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

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38
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 ...
28
votes
6answers
868 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 ...
23
votes
8answers
2k 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 ...
27
votes
4answers
761 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
34
votes
3answers
4k 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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
11
votes
5answers
3k views

Geometry Book Recommendation?

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
12
votes
2answers
1k 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?
11
votes
1answer
2k 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 ...
26
votes
6answers
3k views

Why does volume go to zero?

The volume of a $d$ dimensional hypersphere of radius $r$ is given by: $$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$ What intrigues me about this, is that $V\to 0$ as ...
15
votes
4answers
907 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
808 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 ...
19
votes
3answers
937 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
9
votes
1answer
3k 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
372 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 ...
5
votes
6answers
4k 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 ...
3
votes
1answer
64 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 ...
3
votes
2answers
2k 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 ...
5
votes
1answer
727 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 ...
4
votes
3answers
19k views

Height of a tetrahedron

How do I calculate: The height of a regular tetrahedron, 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 be from ...
2
votes
3answers
1k 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
785 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
1answer
714 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 ...
5
votes
1answer
698 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: ...
5
votes
3answers
3k 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 ...
4
votes
3answers
1k 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 ...
2
votes
1answer
294 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, ...
0
votes
4answers
121 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
480 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
176 views

Find the equation of the line form

Find the equation of the line that goes through the points $(2, 9)$ and $(8, 6)$. Write the equation in standard form.
9
votes
3answers
1k views

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 ...
6
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.
5
votes
1answer
94 views

filling an occluded plane with the smallest number of rectangles

I've got a specific problem which I'll try to describe as clearly as possible. I have a defined rectangular region on a cartesian plane, and within that region there are other given rectangular ...
2
votes
1answer
252 views

Finding equation of an ellipsoid

Consider I have an ellipsoid (let say an egg) lies in a general form in 3D space. Suppose, I have the equations of two projected views of this egg (e.g. one projected view on x-y plane and another one ...
2
votes
3answers
249 views

Vector path length of a hypotenuse

Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to: $ P_n = P_x + P_y = \frac{n}{2}\times w + ...
1
vote
1answer
732 views

Optimal distribution of points over the surface of a sphere

How can one generate a distribution of N points over the surface of a sphere so that the all N voronoi cells have the same area? Which is the best algorithm for this?
0
votes
4answers
4k 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
1answer
258 views

How to calculate the area of a polygon? [duplicate]

Possible Duplicate: How quickly we forget - basic trig. Calculate the area of a polygon For a program I want to calculate the area of a given polygon. It may have every form, the sides even ...
116
votes
14answers
18k 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 ...
54
votes
10answers
11k views

What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point. However, now that I'm in university, I'm not ...
27
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{ ...
35
votes
9answers
26k 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 ...
23
votes
9answers
7k views

What is the (mathematical) point of geometric constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
17
votes
5answers
1k views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
10
votes
5answers
304 views

Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each ...
14
votes
4answers
2k views

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 ...
8
votes
2answers
1k views

What is wrong in my proof that 90 = 95? Or is it correct?

Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true? Definitions: 1. $\angle ABC=90^{\circ}$ 2. $\angle ...
32
votes
6answers
4k views

Pythagorean Theorem Proof Without Words (request for words)

I was intrigued by a book I saw called Proofs without Words. So I bought it, and discovered that the entire book doesn't have any words in it. I figured at least it would have some words explaining ...
15
votes
5answers
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 ...
13
votes
4answers
665 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
15
votes
3answers
14k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...