Pivoting

There are three main functions that "pivot" between target functions. They are softmaxshift, logsumshift, and pivot.

There are two key differences:

1. Pivot does not care about direction (increasing/decreasing), after the pivot it always returns the second function. However, the other two are max/min functions, and so do not strictly pivot. If the second function decreases drastically after the pivot, then the first function may take over again.
2. The shape of the smoothed equation around the inflection point differs. For pivot, it is a hard shift. For softmaxshift, it first decreases and then jumps upwards to the second function. For logsumshift, it smoothly increases from the first to the second function. See https://www.desmos.com/calculator/3ipxfolpzf

Easing

There are several ways to ease functions.

The standard method is via the ease and eased functions. These lerp into a target function and always use time as a controller. When using for movement, use ease for offset and eased for velocity.

Both of these make the movement start and end slowly. Note that the signature is slightly different: ease takes a lambda as argument, since it can be run with any lambda, and eased takes a string as argument, since it can only be run with expression functions in-engine that have well-defined derivatives. (In BDSL1, just the raw text eiosine is used in both cases.)

move 2 nroffset px ease($(eiosine), 2, sine 2 3 t)
move 2 nrvelocity px eased("eiosine", 2, cosine 2 3 t)

Note that these two functions asymptote towards the target function. They do not return a 0-1 value.

You can also invoke easing functions directly. For example, eiosine(t) is a distorted identity function with a slow entry and a slow exit. While ease and eased apply clamping to the time, direct invocations to easing functions do not apply clamping. You can get clamping by doing eiosine(clamp01(t)).

Lasers and Extra Parametrization

Entities may want to parametrize their movement functions by information not contained in the BPI struct. For lasers, the "laser lifetime" is not stored in the BPI struct. (t is set to the time along the draw-path of the laser instead.)

Aliasing provides a solution. The idea is that we take the extra parameters, alias them to fixed shortcuts, and then execute a wrapped VTP. The VTP can access any of the extra parameters via the reference command. There are fixed functions in the engine that retrieve these extra parameters.

Here is a table of automatic aliases:

Entity	Alias	Value
Laser/LPath	LT()	Laser lifetime
Path	CosRot()	Velocity cosine
Path	SinRot()	Velocity sine
Path	Angle()	Velocity angle
Path	Root()	Root location of velocity

Generation Context

When inside a repeater function (GTRepeat/GIRepeat/GCRepeat/GSRepeat), you have access to GenCtx, which is a running accumulator of values modified by the repeater functions.

GenCtx provides these automatic aliases in GCXF<T> functions. Note that the & prefix is only required for BDSL1.

Alias	Description
t (bpi.t)	gcx.i -- iteration number
p (bpi.index)	gcx.index -- firing index as determined by p modifiers
loc (bpi.loc)	gcx.beh.globalLoc -- location of executor
&i	gcx.i -- iteration number
&pi	gcx.pi -- parent interation number
&times	number of times the repeater is running for
&rv2	running RV2 value
&brv2	parent RV2 value
&st	gcx.summonTime
	Automatic Bindings
&lr/&ud	1 when iteration number is even and -1 when odd (bindLR/bindUD)
&rl/&du	-1 when iteration number is even and 1 when odd (bindLR/bindUD)
&axd	Arrow formation X index (bindArrow)
&ayd	Arrow formation Y index (bindArrow)
&aixd	Inverted arrow formation X index (bindArrow)
&aiyd	Inverted arrow formation Y index (bindArrow)
&angle	rv2.a (bindAngle)

Variables inside the GenCtx can be accessed several ways:

• Within GCXF<T> (eg. repeater modifiers), use VARNAME (BDSL2) or&VARNAME (BDSL1). This operates via lexical scoping on the environment frame within the GCX.
• Within BPY/TP/Pred/VTP/etc, use VARNAME (BDSL2) or &VARNAME (BDSL1). This operates via lexical scoping on the environment frame within the ParametricInfo struct.
• Within unscoped functions (eg. bullet controls), use &VARNAME (BDSL2) or &.VARNAME (BDSL1). This operates via dynamic lookup on the environment frame. Variables cannot be exposed automatically since the functions are unscoped.