Articles of geometry

Computing fundamental forms of implicit surface

This question already has an answer here: About the second fundamental form 1 answer

Can someone give me the spherical equation for a 26 point star?

This is the object that I am trying to find the volume of. This can be treated as a “26 point star”. What I need is an equation to describe it. If anyone has that surface in spherical coordinates R(Phi,Theta) that would be awesome. If not, the X,Y,Z solution can also be worked with. This […]

Prove by elementary methods: the plane cannot be covered by countably many copies of the letter “Y”

As a consideration from the post “Prove by "elementary methods": The plane cannot be covered by finitely-many copies of the letter "Y"”, on the basis of the remark made in previous post by the user Moishe Cohen, is it still possible to apply elementary methods to prove weaker results, namely: The plane cannot be covered […]

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 […]