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

learn more… | top users | synonyms

15
votes
5answers
2k views

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 ...
8
votes
3answers
798 views

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 ...
4
votes
4answers
10k views

How do I convert the distance between two lat/long points into feet/meters?

I've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. ...
7
votes
3answers
1k views

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 ...
3
votes
1answer
76 views

Equivalence of the two cosine definitions

There are at least two ways to define the cosine function: You can define it with a right triangle in the unit circle and extend the definition to $\mathbb{R}$. (classic definition) The other ...
5
votes
2answers
2k views

Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
3
votes
3answers
630 views

Area of the field that the cow can graze.

How do we find the area that the cow can graze? The question goes as follows-- There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as ...
15
votes
3answers
636 views

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
6
votes
5answers
1k views

Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse

I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as ...
5
votes
2answers
4k views

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 ...
1
vote
2answers
130 views

For what $n$ does a hyperbolic regular $n$-gon exist around a circle?

Does there exist a relationship in terms of $r$ and $n$ to represent how large $n$ must be if $r$ of the circle is given in the hyperbolic plane? (The edges of the regular $n$-gon are tangent to the ...
122
votes
14answers
19k views

What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 ...
126
votes
12answers
19k views

How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on ...
31
votes
4answers
2k views

Volume of Region in 5D Space

I need to find the volume of the region defined by $$\begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...
43
votes
4answers
2k views

A circle rolls along a parabola

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency ...
24
votes
8answers
3k views

Books on classical geometry

I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd ...
28
votes
6answers
919 views

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 ...
42
votes
10answers
39k views

Why is the volume of a cone one third of the volume of a cylinder?

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by ...
38
votes
3answers
5k views

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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
14
votes
1answer
3k views

Parametrizing implicit algebraic curves

Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker to thank). I learned early on that for the purposes of generating plots on a ...
19
votes
6answers
2k views

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 ...
14
votes
2answers
2k views

Probability that the convex hull of random points contains sphere's center

What is the probability that the convex hull of n+2 random points on n-dimensional sphere contains sphere's center?
5
votes
2answers
4k views

Decomposing an Affine transformation

An affine transformation is composed of rotations, translations, scaling and shearing. In 2D, such a transformation can be represented using an augmented matrix by $$ \begin{bmatrix} \vec{y} \\ 1 ...
42
votes
4answers
3k views

Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what ...
16
votes
4answers
1k views

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 ...
14
votes
4answers
892 views

Coloring the faces of a hypercube

I will restate the 3-D version of the problem. In how many ways can you color a regular cube with 2 colors up to a rotational isometry. The answer is of course a special case of Burnsides Lemma which ...
8
votes
5answers
25k views

Height of a tetrahedron

How do I calculate the height of a regular tetrahedron having side length $1$ ? Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point ...
11
votes
1answer
4k views

Calculating Distance of a Point from an Ellipse Border

I'm thinking about using oriented ellipses to represent curves (dents/bumps etc.) in my physics engine, and have a few questions about working with them: What methods are there to finding the ...
7
votes
2answers
404 views

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 ...
6
votes
6answers
6k views

Visual Ways to Remember Cross products of Unit vectors? Cross-product in $\mathbb F^3$?

Objective to find visual and accessible ways to remember this formula fast $$(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)$$ I have used Sarrus' rule but it is slow, more here. Since it is slow, I have ...
5
votes
3answers
2k views

How do I measure distance on a globe?

I have a $3$-D sphere of radius $R$, centered at the origin. $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are two points on the sphere. The Euclidean distance is easy to calculate, but what if I were to ...
14
votes
6answers
607 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
5
votes
1answer
1k views

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 ...
3
votes
1answer
900 views

How to tile a sphere with points at an even density?

I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon ...
2
votes
3answers
2k views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
2
votes
2answers
1k views

Determing the distance from a line segment to a point in 3-space

Imagine I have a line segment defined by endpoints $p_1$ and $p_2$, and some 3-space coordinate $q$. Is there a robust (in the sense of never giving divide-by-zero errors) way to quickly determine ...
2
votes
3answers
737 views

Inscribing a rhombus within a convex quadrilateral

I was wondering if it is possible to inscribe a rhombus within any arbitrary convex quadrilateral using only compass and ruler? If possible, could you describe the method? If not could you give an ...
13
votes
4answers
707 views

What is so special about triangles?!

Take any random triangle. If we draw internal-angle-bisectors of all its angles, they intersect at the same point. If we draw the perpendicular bisectors of each side (although they aren't ...
8
votes
2answers
10k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
6
votes
1answer
867 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
3
votes
2answers
262 views

Check Points are line, triangle, circle or rectangle

How to determine geometric properties of four distinct points in a plane (x1,y1), (x2,y2), (x3,y3), (x4,y4) represented in the 2-D Cartesian coordinate system, whether these four points are on a ...
3
votes
6answers
455 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
2
votes
1answer
411 views

Trapezoid Root Mean Square

I'm trying to prove that length of the line $AB$, parallel to both bases of a trapezoid, that cuts a trapezoid into two trapezoids of equal area is the Root Mean Square of the bases. In other words, ...
2
votes
2answers
4k views

A line which bisects two sides of a triangle is parallel to the third.

The answers to this question imply that a line which bisects two sides of a triangle must be parallel to the third side. Why is this true? There must be a simple proof. More generally: Let D and E be ...
2
votes
2answers
8k views

How to find the third coordinate of a right triangle given 2 coordinates and lengths of each side

p2 |\ |b\ | \ A| \C | \ |c___a\ p1 B p3 If given point p1 & p2, side A & B how would you find point p3? I know given this information you ...
1
vote
5answers
9k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
4answers
176 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
0
votes
1answer
539 views

Orthonormal vectors in Polar coordinates, show $\hat{e}_R=\frac{(x,y,z)}{r}$

Definitions Unit vector has length 1. Orthonormal vectors are orthogonal and unit vectors. RobJohn's suggestions for the basis in polar coordinates, here, satisfy the criteria but how can ...
0
votes
3answers
190 views

Find the equation of the line in standard form.

Find the equation of the line that passes through the points $(2, 9)$ and $(8, 6)$. Write the equation in standard form.
7
votes
3answers
3k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.