Exponential Spirals for Game Effects May 4th, 2009
Patrick Stein

In earlier posts, I mentioned finding polynomials, riffing off of damped harmonic motion, and then hitting on exponential spirals all trying to come up with a nice looking way to snap game tiles back into place when they are released. I want them to overshoot and then settle into place rather than snap directly into their spot.

The Basic Spiral

I am talking about a simple spiral of the form:

\theta(t) = Kt
r(t) = \alpha\left(Kt\right)^{n}
for some integer n.

That would be an equation for a spiral that starts at the origin and heads outward as t increases. For my application though, I want to end at the origin so I need to substitute 1-t in for t.

\theta(t) = K(1-t)
r(t) = \alpha\left(K(1-t)\right)^{n}

The math is also going to work out slightly better if I use n-2 in place of n in the above equations:

\theta(t) = K(1-t)
r(t) = \alpha\left(K(1-t)\right)^{n-2}

I don’t want the piece to spiral into place when released though. So, really, I am concerned with just the x coordinate from the above equations:

x(t) = r(t) \cos \theta(t) = \alpha K^{n-2} (1-t)^{n-2} \cos \left(K\left(1-t\right)\right)

To normalize everything, I am going to let \alpha = K^{2-n}. And, since I want my interpolation value to go from zero to one instead of one to zero, I am again going to subtract this all from one:

x(t) = 1 -  (1-t)^{n-2} \cos \left(K\left(1-t\right)\right)

Parametrizing by Arc Length

If I just plot the spiral as is, then I am stuck with the same problem that I had with damped spring motion: the frequency is constant. The rate at which it would shimmy would not increase. I want it to really settle into place. So, I have to walk through the spiral at a fixed rate. I need to rescale the (1-t) into something else so that for any given time interval, I cover the same arc length on the spiral.

The first step then is to figure out how much arc length I sweep out with any given t. Call this arc length s(t):

s(t) = \int_0^{t} r(\tau) \theta(\tau) d\theta = \int_0^{t} - \alpha K^{n} (1 - \tau)^{n-1} d\tau
which comes out to:
s(t) = \frac{\alpha K^n}{n} \left(\left(1-t\right)^n-1\right)

Then, to rescale my t, I want to use t' instead so that t' = \frac{s(t)}{s(1)}. So, when t' = \frac{1}{4}, for example, I will need to find the t such that the arc length covered in the first t seconds is \frac{1}{4}-th of the arc length covered in the whole interval. Solving for t in the equation t' = \frac{s(t)}{s(1)} comes out to:

t = 1 - \sqrt[n]{1-t'}

Plugging all of that back into the equation for x(t) gives me:

x(t) = 1 - \sqrt[n]{(1-t)^{n-2}} \cos \left(K\sqrt[n]{1-t}\right)

The simplest spiral here has n = 3, and we will go around four times with K = 8\pi:

You can see how the period speeds up toward the end. This was a good starting point. However, the first oscillation goes almost as far beyond the target as our initial point was. So, I upped the exponent to n = 9.5:


This was pretty much the effect I wanted. Unfortunately, it involves taking an 7.5-th power, an 9.5-th root, and a cosine of calculation. I decided that \frac{7.5}{9.5} was close enough to one to give it a whirl without needing the 7.5-th power.

x(t) = 1 - (1-t) \cos \left(K\sqrt[n]{1-t}\right)

Here you can see the final result.

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