Articles of geometry

Finding the catenary curve with given arclength through two given points

I need to find a general way of expressing a catenary, and I couldn’t find anything online. Is it possible to explicitly find an explicit equation of a catenary which goes through $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ (assuming $x_1 \neq x_2$) so that the length of the curve between the two points is […]

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? Is there a field that studies spaces that are neither globally nor locally Euclidean spaces? If such a field exists, are there […]

The minimum perimeter and maximum height of a triangle under constraints

I’m developing a web application that consists of a calculator triangles. Although I am not a mathematician, with paper, derive and Geogebra I managed to get a lot of formulas to calculate a triangle with the minimum number of data possible. Maybe some is unpublished, and limits to be introduced only correct data. I do […]

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\frac{3}{4} \log_e(2) – \frac{1}{2} \approx 0.01986.$$ But this is also $$\sum_{n=1}^{\infty}\frac{1}{(4n-1)(4n)(4n+1)}.$$ Is there any connection […]

Visualizing the 4th dimension.

In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can atleast see their 3-D cross-sections. A tesseract hypercube would be a good example. Can we conclude that a 3-D cube is […]

A circle rolls along a parabola

I’m thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency of the rotations, C, of course, varies the parabola. Now, if I want the circle to “match up” […]

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: “In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the arc r, and possibly multiplied […]

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, multiply the number of stones in each of the two smaller piles you form, so that if […]

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to start, I have tried to start but don’t think my answer is right.

How to calculate true lengths from perspective projection?

Suppose that I have a single point perspective drawing like . and suppose also that I know some of the real horizontal distances and distances along lines converging to vanishing point. E.g if i know the real length and breadth of a slab of pavement. Is it possible to derive other real distances e.g the […]