then it tails off slower. This is roughly how you convert light linear pixels to gamma corrected pixels. Of course there are also quadratic approximations of sqrt
approxsqrt(t) = t * (27 - 13*t)/14
which I optimized over [0,1]. That's slightly better than (2*t - t^2) but of course that super simple version is okay too. Note that the super simple version there is just the flip & inverse of square(t).
1 - square(1-t) = (2*t - t^2)

Then of course there are various kinds of parameterized warpers that you can expose to prefs to tweak response curves. One is the exponential :

explerp(t,k) = (e^(k*t) - 1)/(e^k - 1)

k > 0
for k very small this becomes a straight line (the identity)
as k gets larger the curve is pushed into the corner in the bottom right
for k very large it's a step function at t=1
and of course 1-exp(1-t) is useful too. (that's the same as using k < 0)

Then there's the quadratic spline remappers.

quad1(t,p) = t*t + 2*p*t*(1-t)

p in [0,0.5] is a true unit remapper
Of course this is just lerp of the identity and the quadratic remappers. More generally :
inline float fquadraticsplinelerper(const float f, const float ptX, const float ptY )
	ASSERT( fisinrange(ptX,0.f,1.f) );
	ASSERT( fisinrange(ptY,0.f,1.f) );
	ASSERT( !fequal(ptX,0.5f) );	//singularity here. 
	float bx = ptX;
	float a = (1 - 2.f*bx);
	float A = bx*bx + a*f;
	float t = (sqrtf(A) - bx)/a;
	float y = t*t + ptY*2.f*t*(1.f-t);
	return y;
will make a curve that goes through (ptX,ptY)

BTW I've only talked about unit remapping, but half-line remapping is very useful too and is a whole other topic. Two of the most useful are tanh(t) , which takes [0,inf] -> [0,1] smoothly, and log(t) which is [0,inf] -> [0,inf] but is very useful for putting things in linear importance scale when the relative magnitudes are what determine importance. For example both sound and image deltas have more linear information content in log scale.

Now we can get to what reminded me of this. Won sent me a link to Herf on Stopping . Mmmm, okay. He comes up with some funny curve. But if you look at the curve he wants, it looks just like a "smooth step" but all squished over to the left. Well, guess what, squishing to the left is exactly the kind of thing that remappers are good at. If we have some plot of a function F(x) in [0,1] and we want to make a new function that has the same endpoints but is squished to the left or the right or up and down - that's exactly what a remapper is. If you imagine having a picture of a checker board pattern, when you run one of these unit warpers, you're stretching some squares and squeezing others.

So, to make the kind of shape Herf wants, we can just use :

stopping(t) = hermitelerp( 1 - cube(1-t) )
and that looks pretty darn good. If you want more control you can use :
stopping(t) = hermitelerp( 1 - explerp(t,k) )

where k = -4 looks good to me

Just for more intuition building I'll show some other ways to get this kind of curve :

squareinv(t) = 1 - (1-t)^2

stopping(t) = hermitelerp( hermitelerp( squareinv(t) ) )
Each time you pass through hermitelerp it makes the ends slower and the mid faster, so doing it twice exaggerates that. squareinv skews the shape over to the left. So we make a shape with really flat ends and a steep middle, then we skew the whole thing to the left and that's the kind of stopping curve we want.
stopping(t) = squareinv( hermitelerp( squareinv(t) ) )
Similar, but the squareinv on the outside takes the shape and skews it upwards. Of course the squares could be cubes to be more extreme, or the squares could be lerped with identities to make them less extreme.

BTW it's a pet peeve of mine when people use physics terms to try to justify something totally hacky, especially when they get it wrong. There's absolutely nothing wrong with being unphysical and just saying "this function looks good". Herf says his "stopping" is based on physical friction; in fact he is not modeling "friction", he's modeling linear drag. Friction is not velocity dependent, it's a constant force (proportional to the normal force, which is proportional to mass in the normal case). Linear drag is a decceleration proportional to your velocity which produces the exponential slowing. Linear drag is the easiest diffeq anyone can solve : a = -k v.

While I'm ranting let me also note that linear drag is not what you usually see in nature. Drag in the air is mostly quadratic drag. That is a = - k v^2. You do see linear drag in the water at low speeds when you have laminar flow. BTW this is why the shape of boat hulls is really crucial - it's not just like a 10% change, it's the difference between laminar and turbulent flow, which changes your drag from linear to quadratic, which is like 1000X difference for fast boats. (it's actually way more complicated than this but that's the rough idea).

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