# Tagged Questions

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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### What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
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### Is there a free subgroup of rank 3 in $SO_3$?

There are known free subgroups of rank 2 in the set of rotations about the origin in $\mathbb{R}^3$, $SO_3$. For instance, the rotations by angle $\arccos \frac {1}{3}$ about the $z$- and $x$-axis ...
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### Tranforming 2D outline into 3D plane

I am writing a program where I would like to allow the user to draw 4 connecting lines, such as: And convert this shape into a 3D plane. Is this possible? Is there an existing algorithm to do so? ...
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### Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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### Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
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### Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
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### How to prove that a torus has the same volume as a cylinder (with the height equal to the torus' perimeter)

I want to find the volume of a torus with a given thickness and a given radius. Let r be the radius of a circle with its midpoint at $M(0|b)$ ($b \geq r$). Now I want to rotate this circle about the ...
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### Flag manifold to Complex Projective line

I am trying to get some intuition about the simplest flag manifold $U(2)/T^2$ which is apparently given by $CP^1\cong S^2$ . I have understood the stereographic projection of $S^2$ onto the ...
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### Triangle Formula for alternative Points

With this formula I tried to calculate the third point in a triangle: http://math.stackexchange.com/a/1553606/126977 I know the length between the two point points and the angles at this corners: ...
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### Why does the focus of a rolling parabola trace a catenary?

I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
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### Equation of a rectangle

I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.
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### Is there a way to rotate the graph of a function?

Assuming I have the graph of a function $f(x)$ is there function $f_1(f(x))$ that will give me a rotated version of the graph of that function? For example if I plot $\sin(x)$ I will get a sine wave ...
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### Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help (by the way- this question does come from home-work, but I've ...
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### How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
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### If point is zero-dimensional, how can it form a finite one dimensional line?

I have extracted the below passage from the wikipedia webpage - Point (geometry): In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. ...
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### The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
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### Constructing the midpoint of a segment by compass

When I am working with my child, I am stuck in this geometry problem. "We have two different points $M, N$ in the plane. Using only compass to construct the midpoint $I$ of the segment $MN$." Thank ...
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### A circle with infinite radius is a line

I am curious about the following diagram: The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and ...
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### Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
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### How to find the center of an ellipse?

I have the following data:- I have two points ($P_1$, $P_2$) that lie somewhere on the ellipse's circumference. I know the angle ($\alpha$) that the major-axis subtends on x-axis. I have both the ...
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### Why is the inradius of any triangle at most half its circumradius?

Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer. I know of two proofs of this fact. Proof 1:...
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### Is there a general formula for $\sin( {p \over q} \pi)$?

Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$ And other values can be calculated through ...
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### Equilateral triangle whose vertices are lattice points?

Is it possible to construct an equilateral triangle with vertices on lattice points? I think the answer is no, but how can I prove this? I started with a triangle with coordinates $(0,0)$ $(a,b)$ ...
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### Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
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### V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
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### How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
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### Is there more than one way to divide the “L”-shaped tromino into four congruent, connected pieces?

Recently my sister-in-law, who is training to become a high school mathematics teacher, asked me the following question: Consider the following polygon constructed by adjoining three squares of ...
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### Prove Existence of a Circle

There are two circles with radius $1$, $c_{A}$ and ${c}_{B}$. They intersect at two points $U$ and $V$. $A$ and $B$ are two regular $n$-gons such that $n > 3$, which are inscribed into $c_{A}$ and ...
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### Covering ten dots on a table with ten equal-sized coins: explanation of proof

Note: This question has been posted on StackOverflow. I have moved it here because: I am curious about the answer The OP has not shown any interest in moving it himself In the Communications of ...
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### What is the difference between a variety and a manifold?

I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ...
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### Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
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### Explaining Horizontal Shifting and Scaling

I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect. For example, ...
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### What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...
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### Finding the circles passing through two points and touching a circle

Given two points and a circle, construct a/the circle through the two points and touching the given circle. I came across this problem in History of Numerical Analysis by H. Goldstein. I spent some ...
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### Generate a random direction within a cone

I have a normalized $3D$ vector giving a direction and an angle that forms a cone around it, something like this: I'd like to generate a random, uniformly distributed normalized vector for a ...
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### why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
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### Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
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### Sheaves and complex analysis

A complex analysis professor once told me that "sheaves are all over the place" in complex analysis. Of course one can define the sheaf of holomorphic functions: if $U\subset \mathbf{C}$ (or \$\mathbf{...