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

10
votes
2answers
345 views

A property of circles

Consider a set of circles on a plane that don't overlap each other and any of them touches at least 6 other circles. Prove that this set has infinite number of circles. Well, there seems to be ...
0
votes
1answer
138 views

“Way” to decide if points are in a rectangle.

Suppose $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ are two points. Also suppose that we have a rectangle which we just know the value of its sides $a$ and $b$. I am looking for some kind of formulation which ...
3
votes
2answers
325 views

Notation for covariant derivative

I'm reading John M. Lee's book " Riemannian Manifolds". On page 57, the covariant derivative of $V$ along a curve $\gamma$ is defined, where $V$ is a vector field along $\gamma$. It is denoted by ...
4
votes
2answers
842 views

Recurrence for number of regions formed by diagonals of a convex polygon.

I've been having trouble with this particular problem, been thinking for it for a good hour or two, but I haven't gotten an explanation to the following question. Suppose $a_n$ be the number of ...
6
votes
5answers
2k views

If a plane is divided by $n$ lines, then it is possible to color the regions formed with only two colors.

I am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese) A plane is divided by many lines. Show that it is possible to color the regions formed with only two ...
2
votes
1answer
330 views

Total number of non-congruent scalene triangles?

What is the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length 11? What does it mean by non-congruent scalene triangles? Do we have to use ...
0
votes
1answer
2k views

Problem of packing spheres of radius $\rho$ into a cylinder

Given a cylinder of radius $R$ and length $L$, I need to find the number of spheres which is possible to pack into the cylinder as a function of the radius $\rho$ of the spheres. I found something ...
3
votes
3answers
155 views

Radius of in-circle

The questions from geometry are most fascinating to me. As a parent I love to learn geometry and shapes from my son's textbook. I came across this statement about the in-circle: Let $m, n$ that are ...
1
vote
3answers
133 views

Which of the circle the line px+qy+r=0 will intersect?

If p, q and r are in arithmetic progression, then the line px + qy + r = 0 necessarily intersects which of the following circles? $$x^2 + y^2 + 4x – 4y + 7 = ...
3
votes
1answer
209 views

Logic proportions problem

What the problem says: When a screen is placed 3 m from a projector, the picture occupies 3 m^2. How large will the picture be when the projector is 5 m from the screen? My direct answer ...
3
votes
3answers
128 views

Showing point is the orthocenter

Given a rectangle WXYZ, let R be a point on its circumscribed circle. Show that, out of the orthogonal projections of R onto WX, XY, YZ, and ZW; one out of these 4 points is the orthocenter of the ...
1
vote
3answers
386 views

Constructing a line with a known line, intersection point and angle.

I am creating a Java game with collisions. I found myself stuck on the following problem. I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, ...
2
votes
3answers
272 views

Find the angle in a triangle if the distance between one vertex and orthocenter equals the length of the opposite side

Let $O$ be the orthocenter (intersection of heights) of the triangle $ABC$. If $\overline{OC}$ equals $\overline{AB}$, find the angle $\angle$ACB.
11
votes
2answers
338 views

How to prove there are exactly eight convex deltahedra?

A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
3
votes
1answer
131 views

Snow Flake Problem: Limit of perimeter & area at $\infty$

I am supposed to find the limits as $n\rightarrow\infty$ of the perimeter & area of a snow flake. $$N_n = \text{Number of sides} = 3\cdot 4^n$$ $$L_n = \text{length of side} = \frac{1}{3^n}$$ ...
1
vote
3answers
6k views

How do I Find all Angles of 4-sided polygon given side lengths?

I have a program that lets users draw custom 4-sided shapes using java 2d. I want to calculate the angles inside the shapes so I can rotate text to the proper angle and label each side. I am trying ...
0
votes
1answer
2k views

Cone created from Sector of Circle

Suppose I use a sector of circle with radius 1 to create a cone (by joining the radius of the sector). How do I express radius of cone in terms of $\theta$? Is it $$\sin{\frac{\theta}{2}} = r, ...
2
votes
2answers
1k views

Calculate radius of variable circles surrounding big circle.

I got a circle, which I know all the details about him. (Radius [100], Diameter [200], Circumference [628.32], Area [31415.93]...) I would like to surround this circle, with smaller circles. I know ...
0
votes
1answer
889 views

Making a circle with paper folding, scissors, pencil, and a straightedge

Can we make a circle using paper folding, scissors, straightedge, anda pencil, allowing an infinite number of operations? I think my chemistry teacher have show me once how to make it during the ...
2
votes
1answer
632 views

Calculating the area of a cross-section of a tetrahedron

Ok, I completely revised my question. For those interested about my purpose with this question, see the older versions. So, I would like to calculate the area of a cross-section of a tetrahedron. The ...
2
votes
2answers
287 views

Application of prime number theorem

In the following I am referring to a small argument made in "A counterexample to Borsuk's conjecture" by Jeff Kahn, Gil Kalai (see http://arxiv.org/abs/math.MG/9307229) In this paper the authors bound ...
1
vote
2answers
4k views

Find the equation of an ellipse given its focus, directrix and eccentricity

Ellipse has a focus $(3;0)$, a directrix $x+y-1=0$ and an eccentricity of $1/2$. Find its equation. I should probably use the fact that $r/d = e$, where $r$ is the distance from the focus to any ...
2
votes
2answers
194 views

What structures does “geometry” assume on the set under study?

The Wikipedia's article for geometry is somehow overwhelming. To make things clear, allow me to ask some questions: I wonder if "geometry" can be defined as the study of a metric space (possibly ...
0
votes
1answer
74 views

Find a rotation where the shape has the least width possible on the x-axis

I am toying around with a shape problem and I am looking for a more clever solution than what I have been able to come up with. Here is the problem: I have a set of points that form an enclosed ...
0
votes
2answers
128 views

Transformation to create a zoom-in effect

I'm sure this is simple, but I can't seem to find any answers on the net (maybe it's just a matter of wording - please redirect me accordingly). Imagine a "camera" with a square viewport zooms in on ...
1
vote
2answers
2k views

Given the vertex angle and side lengths of an isosceles, find the base

I need to be able to do this programmatically, so I'll need to be able to convert an example into algebra, but for the sake of hopefully having it make more sense to me, let's say the two sides are 15 ...
0
votes
1answer
47 views

General form for mapping one disk onto another?

Say I have two disks $A$ and $B$, with two points $a\in A$ and $b\in B$. Is there a way to explicitly construct a linear fractional transformation from $A$ onto $B$ that sends $a$ to $b$? I know a ...
0
votes
1answer
325 views

How to draw a great circle of a sphere?

I am the distance on the sphere? I wonder how to draw a section of a sphere with Sktechpad or other tools? Since the sphere is shown on the plane, how to draw the great circle?
0
votes
1answer
433 views

Plane transformation

I have a plane-A which sits on the origin and where every point on the plane has a z coordinate of 0 (so there is no rotation of the plane). I have plane-B in space and I have a a point (which is the ...
8
votes
1answer
332 views

Space filling with circles of unequal radii

Here is my problem: I have a bunch of circles that I need to display inside a canvas. There are an arbitrary number of circles, each with a predefined radius. The summed area of circles is always ...
3
votes
1answer
678 views

Form of most general transformation of the upper half plane to the unit disk.

In David Blair's book on Inversion Theory, he write that the transformation $$ T(z)=e^{i\theta}\frac{z-z_0}{z-\bar{z}_0} $$ is the most general transformation mapping the upper half plane to the ...
1
vote
1answer
205 views

Is there a Möbius transformation that scales disks to the unit disk?

When working in the complex plane, often times I would like to scale a disk $|z-z_0|<R$ to the unit disk. I would first translate $z_0$ to the origin, but after that, what can we multiply by to ...
1
vote
1answer
134 views

Curvature and Radii

In my handout it is said that a circle $\Gamma(s)$ that is tangent to second order to a curve $\xi:[a,b]\to \mathbb R^2$ with unit speed and with curvature $\kappa$, then the radius of $\Gamma(s)$ is ...
0
votes
1answer
132 views

Function for the upper left part of a circle

What is the function corresponding to the upper left quarter of a circle ? Where $x$ goes from 0 to $x_\text{max}$, and $y=f(x)$ goes from $y_\text{min}$ to $y_\text{max}$.
0
votes
1answer
160 views

Polygonal line connecting z to infinity intersects boundary of rectangle

Show that a polygonal line $\gamma$ connecting $z$ to infinity intersects the boundary of every rectangle $R$ containing $z.$ So we want to consider $t_0 = \sup \{t : \gamma(t) \in R\}$. This ...
0
votes
1answer
289 views

Reference : N dimensional rectangle inscribed in an N dimensional ellipsoid

In specific, given an 5-dimensional ellipsoid with intercepts,(a1, a2, ..., a5), with the Cartesian coordinates, what are the set of linear equations that describe 5 - dimensional rectangle inscribed ...
0
votes
1answer
101 views

A question on triangles

The radii $r_1,r_2,r_3$ of ex-scribed circles of the triangle $ABC$ are in harmonic progression. If the area of the triangle is $24$ sq.cm and its perimeter is $24$ cm, then what is the length of the ...
0
votes
1answer
130 views

Circle : How to get all co-ordinate list of circle parimeter?

I want to find all the co-ordinate of circle. I know the radius of circle and considering center co-ordinate as (0,0). So Is there any equation for finding all ...
0
votes
1answer
2k views

Given two vertices, how to find the other two vertices of a rhombus?

$A\;(-3,-4) $ and $ C \; (5,4)$ are the ends of the diagonal of a rhombus $ABCD$. Given that the side BC has gradient $\frac{5}{3}$; How could we find the coordinates of $B$ and hence of $D$? ...
0
votes
1answer
258 views

First fundamental form

This is a practice exercise from a geometry textbook by P. Wilson. Suppose we have a Riemannian metric of the form $|dz|^2/h(r)^2$ on an open disc of radius $\delta>0$ centered on the origin in ...
0
votes
1answer
434 views

Finding radius r of the overlappable sphere(s) in 3D image

My current problem: I have an input 3D binary image (a 3D matrix that has only 0 and 1) that consists of random numbers of sphere with radius r. We do not know how many spheres are there in the ...
11
votes
1answer
855 views

Can someone please explain the cube to sphere mapping formula to me?

I am wondering if anyone could explain how the following formula works, it is supposed to take the input as a point on a cube then map that to points on a sphere, please go gentle on me, I'm in 9th ...
0
votes
1answer
81 views

Graphs and surfaces

Why is the graph, $G$, of a smooth function $f:\mathbb R^2 \to \mathbb R^3$ necessarily an "embedded surface"? Thank you. Comment: I am starting to thing that there is a typo in the text. I think ...
1
vote
1answer
190 views

Is it possible to have a cosine of 1.0000

I was given a triangle: side opposite of angle A: unknown, referred to as L side adjacent of angle A: 11' hypotenuse: 14' I have to find the cosine of angle A, the degrees, and the length of "L", ...
5
votes
2answers
585 views

21 sided regular polygon and its diagonals

In a $21$ sides regular polygon, how many points inside it are intersection of its diagonal? I found that a polygon with $n$ sides has $\dfrac{n(n - 3)}{2}$ diagonals, but I feel this is not so ...
1
vote
1answer
548 views

Tough Geometry Problem--Regular Polygon inside Circle

$ABCDEFG$ is a regular heptagon inscribed in a unit circle centered at $O$. $\ell$ is the line tangent to the circumcircle of $ABCDEFG$ at $A$, and $P$ is a point on $\ell$ such that triangle $AOP$ is ...
1
vote
2answers
134 views

Triangle Requirements based of triangle Inequality

In a Geometry course we are dealing with triangle inequality and two statements arose: "For any triangle, any side is smaller the the sum of the others." and "For any triangle, the largest side is ...
2
votes
1answer
167 views

Geometrical Drawing

Given are a circumference $(C, r)$, a point $P$, a line $S$ and a length $D$. Determine a line $T$ that passes through $P$ and has intersection with circumference at the points $A$ and $B$ so that the ...
2
votes
1answer
614 views

Volume of a straight beaker with a rounded bottom

I'm creating a simple beaker 3D object in OpenSCAD, and for practical reasons (give the object realistic dimensions, maybe tweak them to meet specific demands) it prints the volume of the top cavity. ...
1
vote
3answers
77 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...