# The Bi-Fold Door Problem

The following is a well-traveled problem that has appeared in both the Mathematics Magazine (June 1993, p193) and The Mathematics Teacher (October 1993 article and November 1993 Letters):

A bi-fold closet door consists of two one-foot-wide panels, hinged at point P. One of the panels is fixed at the point O. Assume that as the endpoint Q moves to the right, the door rubs against a thick carpet. What shape will be swept out on the carpet? Try the video.

To get another perspective, try moving the red point Q in the image below along the horizontal axis.

Figure 1

The video and Figure 1 show that when the door first starts to close, the path of the boundary curve follows the hinge of the door, and therefore the curve is part of a circle of radius 1. After some point, however, the hinge of the door lies below the curve as the carpet is brushed by the right-hand side of the door. In particular, this means that along this part of the curve, the edge of the door is tangent to the curve; that is, the boundary of the brushed carpet is determined by the envelope of tangent line segments represented by the right-hand side of the door.

Figure 2

Consider the position of the door when it has been opened to an angle theta as illustrated in Figure 2. The right side of the door is tangent to the curve we seek at some point $$(x(\theta),y(\theta)).$$ In the figure we have $$t = 2\sin\theta$$ and $$k = \cos\theta$$ and so the slope of the right-hand side of the door is $$-2k/t = -\cot\theta.$$ The equation of the line represented by this side of the door is therefore $y = -\cot\theta (x-2\sin\theta) = (-\cot\theta)x + 2\cos\theta$ Next consider what happens to the door when it is opened by an additional angle h as shown in Figure 2. The equation of the line representing this new door position is $y = -\cot(\theta+h)x + 2\cos(\theta+h)$ The intersection of these two lines occurs at the x-value $x_h = \frac{2(\cos(\theta+h)-\cos\theta)}{\cot(\theta+h)-\cot\theta}$ This intersection point will converge to the point $$(x(\theta),y(\theta))$$ as h approaches 0. Thus $x(\theta) = \lim_{h \rightarrow 0}\frac{\cos(\theta+h)-\cos\theta}{\cot(\theta+h)-\cot\theta} =2 \lim_{h \rightarrow 0}\frac{\frac{2(\cos(\theta+h)-\cos\theta)}{h}}{\frac{\cot(\theta+h)-\cot\theta}{h}} =2\frac{-\sin\theta}{-\csc^2\theta}=2\sin^3 \theta$ (An application of the definition of the derivative!) Of course, this is only valid if the two line segments intersect at an x-value to the right of the first door hinge, which means we must have $$2\sin(\theta)^3 \ge \sin(\theta).$$ Thus this parameterization of the curve is valid for $$\theta \ge \frac{\pi}{4},$$ or when $$x \ge 1/\sqrt{2}.$$

Finally, we get from the equation of the line found earlier that $y(\theta) = -\cot\theta(2\sin^3\theta - 2\sin\theta) = 2\cos^3\theta$ For $$1/\sqrt{2} \le x \le 2$$ the curve is therefore given by $\left(\frac{x}{2}\right)^{2/3} + \left(\frac{y}{2}\right)^{2/3} = 1$ or $y = \left(2^{2/3}-x^{2/3}\right)^{3/2}$ For $$0 \le x \le 1/\sqrt{2}$$, the curve is just part of a circle of radius 1 centered at the origin, and so the boundary of the brushed carpet is given by $y = \begin{cases} \sqrt{1-x^2} & \quad 0 \le x \le 1/\sqrt{2} \\ \left(2^{2/3}-x^{2/3}\right)^{3/2} & \quad 1/\sqrt{2} \le x \le 2 \end{cases}$

1. (1993) Proposals, Mathematics Magazine, 66:3, 192-193, DOI: 10.1080/0025570X.1993.11996117; (1994) Solutions, Mathematics Magazine, 67:3, 225-229, DOI: 10.1080/0025570X.1994.11996220
2. Weiner, Jack L., and G. R. Chapman. "Inflections on the Bedroom Floor."" The Mathematics Teacher, vol. 86, no. 7, 1993, pp. 598-601. JSTOR, www.jstor.org/stable/27968512
3. Derek Seiple, Eugene Boman, and Richard Brazier. "Mom! There's an Astroid in My Closet", Mathematics Magazine, April 2007, 104-111.
4. "Closet Door," The Mathematical Tourist Blog, May 10, 2007.

Larry Riddle, November, 1993

(Converted to HTML, February 1996)
(Revised to use MathJax and add the video animation and Geogebra app, October 2020)