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

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15
votes
1answer
294 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
13
votes
6answers
430 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
12
votes
2answers
797 views

$n$ lines cannot divide a plane region into $x$ regions, finding $x$ for $n$

I noticed that $3$ lines cannot divide the plane into $5$ regions (they can divide it into $2,3,4,6$ and $7$ regions). I have a line of reasoning for it, but it seems rather ad-hoc. I also noticed ...
10
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 ...
10
votes
3answers
9k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
9
votes
1answer
145 views

Find the area where dog can roam [duplicate]

A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam? I tried to find the area of all semicircles and then to ...
9
votes
3answers
214 views

Cutting a sandwich with a crust

Let $S$ be a simple closed curve in ${\Bbb R}^2$ enclosing a convex region $I$. Must there exist a straight line which cuts $S$ into two pieces of equal length and also cuts $I$ into two regions of ...
8
votes
1answer
92 views

For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| ...
7
votes
2answers
778 views

Is there a Möbius torus?

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted ...
7
votes
3answers
9k views

The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
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 ...
6
votes
2answers
10k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
6
votes
1answer
421 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
6
votes
6answers
10k views

Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, ...
6
votes
2answers
1k views

Why doesn't a simple mean give the position of a centroid in a polygon?

I was having a look at this question on SO. From what I know, the centroid is the center of mass of an object. so, by definition its position is given by a simple mean of the positions of all the ...
5
votes
1answer
79 views

Is the non-existence of a general quintic formula related to the impossibility of constructing the geometric median for five points?

In particular, in the Computation section of in the Wikipedia page for geometric median, there is this statement: ...but no such formula is known for the geometric median, and it has been shown ...
5
votes
1answer
126 views

prove that line bisect section

There is incircle $\Gamma$ of triangle $ABC$ tangent to $AB,BC,CA$ respectively at $K,L,M$. Point $D$ is the centre of section $MK$. $|DL|$ is diameter of another circle which intersects with $\Gamma$ ...
5
votes
2answers
192 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
4
votes
2answers
253 views

Inside an equilateral triangle $ABC$,an arbitrary point $P$ is taken from which the perpendiculars $PD,PE$ and $PF$ are dropped onto the sides…

Inside an equilateral triangle $ABC$,an arbitrary point $P$ is taken from which the perpendiculars $PD,PE$ and $PF$ are dropped onto the sides $BC,CA$ and $AB$,respectively.Show that the ratio ...
4
votes
1answer
3k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
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
8answers
531 views

Find the approximate center of a circle passing through more than three points

Consider n point $(x_1,y_1), (x_2,y_2),\ldots, (x_n,y_n)$. For $n = 3$ it is easy to find the center of the circle passing through the three points. I wanted find the approximate center of the ...
3
votes
2answers
130 views

Making cuts on a spiral so that all segments are of the same length

The issue we have is this: We have rolls of magnetic strip (about 2 cm in width) and they are rolled in a roll with about 30 windings and about 10m length. The rolls are about 2cm in width as i said. ...
3
votes
1answer
129 views

rectangularizing the square

There is a square that I want to divide to n people, such that each person gets a rectangular piece with an equal area. An obvious option is to cut 1-by-n rectangles of size n-by-1, but the people ...
2
votes
3answers
964 views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
2
votes
2answers
181 views

Finding a curve that intersects any line on the plane

Question Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ? What are the bounds on the number of such intersections. My question was itself ...
2
votes
0answers
141 views

Equation of an intersection of two cones when the intersection is an ellipse

The two cones with vertex $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and generating angle of two cones is $\alpha$ given. I need to write the equation of the intersection of two cones ...
2
votes
2answers
357 views

Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
2
votes
1answer
292 views

Obtaining Least square adjusted single line by intersecting many 3D planes

I am working with many 3D planes and looking for a Least square solution for below case. IF I am having many number of 3D planes knowing only one point and the normal vector (for eg. O1 and N1), ...
2
votes
1answer
1k views

Is a sphere a closed set?

The unit sphere in $\mathbb{R}^3$ is $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \}$. I always hear people say that this is closed and that it has no boundary. But isn't every point on the sphere a boundary ...
16
votes
3answers
637 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
11
votes
3answers
1k views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given 2 lines, one of the length of the hypotenuse and the other with the length of the sum of the 2 legs, construct ...
11
votes
5answers
3k views

Calculating the area of an irregular polygon

Given the length of the sides of an irregular polygon (no coordinates provided) how do you compute the area of the maximum area of the polygon? Thanks in advance
9
votes
1answer
315 views

Calculating $\sin(10^\circ)$ with a geometric method

Excuse me if this is a simple question: What is a simple geometric method for calculating $\sin(10^\circ)$ using only the sines of $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$? Generally, is ...
8
votes
3answers
172 views

Proof by induction using Fubini's Theorem

I am asked for the volume of the region $x_1+\cdots+x_n\leq 1$ where $x_1,...,x_n\geq 0$. I am proposing that the volume $V(n)$, is given by $$ V(n) = ...
8
votes
2answers
8k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
8
votes
1answer
295 views

How to parameterize an orange peel

I'm trying to parametrize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo: For example, the ideal curve would be inside the unit cube; have ...
7
votes
1answer
234 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
7
votes
4answers
4k views

equilateral triangle with integer coordinates

Is it possible to construct an equilateral triangle with coordinates on a grid of integers? I think the answer is no, but how can I prove this? I started with a triangle with coordinates (0,0) (a,b) ...
6
votes
3answers
173 views

Error measurement between given perfect 2D shape and freeform shape drawn by user

What method should I use to calculate the error between a given perfect shape (e.g. circle, triangle, rectangle etc.) and a freeform shape drawn by the user, which more or less closely matches the ...
5
votes
1answer
453 views

What is “general position” of hyperplanes?

A brief question. I was reading some mathematical writing in which the author makes the following statement: Consider $S$ hyperplanes in general position... What is "general position"? ...
5
votes
5answers
744 views

What is a parallel line?

We are learning vectors in class and I have a question about parallel lines and coincident lines. According to wikipedia a parallel line is: Two lines in a plane that do not intersect or touch at ...
5
votes
1answer
379 views

Ellipse Question

I have only worked with ellipses aligned with the x or y axis. However, how can I approach the following: Suppose we have an ellipse centered at the origin of the following form $$ax^2 + b xy +c y^2 ...
5
votes
2answers
168 views

Warp-like pattern in a closed curve

Given a closed curve in 2D space that intersects itself (transversally, and there's no point in which three paths or more meet), is it possible to look at it as a Celtic knot so when you follow it, ...
4
votes
3answers
308 views

What does the secant value represent?

What does the secant value represent? I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated. ...
4
votes
3answers
279 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
4
votes
3answers
4k views

Deriving the Area of a Sector of an Ellipse

A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. Could you please explain me how to find the area of a sector of an ellipse?
4
votes
3answers
536 views

Heronian triangle Generator

I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm $$a=n(m^{2}+k^{2})$$ $$b=m(n^{2}+k^{2})$$ ...
4
votes
3answers
993 views

Determinants and volume of parallelotopes

The absolute value of a 2 by 2 matrix determinant is the area of a corresponding parallelogram with the 2 row vectors as sides. The absolute value of a 3 by 3 matrix determinant is the volume of a ...
3
votes
3answers
206 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...