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

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0
votes
3answers
896 views

Triangle whose height and sides are consecutive integers

This is probably a old puzzle,and maybe you have seen it somewhere else before.Imagine a special triangle. The height and the three sides of this triangle are 4 consecutive integers.Can you figure out ...
3
votes
1answer
113 views

Eight queens problem, wondering about the non-unique solutions

I've done the code that generates all the solutions. But know I am suppose to filter out any redundant solutions based on symmetry and rotations. I have code for vertical symmetry, horizontal ...
1
vote
1answer
44 views

Geometry (X : G), X isomorphic to G?

Let us have a geometry with a set X and a group G. What if X itself can be endowed with a group structure to be isomorphic to G? Can we gain something from this?
4
votes
1answer
434 views

2D transformation

I have a math problem for some code I am writing. I don't have much experience with 2D transformations, but I am sure there must be a straight-froward formula for my problem. I have illustrated it ...
2
votes
1answer
88 views

Constructable Trigonometric Inverses

By doing some right triangle gymnastics, we can derive things like $\cos(\arctan x) = \frac{1}{\sqrt{1+x^2}}$, for $x>0$ $\cos(\arcsin x) = \sqrt{1-x^2}$ $\tan(\arcsin x) = ...
4
votes
2answers
429 views

Space-filling curve with distance locality

Is there a space-filling curve $C$ that has the property that, if $C$ passes through $p_1=(x_1,y_1)$ at a distance $d_1$ along the curve, and through $p_2$ at $d_2$, then if $|p_1 - p_2| \le a$, then ...
9
votes
1answer
2k views

Why does GPS require a minimum of 24 satellites?

From Wikipedia, The GPS design originally called for 24 SVs, eight each in three approximately circular orbits, but this was modified to six orbital planes with four satellites each. [...] The ...
2
votes
1answer
376 views

Conditions on polygon to ensure equal interior angles or opposite sides

In following, ($x$) denotes condition $x$, and ($x$') denotes condition $\neg x$. For a polygon of $2n$ sides, let it be given that (a) opposite sides are parallel. It is not difficult to find ...
5
votes
1answer
893 views

How to prove that Pi exists? [duplicate]

Possible Duplicate: Proof that Pi is constant (the same for all circles), without using limits How do we prove that the ratio of a circle's circumference to its diameter is a certain real ...
2
votes
1answer
95 views

Angles in $\mathbb{P}^2$

Do there exist a way to define angle between lines at $\mathbb{P}^2(k)$?, where $k$ is an algebraically closed field of characteristic zero.
6
votes
1answer
3k views

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?
5
votes
2answers
1k views

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? ...
5
votes
3answers
378 views

Is it always possible to simply expand a simple 2D polygon with any point?

Given a simple 2D polygon P = ( M1 .. Mn ) and a point M, is it always possible to construct a new simple polygon P' by "adding" M to P as a new vertex? If so, is this always possible without ...
-1
votes
1answer
424 views

Calculating points on the curve

I want to get the x and y coordinates of a curve..How can i do that... In the above image.Is it possible to calculate the intermediate points(one side) by knowing starting and ending point
1
vote
2answers
137 views

Is there a solid where all triangles on the surface are isosceles?

Are there any solids in $R^{3}$ for which, for any 3 points chosen on the surface, at least two of the lengths of the shortest curves which can be drawn on the surface to connect pairs of them are ...
6
votes
3answers
9k views

Orthogonal projection of a point onto a line

please give me a directions how to solve this: find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$
3
votes
3answers
7k views

Convert coordinates from Cartesian system to non-orthogonal axes

I have a 2D coordinate system defined by two non-perpendicular axes. I wish to convert from a standard Cartesian (rectangular) coordinate system into mine. Any tips on how to go about it?
1
vote
0answers
39 views

Software for knotted $\mathbb{S}^2$'s in $\mathbb{S}^4$

According to the work of J. Scott Carter you can draw pictures of knotted surfaces in four-space in several different ways. I know the man is a real artist in this, but did anybody come across some ...
10
votes
1answer
197 views

Dissection of square into triangles

Prove that a square cannot be dissected into an odd number of triangles of equal area. Got to read about the question and its history in "Algebra and Tiling Homomorphisms in the Service of Geometry ...
2
votes
1answer
1k views

When is the determinant of a Hessian matrix positive?

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a $C^2$-function and let $H=\left(\frac{\partial^2f}{\partial x_i \partial x_j}\right)_{1\le i,j\le n}$ be its Hessian matrix. Suppose I know that $ \det ...
10
votes
4answers
2k views

Ten soldiers puzzle

This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each ...
9
votes
3answers
320 views

Infinite sequence of nested, falling, colliding spheres

Imagine an infinite collection of nested, concentric spheres, of radius 1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, and so on. Suppose they are somehow suspended in space, fixed on their common ...
5
votes
1answer
172 views

Given the area and height of a rectangle, what is the width of the base of a circular segment with the same height and area?

Given a rectangle of height $h$ and area $A$, what is the width $c$ of the chord at the base of a circular segment with the same height and area? I've made a diagram of the problem: My progress ...
3
votes
2answers
8k views

How do I rotate a matrix transformation with a centered origin?

This is actually something I'm doing in Objective-C programming, but since it's very math-oriented I thought I'd post it here. I was reading up on linear transformations: ...
1
vote
1answer
624 views

What's the simplest algorithm for resizing an object inside a rectangle so that it's as large as possible?

This is a simple enough problem that I could just cover all corner cases, but I was wondering if there was an elegant way to do this. Here is the starting point. It finds out which side of the image ...
2
votes
1answer
320 views

Finding a homeomorphism guaranteed by Schoenflies Theorem

Assume I have a Jordan curve $C \subset \mathbb{R}^2$. Then by Schoenflies Theorem there exists a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(C)$ is the unit circle. Is ...
2
votes
1answer
240 views

An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
9
votes
1answer
335 views

Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?

Motivation I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
11
votes
2answers
4k views

Finding the intersection point of many lines in 3D (point closest to all lines)

I have many lines (let's say 4) which are supposed to be intersected. (Please consider lines are represented as a pair of points). I want to find the point in space which minimizes the sum of the ...
7
votes
1answer
662 views

Sum of angles | Tetrahedron

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°. Any help on getting me started out here? I am not able to get any idea as ...
1
vote
1answer
1k views

Rotation of a vector distribution to align with a normal vector

I generate a distribution of random points on a unit hemisphere whose pole is on the positive z-axis (the base lies in the x-y plane). Each point represents a directional vector $v$ in which a ray ...
3
votes
4answers
1k views

Best way to find the Coordinates of a Point on a Line-Segment a specified Distance Away from another Point

I have 4 points: $Q, R, S, T$. I know the following Coordinates for $R$, $T$, and $S$; Length of $\overline{RQ}$ That segment $\overline{RT} < \overline{RQ} < \overline{RS}$; I need to ...
0
votes
1answer
399 views

algebraic way to compute intersection of disks

Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
3
votes
2answers
111 views

Creating a special vector from two vectors

I have two vectors (and their two dimensional components): $\vec{AB}$ and $\vec{AC}$ that have the same length. How can I calculate a vector $\vec{AD}$ components that satisfies $\angle {DBA} = \angle ...
1
vote
2answers
2k views

Two plane intersection and angle between 2 planes

I am trying to implement my problems in different ways. So, may be though this question has some relation to some other questions, please answer me. We know; Intersection of two planes will be given ...
1
vote
3answers
697 views

Line/Plane intersection : in high dimension

I have 3 points in $R^d$ defining a triangle, and 2 points (still in $R^d$) defining a line. I would like to compute the intersection of this line and the triangle (or at least the plane defined by ...
4
votes
4answers
1k views

Resizing a rectangle to always fit into its unrotated space

(For those coming here looking for answers to rectangle problems it may help to see the related (and solved) question: Given a width, height and angle of a rectangle, and an allowed final size, ...
3
votes
3answers
196 views

How to calculate the x/y coordinate of F in this diagram (geometry)

In the diagram, I've provided, how do I calculate the $x$, $y$ coordinates of $F$ if the points $A$, $B$, $C$ are arbitrary points on a grid? I'm looking for a formula to solve $F's$ $X$ axis and ...
7
votes
3answers
1k views

What is the connection between linear algebra and geometry?

I am currently studying linear algebra. Yet, I found discussions about linear algebra usually explain things in a geometric fashion. I am quite confused on how to link up these two topics. Can ...
1
vote
2answers
261 views

Calculating the area of a special hexagon

How can I calculate the area of a hexagon which all of it's angles are equal given 3 of it's sides? Edit: I forgot the constraint that opposite sides have same length, e.g. for hexagon $ABCDEF$ $AB = ...
1
vote
4answers
320 views

A Sphere Containing Points of Pairwise Equal Distance

Suppose one has $m$ points in $\mathbb{R}^n$ with the property that the distance between any two of them is some fixed constant $d$. Is it true that there is a sphere (living in $\mathbb{R}^n$) ...
1
vote
2answers
146 views

Orthogonal mapping $f$ which preserves angle between $x$ and $f(x)$

Let $f: \mathbf{R}^n \rightarrow \mathbf{R}^n$ be a linear orthogonal mapping such that $\displaystyle\frac{\langle f(x), x\rangle}{\|fx\| \|x\|}=\cos \phi$, where $\phi \in [0, 2 \pi)$. Are there ...
1
vote
1answer
78 views

Solids produced from finite constructive solid geometry operations

Constructive Solid Geometry is a way of describing/building up solid objects from simpler primitive objects. Let's assume you can perform affine transformations on objects, along with the CSG set ...
5
votes
2answers
2k views

Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area

In other words, if I had a rectangle of $10\times 10$ and an angle of $45$, and the allowed area was $100\times 100$, the rectangle would be about $70\times 70$. The allowed area is $100\times 100$ ...
2
votes
1answer
225 views

What is the most accurate method to get intersection point in 3D?

I have been given 3D point data, belonging to different planar segments. Points are not exactly laid on the planes so that I have fitted best planes using least square solutions. Now, I want to find ...
0
votes
2answers
66 views

How do I apply a speed to make a point travel along a line between two points

I have point $A$ which is traveling towards point $B$. Both points have $x,y,z$ coordinates. Point $A$ has a speed. For a given time period how much would I add to the $x,y,z$ coordinates of $A$ in ...
2
votes
1answer
247 views

How to find a rectangle which is formed from the lines?

I have Cartesian coordinates $A$ and $B$. Line $AB$ is the axis (center) of the rectangle. And I have $H$ (height). I need Cartesian coordinates blue rectangle.
3
votes
1answer
91 views

Name for this triangle centre

Given a triangle I draw circles around each vertex. I chose the radii of these circles so that they are all mutually tangent. There is only one way to do this. I extend these tangents. They concur at ...
0
votes
2answers
341 views

Get Rotation in degrees (0-360) from a rotated angle?

I have a rectangle that is facing up. ($0^\circ$) I'm getting a number bettween $-1000$ to $1000$ or even more, and this number is the angle that is rotating the rectangle. How can I know the ...
1
vote
2answers
210 views

Calculating points on a plane

In the example picture below, I know the points $A$, $B$, $C$ & $D$. How would I go about calculating $x$, $y$, $z$ & $w$ and $O$, but as points on the actual plane itself (e.g. treating $D$ ...