Articles of rectangles

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. Specifically, if S contains small sub-squares, they will be mapped into sub-shapes of C that are […]

Least greedy square

There are $n$ squares of $m$ different colors. Squares of the same color are interior disjoint, but squares of different colors may intersect. For every square, define its “greed” as the maximum number of squares of a single color that it intersects. For example, in the figure below, the top-left red square has a greed […]

Family of geometric shapes closed under division

The family of rectangles has the following nice properties: Every rectangle $R$ can be divided to two disjoint parts, $R_1 \cup R_2 = R$, such that both $R_1$ and $R_2$ are rectangles (i.e. belong to the same family). Moreover, for every number $p > 0$, it is possible to cut $R$ to two disjoint rectangles […]

cutting a cake without destroying the toppings

There is a square cake. It contains N toppings – N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake. I want to divide the cake into 2 non-empty rectangular pieces, by either a horizontal or a vertical cut, such that the number of […]

Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under what conditions will the $M_1M_2M_3M_4$ rectangle be a square?

rectangularizing the square

There is a square that I want to divide to n people, such that each person gets a rectangular piece with an equal area. An obvious option is to cut 1-by-n rectangles of size n-by-1, but the people don’t want such rectangles, they say they are too skinny. They want to get R-balanced rectangles, which […]

Compute ratio of a rectangle seen from an unknown perspective

TL;DR: Given 4 points on a two dimentional plane, representing a reclangle seen from an unknown perspective, can we deduce the width / height ratio of the rectangle ? Details: From a picture, and some opencv work (canny, hough lines, bucketing to tell appart “lines” and “columns”, choosing interesting lines, math to deduce lines intersections), […]

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it is always correct. Here is the heuristic: Try to construct a square whose diagonal is the longest side of […]

Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation for the area to be $A=4xy$ but then when trying to find the derivative I don’t think I’m doing it right.

Relationship of aspect ratio to the homography matrices between a rectangle and an arbitrary quadrilateral

I’ve been reading everything I can on the perspective mapping between a 2D rectangle and the projection onto the plane in 3D space of a rectangle. I’ve learned that any such quadrilateral resulting from the projection can be mapped to any rectangle. I’ve learned that the only constraints are that the quadrilateral must be convex. […]