Folding a corner on a paper to the opposite side

If we bend the $\text{A4}$ paper by connecting two opposite corners, what length is the $\color{red}{\text{bend}}$?

I drew the two blue lines on the last image, which are sides of a rhombus. By the help of that rhombus, I managed to calculate the bend as the length of the shorter diagonal:

$$\frac ab\sqrt{a^2+b^2}$$

Taking the length of paper sides as $1$ and $\sqrt2$, the bend is then $\frac{\sqrt3}{\sqrt2}=\frac{\sqrt6}{2}$.

But what if we bend the point not to the opposite corner, but to the some point $T$ on the oppoiste side of the rectangle with sides $a,b$ where that point is $c$ units away from the bottom left corner?

When $c=0$, we get $B=\frac ab\sqrt{a^2+b^2}$ as the above case.
When $c=b$, we get $B=b$, as you are just folding the paper then.

But how do we solve for the $B$ in general over some $a,b,c$?

We can also extend the line of the bottom side and put our point $T$ outside of the paper, as long as the bottom left point is in the same place; and consider these cases too.

• Surely, in that case, we assume the paper is of actually of size $1$ by $(b-c)$ and then use the equation $B=\frac ab\sqrt{a^2+b^2}$ as above? We're just ignoring the part of the paper left of the point $c$ Jul 17, 2017 at 13:06
• @lioness99a You mean just plug $(b-c)$ into $b$ in the function? I think that works only if $c\le (b-a)$ since a part of the bend gets cut out otherwise. Jul 17, 2017 at 13:21

This answer deals with the case where $b-a\lt c$ since lioness99a has already dealt with the case where $0\le c\le b-a$.

We may suppose that $$A(0,a),\quad B(b,a),\quad C(b,0),\quad D(0,0),\quad E(c,0)$$ where $A,B,C,D$ are the vertices of the paper and we want to bend $B$ to $E$.

The equation of the perpendicular bisector $\ell$ of the line segment $BE$ is given by $$\ell : y-\frac a2=-\frac{b-c}{a}\left(x-\frac{b+c}{2}\right)$$

If we define $F,G,H,I$ as the intersection point of $\ell$ with $y=a,x=b,x=0,y=0$ respectively, we have, for $c\not=b$, $$F\left(\frac{b^2-c^2-a^2}{2(b-c)},a\right),\ G\left(b,\frac{a^2-(b-c)^2}{2a}\right),\ H\left(0,\frac{a^2+b^2-c^2}{2a}\right),\ I\left(\frac{a^2+b^2-c^2}{2(b-c)},0\right)$$ We can see the followings :

• If $c\lt b$, then $F_x\lt b$ and $0\lt G_y\lt a$ always hold and $$F_x\ge 0\iff c\le \sqrt{b^2-a^2}$$

• If $c\gt b$, then $F_x\ge b,G_y\lt a$ always holds and $$G_y\gt 0\iff c\lt a+b\qquad\text{and}\qquad H_y\ge 0\iff c\le\sqrt{a^2+b^2}$$

Here, let us separate it into cases :

Case 1 : $b-a\lt c\le \sqrt{b^2-a^2}$

We see that $F$ is on the line segment $AB$ and that $G$ is on the line segment $BC$.

Therefore, the length of the bend is given by $$FG=\sqrt{\left(\frac{b^2-c^2-a^2}{2(b-c)}-b\right)^2+\left(a-\frac{a^2-(b-c)^2}{2a}\right)^2}=\frac{(a^2+(b-c)^2)^{3/2}}{2a(b-c)}$$

Case 2 : $\sqrt{b^2-a^2}\lt c\le \sqrt{a^2+b^2}$

We see that $G$ is on the line segment $BC$ and that $H$ is on the line segment $AD$.

Therefore, the length of the bend is given by $$GH=\sqrt{\left(b-0\right)^2+\left(\frac{a^2-(b-c)^2}{2a}-\frac{a^2+b^2-c^2}{2a}\right)^2}=\frac{b}{a}\sqrt{a^2+(b-c)^2}$$

Case 3 : $\sqrt{a^2+b^2}\lt c\lt a+b$

We see that $G$ is on the line segment $BC$ and that $I$ is on the line segment $CD$.

Therefore, the length of the bend is given by $$IG=\sqrt{\left(b-\frac{a^2+b^2-c^2}{2(b-c)}\right)^2+\left(\frac{a^2-(b-c)^2}{2a}-0\right)^2}=\frac{a^2-(b-c)^2}{2a(c-b)}\sqrt{a^2+(b-c)^2}$$

Case 4 : If $c\ge a+b$, then we cannot bend the point.

The answer is as follows :

If $c\ge a+b$, then we cannot bend the point.

If $0\le c\lt a+b$, then the length of the bend is $$\begin{cases} \dfrac{a}{b-c}\sqrt{a^2+(b-c)^2} & \text{if 0\le c\le b-a} \\\\ \dfrac{(a^2+(b-c)^2)^{3/2}}{2a(b-c)} & \text{if b-a\lt c\le\sqrt{b^2-a^2}} \\\\ \dfrac{b}{a}\sqrt{a^2+(b-c)^2} &\text{if \sqrt{b^2-a^2}\lt c\le \sqrt{a^2+b^2}}\\\\ \frac{a^2-(b-c)^2}{2a(c-b)}\sqrt{a^2+(b-c)^2} &\text{if \sqrt{a^2+b^2}\lt c\lt a+b} \end{cases}$$

While $0\leq c\leq b-a$, we can just consider this as a piece of paper which has size $a$ by $b-c$:

Therefore, the equation to calculate the length of the bend will be \begin{align}B&=\frac a{b-c}\sqrt{a^2+(b-c)^2}\end{align}

When $c=b-a$ then the bend starts in the bottom right corner of the paper, and so this approach won't work for $c> b-a$ as it no longer forms a rhombus.

[I will update my answer with the other half when I've had time to do the Maths]

• How did you create those images if I may ask? Jul 17, 2017 at 13:28
• I didn't realise it didn't hold for $c>b-a$ at a first glance. I'll update this answer with the whole solution when I've worked it out, but I've covered half of the answer so I didn't want to delete it completely. Jul 17, 2017 at 13:41
• @ImreVégh They look like they were made on Geogebra (however I just saved the image and edited it in Paint!) Jul 17, 2017 at 13:42
• @ImreVégh You can do it in geogebra. There are already online "paper bending simulations" made in it, such as this one Jul 17, 2017 at 13:44