Closing the Closet Problem

(JavaSketch interactive diagrams) 

In this diagram above,  L is the length of each panel, (x’,y’) is the location of the door hinge, and (a,b) is a point in the envelope region swept out by the door.  Imagine looking down onto the floor from above.  The equation for the circle governing the motion of the hinge is:

Hence the point (x’, y’) can be generally expressed as:

This hinge point traces out the path of the curve towards the end of the opening motion.  Precisely, the hinge defines the curve once the door panels are perpendicular.  At this point, the second panel is tangent to the circle, and thus a minute change in either direction moves the hinge parallel to the second panel.  Moving in the direction of closing the closet leads to the condition of the hinge movement having greater slope than the panel, and thus the hinge does not define the curve.  Moving in the opening direction leads to the condition that the slope of the panel is greater than the slope of the hinge movement (keeping the end of the second panel below the circle) and thus the hinge defines the curve.  The x-coordinate of the transition point (actually an inflection point) is L/(2^.5).

The line segment drawn containing (a,b) is contained by the line:

Assuming (a, b) lies on this line, using the above equation yields:

Now, differentiate b with respect to x’, and set this derivative equal to zero.  This will yield the conditions for a maximum value of b.  (Note: this corresponds to the particular second panel alignment that yields the maximum value of b for a given value of a)  The condition obtained is:

Substituting this expression for x’ back into the original expression for b yields:

Which simplifies to:

Below is a graph of this function with L=5

Because the envelope boundary is defined by the circle for values of x smaller than 5/(2^.5) – corresponding to the region where the hinge defines the boundary, the actual envelope, as a piecewise function, is graphed below.

Connections to the Hypocycloid or Astroid

The path of a hypocycloid is defined by tracing the path of a point on a circle as it is rolled along the interior of a larger circle (see diagram below).

The point on the circle traced around is initially (R, 0).  Note that  the smaller circle has ¼ the diameter of the larger.  This arrangement allows for 4 complete rotations of the smaller circle as it traversed the interior of the larger.  The resulting curve is called an Astroid, and is shown below:

The curve of the astroid in the first quadrant is identical to the envelope curve above!!

Let’s see why.  The parameterization of the Astroid is:

                        x = a cos³(t)                             y = a sin³(t)

Or, in Cartesian coordinates, x^2/3+y^2/3 = a^2/3, which is identical to the formulation for b shown above (a = x, b = y, a = 2L).

Below is a diagram of the Hypocycloid in conjunction with the closing closet made using Geometer's sketchpad
(For an animation click here)

Some additional results:

Point T lies on the envelope curve, and hence is the tracing point for the hypocycloid.  For this to be true, arc QT would have to equal arc QO.  Testing these values of SketchPad revealed this equivalence.  There are two ways to prove that it is true, analytically and geometrically.

It is actually quite easy to see why it is true geometrically.  Because DAZM is isosceles, with angle A equal to angle M, ÐRZT must have twice the measure of ÐZAN.  ÐQRT must have twice the measure of ÐRZT (central angle is twice the inscribed angle with the same intercepted arc).  Together, this implies that ÐQRT is four times ÐZAN (as indicated by the SketchPad measurement shown).  Because the circles are in a four to one ratio, the intercepted arcs QT and QO must have the same length.  Interestingly, a similar argument shows that arc QM is also the same length.

Because QT and QO are equal in length, point T must be on the Hypocycloid path and hence the envelope curve.  This provides an easy way to construct, for a given orientation of the door, what point on ZM is part of the envelope (for every orientation only one point on ZM is on the envelope).  To show this analytically requires finding the generalized intersection between the circle centered at R and the line containing ZM.  Both are easily defined in terms of x, y, and q (ÐZAN).  Combining these equations it is easy to find x and y as functions of theta.  This generalized intersection can be shown to be y = 2Lsin³q, x = 2Lcos³q, the parameterized envelope curve!!

COMING SOON: Making a physical gear mechanism to show the connection between a rotating circular gear (hypocycloid), an opening closet, and the envelope trace defined by the Astroid.