Articles of geometry

How to graph in hyperbolic geometry?

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$. How do I go about doing this? I am very new to non-Euclidean geometry. Thank you.

Projection of the XY plane.

Is there a way to project the infinite Complex plane to either the Poincare disk or the unit disk – for all values of x + iy ?

What are these 3D shapes, if anything?

I’m trying to think of a 3D analogue to the ellipse, but not the ellipsoid. A circle is constructed with a piece of string anchored at the radius. So the distance from center to the curve is always the same. An ellipse is constructed with a piece of string ending at two foci. The sum […]

Is there a pure geometric solution to this problem?

$ABCD$ is a square and there is a point $E$ such that $\angle EAB = 15^{\circ}$ and $\angle EBA = 15^{\circ}$. Show that $\triangle EDC$ is an equilateral triangle. Now there is a proof by contradiction to this problem. I was wondering if there is a pure geometric solution too (no trigonometry)?

Possible areas for convex regions partitioning a plane and containing each a vertex of a square lattice.

If the plane is partitioned into convex regions each of area $A$ and each containing a single vertex of a unit square lattice, is $A\in (0,\frac{1}{2})$ possible? If each each vertex is in the interior of its region is $A \neq 1$ possible? More generally if $\rm{ I\!R}^n$ ($n\ge 1$) is partitioned into convex regions, […]

Locus of intersection of two perpendicular normals to an ellipse

I came up with the following question while playing around with Geogebra, and rather than do it myself I figured I’d offer it here. For a given ellipse, the director circle is defined to be the set of all points where two perpendicular tangents to the ellipse cross each other. Suppose we instead consider the […]

Finding vertices of regular polygon

I am trying to find the vertices of a regular polygon using just the number of sides and 2 vertices. After the second vertex, I will make left turns to find each subsequent vertex that follows. For example, If I have 4 sides, and 2 points, (0,0) & (0,10), how would I go about find […]

When does intersection of measure 0 implies interior-disjointness?

If there are two “nice” shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their boundary. My question is: what is a simple term for those “nice” subsets of $R^2$ for which intersection of area 0 implies interior-disjointness? […]

Geometric interpretation of Pythagorean_theorem in complex plane?

The Pythagorean theorem written as $$ a^2 + b^2 = c^2 $$ has the simply geometric meaning that the sum of the areas of the two squares on the legs ($a$ and $b$) equals the area of the square on the hypotenuse $c$ But algebraically the Pythagorean theorem can also be written using complex numbers […]

Point on circumference a given distance from another point

I am writing a game and need to figure out some math. If I have a circle with the equation $r^2 = (x-d)^2+(y-e)^2$, where $r$, $d$, and $e$ are constants, and a point $A(a,b)$, how do I find the point(s) on the circumference of the circle that are a given straight-line distance $D$ from $A$? […]