Articles of geometry

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It’s the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn’t posed correctly and ended up generating responses that may be helpful for some people who stumble upon the question, but don’t address […]

Does there exist a constructible (by unmarked straightedge and compass) angle that cannot be quintsected?

I know that for example an angle of $20^\circ$ cannot be quintsected because an angle of $4^\circ$ cannot be constructed (I’m thinking in terms of (unmarked) straightedge and compass. But an angle of $20^\circ$ cannot be constructed (as above) and I would be interested to see an example of a constructible angle that cannot be […]

Longest pipe that fits around a corner.

This question already has an answer here: Intuitive explanation for formula of maximum length of a pipe moving around a corner? 3 answers

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+…..+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using the product-to-sum formula for $\cos(x)+\cos(y)$?

Prove that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides

Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides. My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}…………….(1)$ […]

Why do lattice cubes in odd dimensions have integer edge lengths?

This is a spinoff from Characterization of Volumes of Lattice Cubes. That question claims a number of facts as being proven, but doesn’t include the full proofs. That’s fine for the question as it stands, but I find the subject interesting enough to wonder about the details. So here I’m asking. Let’s start by defining […]

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the Roll and it is \begin{pmatrix} cos(Pitch) sin(Yaw)\\ cos(Yaw) cos(Pitch)\\ sin(Pitch) \end{pmatrix}. How should it […]

Pair of straight lines

Question: Find the equation of the bisector of the obtuse angle between the lines $x – 2y + 4 = 0$ and $4x – 3y + 2 = 0$. I don’t even know how to proceed here. I know how to find the angle between two lines, but not sure whether that would help in […]

Infimum over area of certain convex polygons

Let us identify the Euclidean plane with the complex plane $\Bbb C$. For every $n\ge 1$, consider the following collection of finite sets in $\Bbb C^{n+1}$: $$S_n:=\{(z_0,z_1,\dots,z_n)\in\Bbb C^{n+1} : |z_i-z_j|\ge 2,~\forall i\ne j~\},$$ and for every $v=(z_0,z_1,\dots,z_n)\in S_n$, define $$P(v)=\{z\in\Bbb C:|z-z_0|\le |z-z_i|,~i=1,\dots, n\}.$$ By definition, $P(v)$ is the intersection of finitely many closed half-planes, so […]

4-ellipse with distance R from four foci

I’m trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic curve without Square roots Will reward bounty to anyone who gives me the equation asked for above as well as a generalized equation for […]