What is this method of interpolation / blending?

Question

I am reverse-engineering two classes from a game. The two classes are responsible for generating sets of t values for interpolating / blending between two points through the formula:

P = P0 * (1 - t) + P1 * t

Both classes have the same interface, that is:

class CalcRatio:
    start(blend_duration)  # In frames
    calc()                 # Called once every frame

The first class is implemented as follows:

start(blend_duration):
    t = 0
    counter = 0
    inv_duration = 1 / blend_duration

calc():
    if t == 1:
        return

    counter += inv_duration
    if counter >= 1:
        t = 1
        counter = 1
        inv_duration = 0
        return

    t += inv_duration

This is very clearly plain linear interpolation where P = P0 * (1 - t) + P1 * t, t = frame / duration and frame is the current frame number since calling start(). (I do not even know why the game even needed this class.)

However, I do not understand the second class. It is implemented as follows:

start(blend_duration):
    t = 0
    counter = 0
    unk = 1
    inv_duration = 1 / blend_duration

calc():
    if unk == 0:
        return

    counter += inv_duration
    if counter >= 1:
        t = 1
        counter = 1
        unk = 0
        return

    prev_unk = unk
    unk -= prev_unk * counter * counter
    unk2 = 1 / (unk / prev_unk + (1 - unk))
    t = (1 - unk) * unk2
    # Interesting fact: 1 - t == (unk / prev_unk) * unk2

I do not understand what this is doing. The new point is still calculated as P = P0 * (1 - t) + P1 * t, but any idea what t is in terms of frame? What are unk and unk2 supposed to represent?