Articles of geometry

I'm looking for the name of a transform that does the following (example images included)

I’m in the usual situation that if I would know what the name of the thing was, then I could find the answer. Since I dont know the name, here is what I’m looking for: Suppose I have the following “snake” of 10 quadrilaterals: I now want to apply a transformation to each of these […]

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: 1) a triangle can be divided into 3 congruent shapes. Is it equilateral? 2) a shape in the plane can be divided into n congruent shapes for any positive integer n. What can it […]

Form of most general transformation of the upper half plane to the unit disk.

In David Blair’s book on Inversion Theory, he write that the transformation $$ T(z)=e^{i\theta}\frac{z-z_0}{z-\bar{z}_0} $$ is the most general transformation mapping the upper half plane to the unit circle, provided $z_0$ is in the upper half plane. If $z_0$ is in the lower half plane, then the upper half plane is mapped to the exterior […]

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle’s center in its interior?

How to compute the area of the shadow?

If we can not use the integral, then how to compute the area of the shadow? It seems easy, but actually not? Thanks!

Sphere on top of a cone. Maximum volume?

I received an interesting problem which I can not figure how to solve. It follows: An ice cream has the shape of a sphere and a cone like the image below. What is the maximum volume that the sphere can occupy out of the cone? Note that $M$ is the center of the sphere, not […]

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at $(a,0)$, find parametric equations for the curve traced by $P$, using angle $t$ from the positive $x$-axis to the center […]

Finding points on two linear lines which are a particular distance apart

I have two linear, skew, 3D lines, and I was wondering how I could find a points on each of the lines whereby the distance between the two points are a particular distance apart? I’m not after the points where the lines are the closest distance apart, nor the furthest distance apart, nor the point […]

Functions determine geometry … Riemannian / metric geometry?

Given a compact Riemannian manifold $(M,g)$, is there a subring of $C^{\infty}(M)$ that determines the isomorphism class of $(M,g)$? (In the same way that $C^{\infty}(M)$ determines the diffeomorphism class via the max spectrum.) Same question for compact metric spaces, more generally. I am not sure what condition to ask for on these functions… maybe something […]

Tarski-like axiomatization of spherical or elliptic geometry

Preamble Tarski’s axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive) and congruence $xy\equiv zw$ (meaning the line segment between $x$ and $y$ is the same length […]