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Is there a formula/calculation to work out how to find the other 2 coordinates for a rectangle if you only have the bottom left and top right coordinates?!

e.g. My bottom-left coordinate would be (-1,0) and top right would be (3, -2), I can work out the midpoint but I can't seem to get my head around how to work out the other 2 coordinates?!

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Have you drawn a picture? – Ragib Zaman Nov 24 '11 at 13:43
I think that solution isn't unique... – pedja Nov 24 '11 at 13:47
@pedja If it is given that the two points given are diagonal such as in the OP's example, then a unique rectangle is determined. – Ragib Zaman Nov 24 '11 at 13:53
Its a formula I need for programming, obviously drawing it out I can work it out but was wondering if the was a general rule of calculation to work it out?! – r0bb077 Nov 24 '11 at 13:59
@RagibZaman,I am not sure about that..draw the circle with radius half of the diagonal and midpoint as center then each point on the circle can be vertice point of the rectangle – pedja Nov 24 '11 at 14:01

2 Answers

up vote 1 down vote accepted

Let's observe picture below. Each point $(x,y)$ of the circle that satisfy following equality can be vertice point of the rectangle :

$\left(\sqrt{(x_A-x)^2+(y_A-y)^2}\right)^2 + \left(\sqrt{(x_C-x)^2+(y_C-y)^2}\right)^2=\left(\sqrt{(x_A-x_C)^2+(y_A-y_C)^2}\right)^2$ ,

where $A(-1,0)$ , and $C(3,-2)$ are given vertices.

For example if you choose point $B_1$ you can calculate coordinates of $D_1$ by using equalities :

$x_{D_1}=2x_O-x_{B_1}$ , and $y_{D_1}=2x_O-y_{B_1}$

Answer to sub question :

If you have length of one edge , let's say $|\bar{AB_1}|$ then you can write an extra equality :

$ |\bar{AB_1}|= \sqrt{(x_A - x)^2+(y_A-y)^2}$ , and find relation between $x$ and $y$ coordinates.

enter image description here

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Hey pedja, thanks for the graphics but how would I find out what B1 actually is?! – r0bb077 Nov 24 '11 at 15:07
@robbo77,coordinates of $B_1$ must satisfy square-root equality above... – pedja Nov 24 '11 at 15:12
You can chose $B_1$ freely as any point on the circle. The last corner is that directly opposite it. In Pedja's graphic, $AB_1CD_1$ and $AB_2CD_2$ are two different rectangles that both have AC as diagonal, so you won't get a unique solution without adding more constraints (such as "the edges must be parallel to the coordinate system axes). – Henning Makholm Nov 24 '11 at 15:13
Ahhh, now I've got there!! SO basically I need to know another coordinate or the length of the sides. Thanks guys! – r0bb077 Nov 24 '11 at 15:18
Sorry to be so pre-school but if I did set a length for one of the edges how would I then find the coordinates?! – r0bb077 Nov 24 '11 at 15:25
up vote 2 down vote

In your example, the bottom left coordinate is higher up than the top right coordinate.

Suppose that $b<d$ and $a<c$. If the bottom left coordinate is $(a,b)$ and the top right is $(c,d)$, then the top left coordinate would be $(a,d)$ and the bottom right would be $(c,b)$.

Other cases can be worked out similarly.

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Thanks Joe, just noticed I had my numbers mixed up!! – r0bb077 Nov 24 '11 at 14:36

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