FORMULA: Scale To Fill, Given Rotation

Formula + Animation

\text{scale factor} = r\abs{\sin\theta} + \abs{\cos\theta}

After Effects Code

r = degreesToRadians(transform.rotation);
s = thisComp.width/thisComp.height*Math.abs(Math.sin(r)) + Math.abs(Math.cos(r));
[s,s]*100

Derivation

Overview

To find the scale factor, we will first construct a rectangle (black) with half-height of $1$ and half-width of $r$ (the aspect ratio). Next, we will construct an identical rectangle (red) with rotation $A$ about its center. Finally, we will construct a third rectangle (blue) identical to the second but scaled to circumscribe the first rectangle. See the figure below.

To find the scale factor, we must find the length of the half-height of the blue rectangle, that is, $\text{scale factor} = c+e$ .

Setup

We will make use of the Law of Sines. Recall that

\frac{x}{\sin(X)} = \frac{y}{\sin(Y)}

where $x$ and $y$ are side lengths of any triangle, and $X$ and $Y$ are their corresponding opposite angles.

Rearranging

x = \frac{y\sin(X)}{\sin(Y)}

Proof

Using the Law of Sines, find $c$ and $e$ .

c = \frac{1\sin(90)}{\sin(B)}

c = \frac{1}{\sin(B)}

e = \frac{d\sin(E)}{\sin(90)}

e = d\sin(E)

Now, find $d$ . Recall that $r$ is the half-width of the rectangle.

r = a+d

d = r-a

d = r – \frac{1\sin(A)}{\sin(B)}

Plug into $e$ .

e = d\sin(E)

e = (r-\frac{1\sin(A)}{\sin(B)}) \sin(E)

e = r\sin(E)-\frac{\sin(A)\sin(E)}{\sin(B)}

Notice that $A \cong E$ .

e = r\sin(A)-\frac{\sin^2(A)}{\sin(B)}

Combine

\text{scale factor} = c+e

[\frac{1}{\sin(B)}] + [r\sin(A) – \frac{\sin^2(A)}{\sin(B)}]

r\sin(A) + \frac{1-\sin^2(A)}{\sin(B)}

r\sin(A) + \frac{\cos^2(A)}{\sin(B)}

Notice that $\cos(A) = \sin(B) = \frac{1}{c}$ .

r\sin(A) + \cos(A)

Finally, substitute $A$ with $\theta$ , and take the absolute value.

\text{scale factor} = r\abs{\sin\theta} + \abs{\cos\theta}