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

20
votes
2answers
720 views

What is (a) geometry?

There is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it. There is also no question what (universal) algebra is and what it's about. ...
3
votes
3answers
41 views

Distance between the nail and the center of the disk

Suppose you have a disk with radius $r$ and a string of length $2 \pi r+l$, i.e. longer than the perimeter of the disk. Hang the disk (of center $O$) from the nail at $A$ using the string as shown ...
1
vote
2answers
27 views

Space formed by dot products of three vectors

Suppose I have 3 3D unit vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{V}$. I define the three corresponding scalars $u_1=\mathbf{v}_1 \cdot \mathbf{V}$, $u_2=\mathbf{v}_2 \cdot \mathbf{V}$, ...
0
votes
1answer
14 views

Geometry - Volume of a distorted tent

How would one calculate the volume of a tent shaped object with the upper edge not parallel with the base plane of the tent?
1
vote
1answer
277 views

Spherical coordinates of a unit vector around a normal $N$

So if I have a unit normal for a surface $N(x,y,z)$ and an incident unit vector $V(x,y,z)$ to that surface, how would I represent the vector V in spherical coordinates relative to the normal?
2
votes
1answer
15 views

What is the solid angle of the intersection loop between a cone and an off-axis sphere?

An upright (green) cone with opening angle $2a < \pi/10$ has its vertex at point O with cartesian xyz coordinates $(0,0,0)$. The cone axis (dotted line) lies in the plane $y=0$ and is parallel to ...
0
votes
4answers
36 views

Ellipse focal proof

In the ellipse with equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ How to proof the production of distances between focal and a random tangent is $b^2$ $F1A * F2B = b^2$
3
votes
3answers
56 views

How to solve for $\theta$ in an expression involving linear and $\sin$ terms

While trying to solve a spatial geometry problem, I came across the expression: $$156θ-36\sin\theta=554.8$$ And I have no idea where to even begin.
1
vote
2answers
276 views

Constructing a Cone and its Normal Vectors in Spherical Coordinates

I am attempting to construct a right circular cone of maximum radius $R$ and angle $\theta$ in spherical coordinates, then find the normal vector of the surface of this cone at all points. Here's what ...
1
vote
1answer
28 views

How can you prove that perimeter of right triangle equal to diameter of incircle and twice the diameter of circumcircle? [on hold]

How can you prove that perimeter of right triangle equal to diameter of incircle and twice the diameter of circumcircle?
2
votes
1answer
21 views

Are the principal congruence subgroups of SL(2,Z) normally generated by a single element?

Let $N\ge 3$, then would I be correct in saying that the principal congruence subgroup $\Gamma(N)$ (defined to be the 2x2 matrices in $SL(2,\mathbb{Z}$) congruent to the identity mod $N$) is the ...
11
votes
0answers
89 views
+50

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
3
votes
1answer
503 views

Mapping random points on a sphere onto a uniform grid

Say I had an arbitrary sphere that is covered in a uniform triangle mesh of N elements with each element having a unique sequential index. If given the coordinates of a random point on the surface of ...
0
votes
0answers
30 views

A question on matching points in the plane

Let $A,B\subset\mathbb{R}^2$ with $|A| = |B| = 5$. For any $x\in \mathbb{R}^2$ denote by $A_x\subseteq A$ the set of points $a\in A$ such that $a\leq x$ (product order). We know that the following ...
3
votes
3answers
404 views

How did the Ancient Greeks know that the circle method of finding square roots was mathematically valid? How do we know that?

The Ancients used this method. (or at least James Grime said in a numberphile video) To construct the square root of a number, draw an interval of length $a+1$, and then draw a semi-circle with the ...
2
votes
0answers
25 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
6
votes
4answers
12k views

finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in x-y plane? One approach is to find the length of each side from the coordinates given ...
2
votes
1answer
75 views

Surface normal to point on displaced sphere

I want to calculate the surface normal to a point on a deformed sphere. The surface of the sphere is displaced along its (original) normals by a function $f(\vec x)$. In mathematical terms: Let ...
8
votes
4answers
341 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
2
votes
1answer
85 views

Problem of a circle tangent to three other circles

Two circles with centres A and B and radii 14 and 7 units respectively touch each other externally. M is the mid point of segment DE and is the centre of the circle with radius 21 units. The two ...
0
votes
0answers
21 views

Isoceles Triangles on a Grid Proof

Given: A Finite Set of Unit Squares on a Large Grid. If we were to choose one of those sets of unit squares, we see that the squares of the set are tiled with isoceles right triangles, each with a ...
-1
votes
0answers
20 views

Convex basis and conical basis (how to draw?)

There is a question, I'm struggling with: Find vertices of the following described polyhedron, $P:=P(A,b)=conv(V)+cone(E)$ where $V$ is the set of all vertices of $P$ and $E$ is the set of all ...
0
votes
1answer
15 views

Calculating my location based on known location

This question is linked to Can known object be used to back-calculate my location? (been almost a month, figured it would be best to start a new question.) I have a map, and I know which way true ...
1
vote
2answers
2k views

How to prove volume and surface area of sphere [duplicate]

Possible Duplicate: Why is the volume of a sphere $\frac{4}{3}\pi r^3$? We know that the surface area of a sphere is $4\pi r^2$ and the volume is $\frac{4}{3}\pi r^3$, where $r$ is the ...
2
votes
1answer
21 views

Does there exist a 4D torus with a spherical cross-section, analogous to a circle for the 3D case?

I don't mean to be a bother. It seems as though the answer may be obvious, but then, seemingly simple math questions can have surprising answers. I should also like any pointers re: the general ...
2
votes
2answers
37 views

Jacobi Elliptic Functions Special Case

I have spent some time analysing the pendulum problem, and hence the Jacobi elliptic functions recently, and have come across what seems to me to be a slight inconsitency. I define my $am(t|k)$ as the ...
1
vote
0answers
38 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
0
votes
1answer
19 views

Polygonal sides when interior angle relationship is given

The difference between any two consecutive interior angles of a polygon is 5°. The smallest angle is 120°. Find the number of sides. I know that the sum of interior angles of a polygon is ...
-2
votes
0answers
31 views

Show that the Axioms are satisfied [on hold]

A real number x is called dyadic, if it can be written as $x = a/2^n$, for some integers $a,n \in \Bbb Z$. In particular, all the dyadic numbers are rational numbers. Let $\Pi = \{(a/2^m, b/2^n)|a, b, ...
-1
votes
1answer
25 views

Show that the midpoint of $AB$, $AC$, and $DE$ are aligned.

Let $ABC$ be a rod, $D$ and $E$ two points such as: $\vec{EC} = k \cdot \vec{EA} / \vec{DA} = k \cdot\vec{DB}$. How can I show that the midpoint of $AB$, $AC$, and $DE$ are aligned?
2
votes
1answer
76 views

Finding slope from straight line equation

Line $k$ lies in the $xy$-plane. The x-intercept of line $k$ is $−4$, and line $k$ passes through the midpoint of the line segment whose endpoints are $(2, 9)$ and $(2, 0)$. What is the slope ...
-1
votes
0answers
48 views

Bisectors of opposite angles of a circular quadrilateral meet at the diagonal.

Let ABCD be a circular quadrilateral so that the bisectors of angles ABC and ADC meet at the diagonal AC. Let M be the midpoint of AC. Let q be a line parallel with the side BC so that q passes ...
1
vote
1answer
262 views

shorter of shortest paths between two points via a pair of lines

The following exercise is from the book What is Mathematics by Richard Courant and Herbert Robbins: Given two lines L, M and two points P, Q situated inside the angle formed by the two lines, the ...
-5
votes
1answer
71 views

Can anyone tell me how to factor this expression? [on hold]

...$\dfrac{2x^2-4x}{x+10}$ this is in response to a side splitting theorem question where these values are part of the proportions. Here is the complete question: We have 2 similar right triangles. ...
-2
votes
1answer
32 views

Proof involving circumradius of triangle and Law of Sines

Show that in any triangle, we have $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right), $ where $R$ is the circumradius of the triangle. I'm not quite ...
0
votes
1answer
15 views

isoperimetric inequalities in permutohedron

Consider the graph whose vertices are all n! permutations of numbers 1..n and there is an edge between two vertices iff we can get from one to another by an adjacent transposition. We call this graph ...
2
votes
4answers
10k views

How to find length of side of equilateral triangle when have radius of circle inscribed inside?

If I have a circle inscribed inside an equilateral triangle, and I know the radius of the circle, what is the formula to determine what the length of the side of the triangle is?
0
votes
1answer
29 views

Fattest scalene quadrilateral

What angles of a plane scalene quadrilateral maximize its area? By 'scalene' I mean the four lengths are unequal. It is known that if a quadrilateral has opposite sides equal and parallel as a ...
0
votes
0answers
26 views

Geometrical interpretation of an overidentified linear system

In my econometrics class we talked about Instrumental Variables. Suppose one has a $n\times k$ matrix $X$ of regressors and a $n\times m$ matrix $Z$ of instrumental variables. Given the matrices are ...
3
votes
1answer
46 views

Rectangle circumscribed to an ellipse of max area/perimeter

I could solve the classical problem of maximizing the area (fixing the perimeter) or maximizing the perimeter (fixing the area) of an inscribed rectangle, but I don't know how to solve ...
-2
votes
0answers
20 views

hollow sphere problem [on hold]

A hollow space on earth surface is to be filled. Total cost of filling is 20000 Rupees. The cost of filling per mt3 is 225 Ruppes. How many times a size of 3 mt3 soil is required to fill the hollow ...
0
votes
1answer
49 views

Books Authored by P.S. Modenov

Does anyone have any digital copies of the English translated versions of the books written by Soviet mathematician, Peter S. Modenov?
1
vote
5answers
301 views

Relation between edgelengths in a tetrahedron with two right angles and three equal edges

I have got a problem I can't solve myself. I had an attempt, but it's wrong. I was told to draw a grid of this tetrahedron and then it's easier to find a solution (I tried it, but I don't see ...
37
votes
3answers
2k views

Cutting up a circle to make a square

We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if ...
1
vote
2answers
675 views

What's wrong with this solution of Tarski's circle-squaring problem?

Tarski's circle-squaring problem asks whether it is possible to cut up a circle into a finite number of pieces and reassemble it into a square of the same area. Note that this is different from the ...
1
vote
1answer
29 views

Geometrical calculation to determine size difference between two rectangles when rotating one

I've asked a programming question on StackOverflow here which should give you a good understanding why I'm trying to do this. I'm asking it here because it's now down entirely to the mathematics of ...
1
vote
1answer
32 views

Find part of segment between two circle centers

I drew the following image to help me explaining the question: Having two circles Source and Target, I want to build an arrow like in the image. The Source has coordinates $Source(sx, sy)$ and ...
3
votes
1answer
42 views

Inequality between area and boundary length, $4\pi A \leq L^2 $

Suppose we have a simply connected region $R$ in $\mathbb{R}^2$ with area $A$ and the boundary of $R$ is a curve sufficiently well behaved (say piecewise $C^1$) that we can say it has length $L$. Then ...
-7
votes
0answers
44 views

An Interesting Areas Question [on hold]

Let the area of the triangle $ABC$ be $x$. The points $A_1$, $B_1$ and $C_1$ are the mid points of the sides $BC$, $CA$, and $AB$ respectively. The point $A_2$ is the mid point of $CA_1$. Lines ...
0
votes
1answer
33 views

Ratio of squares touching edge of circle?

Consider an infinite amount of squares stacked on top of each other where the top left corners are touching the edge of a circle: Call the largest blue square x. How would I find the ratio of ...