Articles of geometry

Very special geometric shape – parallelogram (No name yet?)

I suppose this geometric shape is something very ‘special’. I cannot clarify in short about being ‘special’, but I think this shape stands together with such special shapes like the square and the regular hexagon. Here it is: So it is a parallelogram based upon a square. Its height is equal to the side of […]

Changing a bezier curve by dragging a point on the curve itself rather than a control point

I’m developing an iPhone app that allows users to cut out part of an image from its background. In order to do this, they’ll connect a bunch of bezier curves together to form the clipping path. Rather than have them move the control points of the curves in order to change the curve, I’d like […]

Understanding Norms on Vector Spaces

Let $\|\cdot\|$ be a norm (not necessarily the standard norm) on $\mathbf R^2$ and $S$ be the set of all the vectors $v$ such that $\|v\|=1$. For any point $p\in S$, let $\ell_p$ denote the line joining origin and $p$. I think the following should hold: For any point $p\in S$, there is a neighborhood […]

Calculate rectangle vertices

I have a programming problem and I’ve forgotten some of my math rectangle formulas. I have to make a program that creates a rectangle that isn’t parallel to the X and Y axis, I assume this means a diagonal rectangle in the first quadrant of a Cartesian graph. My inputs are the following: coordinates of […]

Maximizing area of a rectangle inscribed in a semicircle

A rectangle of largest area is inscribed in a semicircle of radius $r$. What is the area of the rectangle? I just need the hint to solve it. How can I get length and breadth of rectangle in terms of radius $r$? If I can get length and breadth in terms of radius $r$, then […]

Cyclic quadrilateral with equal area and perimeter

I was studying about cyclic quadrilaterals , and a thought came that are there infinite number of cyclic quadrilaterals having perimeter equal to its area or if they are finite and how many are there in total? I just cannot understand how to proceed to solve this or how to count them ? Also , […]

notion of the minimum of a function over a polytope

Let $P$ be a polytope with $M$ vertices. (The polytope $P$ is the intersection of the hypercube $0≤x _j ≤1$ with the hyperplane $\sum_{j=1}^nx_j=t$, $0\leq t\leq n$). Suppose that the volume of $P$ is $Vol(P)=A$. We subdivide $P$ into $M$ pieces each of the volume $A/M$. Let $f$ be a function, such that the integral […]

Vector placement for optimal projections

Let $n_1,\cdots,n_m$ be unit vectors arbitrarily chosen in $\mathbb{R}^3$. Then, there exist some optimal unit vector $n$ such that $$|n^Tn_i|\geq \delta(m)>0 \quad \forall i=1,2,\cdots,m$$ I am looking for a lower bound on $\delta(m)$ for all possible configurations of $n_1,\cdots,n_m$. For example if $m=2$, then it appears that $\delta(2)\geq \sin(\pi/4)$ (the worst case is when they […]

Find the co-ordinates of the point on the join of two points which is nearest to the intersection of two planes

Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$. Please give me an outline to solve the problem. Thanks.

How to raise a number to a power geometrically.

There are methods to add two lines of arbitrary lengths or multiply them together known since Greek times; and more advanced methods based on the concepts of bases and units. But, I have not been able to find a way to exponentiate a number geometrically without using algebra. I would love if someone could somehow […]