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

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13
votes
3answers
4k views

How to intuitively see that the volume of a pyramid = 1/3 x area of base x height

I'm interested to know if anyone can point me to a non-calculus way of seeing that the volume of a pyramid = 1/3 x area of base x height. Yes, I've googled.
13
votes
4answers
750 views

Developing the unit circle in geometries with different metrics: beyond taxi cabs

My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" ...
12
votes
5answers
867 views

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
18
votes
2answers
829 views

Reuleaux Rollers

The Reuleaux polygons are analogs of the regular polygons, except that the "sides" are composed of circle arcs instead of lines. It is known that for an odd number of sides, e.g. the Reuleaux ...
3
votes
1answer
448 views

Algorithm for shortest path on manifold

What are some algorithms used for finding the shortest path between 2 points on a (riemannian) manifold (the manifold may have a smooth boundary)? So far I've had 3 ideas, none of which seem that ...
1
vote
1answer
1k views

Can two angle bisectors of a triangle be perpendicular?

Interesting problem... I tried solving it through isosceles triangles, and I've proved it doesn't work on equilateral triangles. Can anyone give me some hints towards a geometric proof?
2
votes
1answer
1k views

Calculating the height of a circular segment at all points provided only chord and arc lengths

Please imagine that we have a circular segment with some arc length 's' and chord length 'a' (using notation from http://mathworld.wolfram.com/CircularSegment.html). Provided only 'a' and 's', and ...
2
votes
1answer
429 views

“circumference” of a sphere

What is the name for I can only describe as the circumference of a sphere. I mean like the equator on a planet. What is the line called which goes around the entire sphere, it can be anywhere (so not ...
5
votes
2answers
833 views

Splitting equilateral triangle to 5 parts

Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable from each other by a rigid motion) parts?
2
votes
5answers
194 views

Existence of lines not containing given points in general position

I remember seeing something like the following problem in the past and would like to know if it has a solution (or if I can find a source for it). Problem Given a finite set of points in the plane in ...
6
votes
3answers
247 views

Uniqueness of length minimizing geodesic

In a compact hyperbolic Riemann surface without boundary tbe length minimizing geodesic between two points $p$ and $q$ is unique.
1
vote
3answers
213 views

How to write the equation of a line in $\mathbb C^n$

I want to write the equation of a line in $\mathbb C^n$ passing through a point $(z_1,z_2,...,z_n)$. Actually I have a set of points and I suspect they all lie in the same line which passes through ...
7
votes
3answers
308 views

Lower bound on product of distances from points on a circle

Let $C$ be a circle of radius $r$ with $n$ points. Prove that there is a point on the circle such that the product of the distances from this point to the other $n$ points is greater than $r^n$. So we ...
1
vote
1answer
355 views

How to calculate a Bézier curve with only start and end points?

This animation from Wikipedia shows basically what I want to accomplish, however - I'm hoping to have it flipped around, where it starts progressing more towards the destination and "up" (in this ...
4
votes
1answer
543 views

Hyperbolic geometry. 3 dimensions. What is not well understood?

According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions. http://mathworld.wolfram.com/HyperbolicGeometry.html What isn't well understood about ...
1
vote
1answer
621 views

Annuloid (Torus)-line intersection

I need calculate ray (line) intersection with torus for my ray-tracing program (I know, its to graphics, but i need math behind it). I can solve equation of order x^4, but thats too way slow ...
20
votes
7answers
3k views

How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
2
votes
3answers
795 views

Nice geometric parallelepiped proof?

Question 1: The volume of a parallelepiped in $\mathbb{R}^n$ with n sides given by the vectors $(x_{1_1}, x_{1_2} ... x_{1_n}), (x_{2_1}, x_{2_2} ... x_{2_n}) ... (x_{n_1}, x_{n_2} ... x_{n_n})$ and ...
7
votes
2answers
896 views

Penrose Tile generator

Does anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site? I found this ...
12
votes
2answers
434 views

Decomposing the plane into intervals

A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to ...
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 ...
2
votes
3answers
206 views

Find all points with a distance less than d to a (potentially not convex) polygon

I have a polygon P, that may or may not be convex. Is there an algorithm that will enable me to find the collection of points A that are at a distance less than d from P? Is A in turn always a ...
1
vote
3answers
4k views

How to find rectangle intersection on a coordinate plane

Given the coordinates of two rectangles on a coordinate plane, what would be the easiest way to find the coordinates of the intersecting rectangle of the two? I am trying to do this programatically.
1
vote
2answers
194 views

What automorphisms of R^n preserve convexity?

I believe I'm right in saying that an isometry of a convex figure will be convex. There are other automorphisms that have this property, for example "stretching along one axis" (e.g. $f(x,y) = ...
0
votes
1answer
90 views

is surface always locally connected area in euklidian 3d?

I have some geometry in 3d (like road surface or terrain surface) and there could be several not connected parts of a road. So I'd like to know if I may unite this two not connected pieces of road in ...
3
votes
3answers
403 views

θ = (length of arc)/(angle subtended by it). How?

Thats it! Thats what I want to know. If θ is the angle subtended by an arc of length L at its center with radius R. We know, θ = L/R. How did we get this? Please don't say we got it from ...
3
votes
3answers
274 views

Riemannian 2-manifolds not realized by surfaces in $\mathbb{R}^3$?

A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$ (i.e., distance measured by shortest paths on $S$) is a Riemannian 2-manifold: differentiable because ...
9
votes
2answers
2k views

Finding an angle within an 80-80-20 isosceles triangle

The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it. So ...
0
votes
2answers
49 views

What is the number of sides of the hollow area generated by juxtaposition of four $2n$-sided regular polygons?

Here's an horrible drawing that tries to explain what I'm asking: Trying this with small numbers gives me $f: 4 \to 0, 6 \to 4, 8 \to 4, 10 \to 8, 12 \to 8.$ This suggests that $$f(2n) = 4 \times ...
11
votes
3answers
767 views

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 ...
-2
votes
1answer
352 views

Greek Geometry Algebra [closed]

Tell me everything you know about Euklid, Pythagoras etcetera. I'm about to begin reading about greek algebra used in calculating geometric figures. After reading I'm going to write a 15-20 pages ...
1
vote
1answer
1k views

Calculate Euler yaw pitch and roll for a 3d point to face another

This may be a really basic question but the typical atan2 method isn't working when I write it in a program. I need a camera object to face a certain target in 3D space. I need to calculate the 3 ...
3
votes
1answer
422 views

Is there an algorithm out there that will show you how to fit the most rectangles into a circle?

Rectangles will all be the same size, circle may vary. I was discussing this with a friend and he said to just start placing them in the middle and moving outwards but it seems that you could free ...
2
votes
0answers
1k views

How to extract Euler angles from a a point in a plane?

Given a certain coordinate frame, I can compute a new one by applying a set of rotations in a given order (what I call Euler Z-Y-X). So I yaw, then pitch then roll. Now imagine that I want to do ...
1
vote
1answer
152 views

Properties of a trisected rectangle?

This is part of a homework question I'm trying to figure out the perimeter of the black-filled object below (sorry, it's crudely drawn in paint, but I think you get the idea). To find the perimeter, ...
5
votes
1answer
189 views

Area of a sector between a point and a function determined by an angle

I've been trying to find a way to do this: given a point $P(\alpha,\beta)$, a function $f(x)$, and an angle $\theta$, find the area of the sector determined by extending a horizontal line from $P$ to ...
0
votes
1answer
446 views

Coordinate substitution between two cartesian systems

I have multiple objects in my lab which are fully described in two coordinate systems. Both systems are Cartesian and in meters. If they had a common origin but were rotated relative to each other I ...
3
votes
3answers
437 views

Tangent to two disks: Roots of a 4th-degree polynomial?

Suppose you would like to find the two tangent lines that support two given disks in the plane to the same side. Parameterizing the circles using $( \cos \theta, \sin \theta )$, I find that ...
1
vote
0answers
72 views

Algorithm for approaching a position

I'm looking for an algorithm, that can process an amount of (GPS-) positions and determine, if the actual position gets closer to a way point. What is the "best" way to filter variability in the ...
2
votes
2answers
288 views

Convex polyhedron with open faces

Convex polyhedron $P$ is a subset of $\mathbb{R}^n$ that satisfies system of linear inequalities \begin{align} a_{11}x_1 + \cdots + a_{1n}x_n & \sim_1\, c_1 \\ & \vdots \\ a_{p1}x_1 + \cdots + ...
1
vote
2answers
2k views

how to find point lie on the arc

i have arc A ,i know startangle,endangle,startPoint,Endpoint,centre,radius of arc and i have point B, i like to find point B lies or not in arc A , i need formula or algorithum for this
2
votes
1answer
170 views

What is the name of this lattice?

Suppose we have an atom at every point with integer coordinates in $\mathbb{R}^d$. Take a ($d-1$)-dimensional hyperplane going through $\mathbf{0}$ and orthogonal to $(1,1,1,\ldots)$. What is the name ...
2
votes
4answers
146 views

how do find the line lies on another line

I have Line $A$ with points $(1,1)$ and $(8,8)$ and another Line $B$ with points $(2,2)$ and $(4,4)$. I would like to prove Line $B$ lies on Line $A$. How can I do this?
8
votes
4answers
653 views

Orientability of $\mathbb{RP}^3$

I was wondering if there is a nice way to see that $\mathbb{RP}^{3}$ is orientable without using tools of algebraic topology, like homology. The only think I could think of was to argue that ...
6
votes
2answers
493 views

Vortex Voronoi diagram?

Suppose there are a finite number of disjoint unit-radii disks in the plane, each spinning clockwise or counterclockwise at the same angular velocity. The plane is filled with a thin fluid layer, and ...
0
votes
3answers
448 views

Graphing a Parametric Polynomial based on a given set of points

I have been tasked with creating a C++ program (with GDI+ for graphics) that takes a set of user defined points and creates polynomial curve through them. For extra credit, I have to support a ...
6
votes
3answers
1k views

geometry and topology

I was wondering what are the differences and relations: between geometry and topology; between differential geometry and differential topology; between algebraic geometry and algebraic topology? ...
1
vote
4answers
158 views

Finding a line that satisfies three conditions

Given lines $\mathbb{L}_1 : \lambda(1,3,2)+(-1,3,1)$, $\mathbb{L}_2 : \lambda(-1,2,3)+(0,0,-1)$ and $\mathbb{L}_3 : \lambda(1,1,-2)+(2,0,1)$, find a line $\mathbb{L}$ such that $\mathbb{L}$ is ...
0
votes
3answers
892 views

Intersection of Cubic curves

This is the question which i am attempting to solve, and it seems to difficult to get rid of the exponents. Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect in ...
4
votes
3answers
3k views

Finding point coordinates of a perpendicular bisector

Given that I know the point coordinates of A and B on segment AB and the expected length of a perpendicular segment CD crossing the middle of AB, how do I calculate the point coordinates of segment ...