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

4
votes
2answers
155 views

Changing the manifold, preserving the discrete spectrum

On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined. If $M$ is not compact, then $L$ admits a continuous spectrum. Is there a way of "changing" $M$ and/or $L$ in say $M'$ ...
0
votes
2answers
172 views

some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...
0
votes
1answer
51 views

How can I count waypoints between a curve?

I have curve that is drawn between point A and B. I want to divide this curve to 100 smaller waypoints. How can I determine what these 100 waypoints are as coordinates, when I only know points A and ...
1
vote
1answer
152 views

Decomposition of any point in the unit hypercube as a positive linear combination of polynomial number of vertices

I have function values at each of the vertices of the hyper cube. What would be a natural interpolation of the function to each point on and inside the cube that can be written as a positive linear ...
3
votes
1answer
73 views

Intersection of Sphere and Line in $\mathbb{R}^n$?

This seems to me as a very simple and basic question, though I'm having trouble with it. The Problem Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
2
votes
1answer
336 views

Finding the radius of the largest sphere possible between a corner and another sphere

In a 3 dimensional Cartesian plane there is a sphere A that is in the first octant and is tangent to all coordinate planes. Now, imagine we want to find the another sphere B also tangent to all ...
16
votes
3answers
2k views

Proving Stewart's theorem without trig

Stewart's theorem states that in the triangle shown below, $$ b^2 m + c^2 n = a (d^2 + mn). $$ Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
1
vote
1answer
501 views

Coordinates of point on a line defined by two other points with a known distance from one of them

I have two points in 3D space; let's call them $A=(a_x, a_y, a_z)$ and $B=(b_x, b_y, b_z).$ Now, I need to place a third point, let's call it $C=(c_x, c_y, c_z)$, which lies on the line between $A$ ...
3
votes
0answers
201 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
1
vote
2answers
548 views

How can I learn de casteljau algorithm? (from calculus)

I'm an highschool graduate who is currently waiting for college. Meanwhile, I'm trying to do a little project by myself. (Computer stuff) And yesterday, I found that I needed to deal with something ...
14
votes
1answer
376 views

Two tetrahedra are congruent given a certain condition

This question is inspired by a Miklos Schweitzer problem, namely Problem 9./2007 Let $A$ and $B$ be two triangles in the plane such that the interior of both triangles contains the origin, and for ...
7
votes
2answers
468 views

Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and ...
2
votes
1answer
155 views

Divisors over a projective curve

I have a few doubts about some algebraic geometry problems. This is the situation: X is a (smooth) curve on the projective plane $P^2(F)$, F being an algebraic closure of $F_2$ $X = ...
1
vote
1answer
258 views

Projecting a nonnegative vector onto the simplex

Given an elementwise nonnegative vector $y$, I'd like to find the projection of $y$ onto the simplex $S: \{ (x_1, \ldots, x_n) ~|~ \sum_{i=1}^n x_i=1, x_i \geq 0 \mbox{ for all } i \}$. Is there a ...
2
votes
1answer
115 views

Join of simplices and spheres

Suppose $c$ a simplex such that it is the join of two of its faces, say $c=a\ast b$. Let $S$ be defined as $$ S=\left\{\sigma\in \partial(c')\big\vert \ \sigma \textrm{ has no vertex in } a\cup ...
1
vote
1answer
324 views

why does multiplicatively weighted voronoi diagram (mwvd) with 2 sites create a circle?

I want to understand the structure of a multiplicatively weighted voronoi diagram. I found that the bisector between 2 sites is circle shaped, but couldn't formally ...
1
vote
0answers
162 views

Distance between two unbounded sets

How to calculate the distance between two (possibly unbounded) ranges of positive real numbers? For example, if three guys specify their prices they would pay for a product: ...
7
votes
3answers
416 views

Reconciling 'intersecting planes' and 'linear transformation' interpretations of matrices

I've learned in linear algebra class that an $n \times m$ augmented matrix can be thought of as a collection of n planes in $\mathbb {R}^m$ . If the matrix is invertible, the planes all intersect at a ...
12
votes
2answers
388 views

Why isn't $ SA_{\text{cube}} = 3x^2 $?

Related to "volume of sphere", why is the surface area of a cube not equal to the derivative of its volume? If you think about a sphere, it makes sense that the rate of change of the volume (with ...
8
votes
1answer
412 views

hatchet planimeter

How does this instrument work? Here is a video that demonstrates its use. After reading the wikipedia page, I still have no idea how it works. Any explanations that are easier to comprehend?
21
votes
3answers
3k views

Is a line parallel with itself?

Simple Question, but I'm finding a lot of dispute on the "lesser" internet. Basically, given a line, is it parallel with itself?
1
vote
2answers
317 views

Volume of a hypercone with cap

What is the volume of this region: $ C(x,\eta(x),\theta,r) = \{{x+\lambda y\in\mathbb{R}^d: y\in\mathbb{R}^d, ||y||_2=1, y^T\eta(x)\geq \cos\theta, \lambda\in[0,r]\}} $ where ...
1
vote
2answers
96 views

Drawing an altitude from origin to the opposite side of a triangle

If I know the vertexes of a triangle and one of them is origin O(0, 0). Then how can I draw altitude from origin to the opposite side of the triangle.
1
vote
3answers
969 views

Isosceles trapezoid with semicircle

I have an isosceles trapezoid, with a semicircle in the middle. I need to know the difference in area of the two shapes. I Radius of the semicircle is 6cm, and the longest base is 14cm.
3
votes
2answers
286 views

How to get 'rectangular size' of arbitrary circular sector?

Given a circular sector defined by sweeping from a 'start' to a 'stop' angle (see diagram below) and a radius, how do you compute the bounds of the rectangle that fits to the edges of the sector? ...
2
votes
2answers
741 views

How to get the cardinal direction from one location to another?

Given are two geo locations, each with latitude and longitude. One is the current location, the other is a target location. is there a formula for calculating the target's cardinal direction for 0 ...
3
votes
2answers
3k views

Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
1
vote
0answers
229 views

Is it possible to make pseudo-perspective out of an isometric image?

Is there a trick to take an isometric image of cubes - that can be thought of as a set of equidistant points, triangles or planes - and in a trivial fashion create a vanishing point -type perspective ...
3
votes
2answers
111 views

length minimizing special linear transformation

Suppose $\gamma:\mathbb{S}^1\to\mathbb{R}^2$ is a smooth origin symmetric strict convex curve. Is there any special linear transformation $A\in SL(2,\mathbb{R})~$ such that the length of $A\gamma~$ ...
9
votes
3answers
1k views

The vertices of an equilateral triangle are shrinking towards each other

For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
6
votes
4answers
3k views

Can a circle truly exist?

Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says... A circle is a simple shape of Euclidean geometry ...
6
votes
3answers
93 views

Algorithm to determine if a collection of unit discs covers the unit disc centered at the origin?

I have a list of points $ (x_i, y_i) $ for $i = 1...n$. Is there an algorithm to determine if the union of the unit discs centered at these points is a superset of the unit disc centered at $(0, 0)$? ...
4
votes
2answers
349 views

Why are only singly and doubly ruled non-planar surfaces found? Why not triply ruled?

Is there a reason why there are no triply-ruled surfaces found in spatial geometry? Does it have to do with the fact that there are at most two dimensions/parameterizations for a surface? If that's ...
0
votes
1answer
3k views

Intersection of two vectors using perpedicular dot product

I am currently working my way through a book on Game Programming in Actionscript 3, and have gotten to a formula required for collision detection on finding the intersection point of two vectors. Im ...
5
votes
2answers
661 views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
28
votes
5answers
3k views

How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I ...
75
votes
4answers
4k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
5
votes
3answers
775 views

Largest Triangle with Vertices in the Unit Cube

How would one find a triangle, with vertices in or on the unit cube, such that the length of the smallest side is maximized? And what is that length? A lower bound for the length is $\sqrt{2}$, by ...
1
vote
1answer
416 views

Foci of a general conic equation

The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF. Is there a similar equation for getting ...
4
votes
1answer
2k views

Find the point in a triangle minimizing the sum of distances to the vertices

Given a triangle in a plane with vertices A, B, C, find the point T that minimizes the sum of distances between A-T, B-T, and C-T. I can experimentally determine this point by sampling the space and ...
2
votes
0answers
240 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
3
votes
1answer
82 views

Construction of representations

Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?
4
votes
1answer
514 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
0
votes
1answer
495 views

Is there a relation of fixed points and eigenvalues of a $3\times 3$ matrix

Let $v \in \mathbb R^3$. Given a matrix $M : \mathbb R^3 \mapsto \{v\}^\perp$, that is, there is one vector $v$ such that $\forall x \in \mathbb R^3\setminus\{v\}$: $\langle Mx,v \rangle = 0$. ...
1
vote
1answer
201 views

How to draw triangle in the middle of the line?

I need to draw a triangle like an arrow in the middle of the line. How can I calculate triangle's coordinates in order to draw it in the middle of the line PLEASE? ** UPDATE ** Here I have found ...
1
vote
2answers
2k views

How to calculate the middle of a line?

My question is following. I have a line with a given (X1, Y1) and (X2, Y2) coordinates (see figure below). I need to calculate ...
2
votes
0answers
390 views

algorithm for the intersections of a line and an ellipse in 2D

I am looking for an algorithm for finding the intersection of a line and an ellipse. I have the line in the form $ax+by+c=0 \qquad(1)$ and the ellipse in the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0 ...
5
votes
4answers
1k views

Find the centre of a circle passing through a known point and tangential to two known lines

I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines. The only knowns are: Line 1 (x1,y1) (x2,y2) Line 2 (x3,y3) ...
1
vote
3answers
675 views

Inscribed kissing circles in an isosceles trapezoid

5 equal circles in an isosceles trapezoid. Radius of circle is 4. Find black colored area. I don't have any ideas, could you give me a hand? Thanks.
2
votes
1answer
96 views

Two equilateral triangles

In an old IMC Shortlist, I found the following problem: Given a triangle $T$, consider the equilateral triangles $T_1\subset T\subset T_2$ such that $T_1$ is the greatest equilateral triangle ...