Total area of rectangle rotation

Question:

The rectangle $$ABCD$$ is rotated into a rectangle $$A′B′C′D′$$.

Initial state $$A$$ is at the origin, $$B$$ is on the y-axis $$D$$ is on the x-axis.

During the rotation, $$A$$ travels vertically along the $$y$$ -axis until it meets $$A′$$.

While $$D$$ simultaneously travels horizontally along the $$x$$ -axis until it reaches the origin.

(The x-axis and y-axis can be understood as impenetrable walls).

The length of the short side is $$h$$ and the length of the long side is $$w$$.

Find the total area swept during this process.

My Attemptions

I drew the following diagram, $$ABOC'$$ area is $$h^2$$.

But I don't quite understand what the arc $$\overset{\frown}{B'C}$$ looks like, I don't know how to get an expression for this curve, either in parametric equation form or implicit equation form.

• You need to specify exactly how the rectangle moves from the initial position to the final position. All you have said is $A$ is "close to" the $y$-axis and that description is mathematically imprecise. An example of a precise statement would be to say that $A$ travels vertically along the $y$-axis until it meets $A'$, while $D$ simultaneously travels horizontally along the $x$-axis until it reaches the origin. Such a motion is a combined rotation and translation, but it is not the only possible way in which the rectangle can move, which is why you need to unambiguously specify it. May 28 at 3:37
• @heropup I suppose the part '$A$ is close to the y-axis, and $D$ is close to the x-axis' means the two points slide along indicated lines, exactly as you described. Possibly OP is not a native English speaker (neither am I), hence the unfortunate expression 'close to an axis'. :) May 29 at 12:21
• What is the education level, at which you get this problem to solve? May 29 at 12:44
• @CiaPan I have given two interpretations in my "answer" below. As the candidate mentions "envelopes", I think the issue should be treated at undergraduate level. May 29 at 14:09
• @GalAster: Referring to Jean Marie's first sketch. If I understood the motion ok, it is a sliding rectangle instead of a ladder. $a=3,h=2.$; HINT: The curves have equations;$\sqrt {x}+\sqrt {y}=\sqrt {a}$ and $(X-x)^2+(Y-y)^2 =h^2 .$ Asteroid and " Asteroid Bertrand Parallel"; A segment length ( width of rectangle) 2 along common normal can be dragged on a curve midway between the two curves to find swept area by Pappu's theorem. May 29 at 16:34

2 Answers

After our exchange, I have understood you mean the type of "rotation" given in the figure below. Caution : it isn't a rotation in the usual sense of the word because this transformation hasn't a center (which should be an invariant point) : see the second interpretation in the "Comment" below.

Fig. 1 : The sides of the moving rectangle "envelope" curves. The North-East curve generated by line segment $$[B_tC_t]$$ is a piece of astroid : see the classical "sliding ladder image" for example here. The North curve is divided into two parts : an arc of the black curve with equation given below in (1) and an arc of the red curve which is an ellipse with parametric equations given in (2).

Using the technique of envelopes it is possible to get the equation of the envelope of line $$A_tB_t$$ (black curve on fig. 1). Here is the way I have proceeded :

I start from the equation of line $$A_tB_t$$ which is :

$$x \cos t - y \sin t = -w (\sin t)^2 \tag{a}$$

I differentiate (1) wrt parameter $$t$$, considering variables $$x$$ and $$y$$ as constant, giving, after a sign change :

$$x \sin t + y \cos t = 2 w \sin t \cos t \tag{a'}$$

Solving system (a) + (a') gives the parametric equations of the envelope (black line) of moving line $$A_tB_t$$ :

$$\begin{cases}x&=&\tfrac12 w \sin(t)\sin(2t)\\y&=&\tfrac12 w \sin(t)(3+\cos(2t))\end{cases}\tag{1}$$

But in fact, we deal with a moving segment which is not the same as a moving line ; as a consequence, there is a kind of "competition" (I haven't found a better word) with the (small) elliptic arc issued from the locus of $$B_t$$, itself a quarter of an ellipse (red curve) with equation :

$$\begin{cases}x&=&h \sin(t)\\y&=& w \sin(t)+ h \cos(2t)\end{cases}\tag{2}$$

Please note that in (2), both parameters $$w$$ and $$h$$ are present whereas in (1), only $$w$$ is present.

In fact, we have to do the same kind of calculations for $$B_tC_t$$. Can you take it from here ?

Fig. 1 has been generated by the following Matlab program :

 clear all;close all;hold on;
axis([0,2.4,0,2.4]);
axis equal tight
w=2;h=1; % rectangle's height and width
for t=0:pi/50:pi/2
b=w*cos(t);c=i*w*sin(t);d=h*i*exp(-i*t);
R=[b,b+d,c+d,c,b]; % rectangle : complex coord.
c='g';
if t==0;c='b';end;
plot(real(R),imag(R),'color','c','linewidth',2);
end
t=0:pi/50:pi/2;
plot(w*(sin(t).^2).*cos(t),w*sin(t).*(1+cos(t).^2),'k','linewidth',8);
plot(h*sin(t),w*sin(t)+h*cos(t),'r','linewidth',8)
text([0.,0.,w,w,h+0.05],[0.1,1.1,1.1,0.1,w],{'A_0','B_0','C_0','D_0','B_1'})


Comment : here is a different interpretation of your question with a true rotation around center $$(w/2,w/2)$$ :

Fig. 2 : Axes are in black. The center of rotation is materialized by a little circle.

The swept area can be computing rather easily because it can be decomposed into circular sectors.

(in fact, I don't think it is the kind of transformation you had in mind).

Matlab program having generated figure 2 :

 clear all;close all;hold on;axis equal;
w=3;h=1; % length and width of rectangle
C=[w/2;w/2]; % center
plot(C(1),C(2),'ok')
R=[0,w,w,0,0
0,0,h,h,0]; % rectangle's vertices
for t=0:-pi/50:-pi/2
M=[cos(t),-sin(t);sin(t),cos(t)]; % rot. matrix
X=M*R+((eye(2)-M)*C)*ones(1,5); using transl. by C
plot(X(1,:),X(2,:),'linewidth',4);
end
plot([0,w],[0,0],'linewidth',8,'color','k');
plot([0,0],[0,w],'linewidth',8,'color','k'); % axes' plots

• It is your first diagram. It seems that the next step is to find the boundary conditions of Astroid. Should it be integral between $h-\operatorname{arccot}\frac{w}{h}, w+\operatorname{arccot}\frac {w}{h}$. May 31 at 4:12
• Or should I use the polar coordinate equation? The graph is symmetric, and the polar coordinates just need to be integrated in two segments and then multiplied by 2 May 31 at 4:14
• Happy you have reacted and said in particular which interpretation was the good one. I don't think polar equations are the good solution. I think you should parametrize for example line segment $A_tB_t$ for $0<t<1$ and use the technique of derivation wrt to parameter $t$ as described for example in my answer here. Same operation for line segment $B_tC_t$. May 31 at 7:06

The figure below is a .GIF image showing the sliding/rotation of the a rectangle that has $$w = 10$$ and $$h = 5$$.

The upper black curve is the trajectory of vertex $$B$$ and the right black curve is the trajectory of vertex $$C$$. The green curve is the envelope of $$AB$$ while the pink curve is the envelope of $$CD$$. The blue curve is the envelope of $$BC$$ while the red curve is the envelope of $$AD$$.

If $$t$$ is the angle (in radians) that side $$AD$$ makes with the $$x$$ axis, then the equation of the trajectories of $$B$$ and $$C$$ are

$$B(t) = (h \sin t , h \cos t + w \sin t )$$

$$C(t) = (w \cos t + h \sin t , h cos t )$$

The green curve is given by

$$P_{AB}(t) = (w \cos t - w \cos^3 t , 2 w \sin t - w \sin^3 t )$$

Then pink curve is given by

$$P_{CD} = (2 w \cos t - w \cos^3 t , w \sin t - w \sin^3 t )$$

The blue curve is given by

$$P_{BC} = (h \sin t + w \cos^3 t , h \cos t + w \sin^3 t )$$

To find the area swept by the sliding/rotating rectangle, we need to find numerically two points:

1. The intersection point between $$P_{AB}(t)$$ and $$B(s)$$. I got the intersection point numerically to be $$(3.828361, 10.87288)$$ at $$t = 0.997414, s = 0.872086$$

2. The intersection point between $$P_{BC}(t)$$ and $$B(s)$$. I got the intersection point numerically to be $$(4.981225, 10.39535)$$, at $$t = 1.358781, s = 1.484109$$.

The area enclosed is determined by the integral

$$A = \dfrac{1}{2} \displaystyle \int x(t) y'(t) - y(t) x'(t) \ dt$$

where the integral is segmented into three parts corresponding to $$P_{BC}(t), B(t)$$, and $$P_{AB}(t)$$.

With the found values of intersection between the curves, the area integral is

$$\text{Area} = \displaystyle \int_{\pi/4}^{1.358781} (-h^2 + 3 w^2/8 ) + hw \sin(2 t) \\ \displaystyle - 3/8 w^2 \cos(4 t) \ d t \\ \displaystyle + \int_{1.484109}^{0.872086} - h^2 \ dt \\ \displaystyle + \int_{0.997414}^{\pi/2} w^2 (\frac{1}{8} - \frac{1}{2} \cos(2t) + \frac{3}{8} \cos(4t) ) \ dt$$

This integral evaluates to

$$\text{Area} = 89.26953$$

• [+1] I hadn't noticed your answer. A caveat : I have remarked that the figure is case-dependent; according to the ratio w/h, the relative positions of the green, black and blue arcs can change. Jun 4 at 22:17
• Thanks. Yes. This is indeed the case. Jun 4 at 22:21