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

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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 ...
0
votes
1answer
126 views

Show that (vector) subspaces of $\mathbb{A}^n$ are algebraic sets

i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of ...
1
vote
0answers
146 views

Iterate the points around the edge of a rectangle

With an arbitrary starting location, that can be any point along the edge of a rectangle, how would you iterate over every point in a clockwise order? So for example if the starting point happens to ...
5
votes
1answer
368 views

What is the path equation that is created with the middle point of a fixed length line segment that touching both ends to an ellipse.

Ellipse equation is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ and the length of line segment is $2k$, if we move the line segment all around of the ellipse while touching both ends to the ellipse. What is ...
0
votes
1answer
589 views

Mensuration- Conversion of Solid from One Shape to Another

A solid Aluminium metal hemispherical bowl of radius 30 cm. is melted down and cast into cylinder of radius 5 cm. and height 2 cm. How many cylinders can be cast from the solid metal?
1
vote
1answer
906 views

Projection to the plane in the direction of the vector?

Just a small thing 762 here,the question is about projection to the plane $T: x+2z-3y+3=0$ in the direction of the vector $\bar{i}+\bar{j}-\bar{k}$. Now I understand this in a way where ...
1
vote
1answer
111 views

A simple question about vector and geometry

My question is why vector$OM={1\over 3} (OB+OC+OD)$and $OA '$ can be expressed as the form $2OM-OA$
1
vote
2answers
476 views

Rotation around a point?

I know that rotation can be understood by simple complex transformation (as shown on 758) $$\begin{align*}y_{1}+iy_2 &= \left( \cos(\alpha) + i \sin(\alpha) \right) \left( x_{1}+ix_{2} \right) ...
2
votes
1answer
1k views

How many geosynchronous satellites does it take to cover all earth surface?

Geosynchronous satellites orbit at an altitude of 35,786 km, directly above the Equator, and assume that earth is a perfect sphere. Please any hints or detail computation would be really appreciated. ...
3
votes
1answer
180 views

proving that four axis-parallel rectangles whose intersection graph is a cycle delimit another rectangle

Working on the 2D plane, I'm looking for an elegant proof of the fact that if four regions $A$, $B$, $C$, $D$ delimited by axis-parallel rectangles are such that: $A$ intersects $B$, $B$ intersects ...
3
votes
2answers
2k views

Calculate coordinates of a regular polygon

Given the regular polygon's side count $n$, the circumscribed radius $r$ and the center coordinates $(x,y)$ of the circumscribed circle, How to calculate the coordinates of all polygon's vertices if ...
5
votes
0answers
235 views

Points in the plane at integer distances

Does there exist a set of $n$ points $p_1,p_2,...,p_n$ in the plane, all at mutual integer distances to each other, and an $e>0$, such that the following statement holds: For all $a,b$ with ...
1
vote
1answer
309 views

Complex numbers as coordinates

Is it possible to have an n-dimensional geometry where each coordinate can be a complex number, or would it make no sense, i.e. lead to contradictions? Spacetime, can be described as having 4 ...
2
votes
3answers
2k views

Maximizing area of a rectangle inscribed in a semicircle

A rectangle of largest area is inscribed in a semicircle of radius $r$. What is the area of the rectangle? I just need the hint to solve it. How can I get length and breadth of rectangle in terms ...
0
votes
2answers
131 views

What creates a unique quaternion?

You can create a unique point N from point M and vector V. You can create a unique vector B from vector A and quaternion Q. Can you create a unique quaternion Y from a another quaternion X, given ...
1
vote
1answer
206 views

Number of bounded volumes in M C Escher's Woodcut Waterfall

When 3 cubes interpentrate in an optimal way they create dozens of smaller closed bounded volumes ... like M. C. Escher's Waterfall picture with the cube-3 compound. For Escher's 3 interpenetrating ...
0
votes
3answers
126 views

Altering the center of mass with an iterative process

We have a system of $n$ particles, and the particle $i$ has a point mass $m_i$. The center of mass is then given by: $$X = \frac{\sum_i^nm_ix_i}{\sum_i^nm_i}$$ $$Y = ...
1
vote
0answers
52 views

Altering the shape of a Gaussian curve

I just posted this on Stack Overflow but then I found out about this forum. I hope I'm not breaking any policies by posting the same question here. I am not allowed to post images here yet, so please ...
4
votes
1answer
1k views

Counting integral lattice points in a triangle that may not have integer coordinates?

I have a triangle with one vertex on 0,0, another at 0,Y, and the third at X,Y, where Y is a positive integer and and X is any positive number (can be irrational/decimal/integral/etc). I tried using ...
0
votes
0answers
49 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
2
votes
1answer
496 views

Having two points of a square and only a compass, how to find the remaining two?

I remember being presented a mathematical puzzle some years back that I still can't solve. The problem is defined as follows: We have two points on a plane, and using only a compass, how do we find ...
8
votes
1answer
303 views

Dividing a square with a hole into two

I was asked the following puzzle for an interview. There is a square sheet. A smaller square hole is made on it (at a random place). How can I divide the rest of the sheet into two halves (in terms ...
2
votes
0answers
318 views

Definition of face of a polyhedron

Let $P$ be a polyhedron in $\mathbb{R}^n$ and $\omega \in \mathbb{R}^n$, viewed as a linear functional $\text{face}_{\omega}= \{ u \in P : \omega\cdot u \geq \omega\cdot v\mbox{ for all }v \in P \}$. ...
0
votes
1answer
90 views

Using GeoGebra for Web based Graphical Problems [closed]

How can I make my documents available those I produced with GeoGebra on this site to share or ask for example? Which format I should choose to save or convert?
11
votes
4answers
4k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
1
vote
2answers
2k views

Finding the component of a vector tangent to a circle

Problem Given a vector and a circle in a plane, I'm trying to find the component of the vector that is tangent to the circle. The location of the tangent vector is unimportant; I only need to know ...
1
vote
1answer
160 views

From $a + \sqrt{b}$ how to find $a+b$?

In a quadrilateral $ABCD, BC =10, CD = 14$ and $AD = 12$, it is also known that $\angle A=\angle B=60^\circ$. Now, if $AB = a + \sqrt{b}, \text{ where } a,b \in \mathbb{N}, $ how could we find $a+b$? ...
1
vote
1answer
904 views

Calculate volume of sphere cap

I found few equations here: http://mathworld.wolfram.com/SphericalCap.html Developing one ($V_{\text{cap}}=\frac{1}{3}\pi h^{2}(3R-h)$) and considering that $R$ in my case is equals to $1$ and ...
0
votes
1answer
81 views

Bijection from projective space to a set of matrices

Let $M$ be the set of $(n+1)\times(n+1)$ symmetric, idempotent matrices of trace 1. What is the inverse function of $f:\mathbb{R}P^n\rightarrow M$ defined by ...
1
vote
1answer
638 views

Find 3D rotation vector and angle to transform a rectangle into a given quadrilateral

I have a given rectangle that I need to transform into a given quadrilateral shape that resulted from a rotation and translation in 3D space, and subsequent projection. ...
3
votes
1answer
238 views

Heronian triangles

How to prove that all Heronian triangles can be found using formulas described here? I understand that the described substitution will give Heronian triangle, but how to prove that using the ...
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
7
votes
4answers
2k views

I need a proof that a line cannot intersect a circle at three distinct points

I need a simple proof that a line cannot intersect a circle at three distinct points.
1
vote
1answer
141 views

Areas of quadrilateral and parallelograms and triangles.

The diagonals of a quadrilateral $ABCD$ meet at $P$. Prove that $ar(APB)*ar(DPC)=ar(ADP)*ar(BPC)$ Please solve this question. I have tried a lot on this question. Please do not use trigonometry, but ...
7
votes
3answers
296 views

Elementary Geometry

The side of the square measures $1\ \mathrm{cm}$ , and $AC = 1\ \mathrm{cm}$, find the value of $AB$
17
votes
2answers
674 views

Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$

Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer. Below is an image. With a given n, how ...
3
votes
1answer
112 views

A vector calculus question

I realize that this sounds like a physics question, but what I am stuck on is a mathematical issue, so I hope you won't mind me posting this question here. I have a cylinder given by the equation ...
2
votes
0answers
107 views

action of a torus on a projective space

How can a torus $T^n$ acts on the projective space $\mathbb{P}^n=\mathbb{P}(\mathbb{C}^{n+1})$? Is it possible or I'm doing a mistake because I need to consider $\mathbb{P}^{n-1}$ ? Thanks
1
vote
1answer
98 views

What's the minimum number of points I need to measure to find out how much two rectangles are overlapping?

This is for a web app, but I think it's a general geometric problem, and almost certainly a solved one: I have a rectangle $(A)$ and I know its size. I'm trying to figure out if there's another ...
1
vote
1answer
48 views

Variant of the modal average

I know that there are variants of the arithmetic mean and the median that are applicable to three dimensional data (centroid and medoid), but have not been able to find such a thing for the modal ...
2
votes
3answers
8k views

Prove the opposite angles of a quadrilateral are supplementary implies it is cyclic.

There is a well-known theorem that a cyclic quadrilateral (its vertices all lie on the same circle) has supplementary opposite angles. I have a feeling the converse is true, but I don't know how to ...
3
votes
0answers
76 views

How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of ...
3
votes
2answers
159 views

Explicitly writing out all elements of $\mathbb{P}^{1}(\mathbb{F}_{n})$

Explicitly the elements of $\mathbb{P}^{1}(\mathbb{F}_{3})$ are $[1:0], [0:1], [1:1]$, and $[1:-1]$. Why is this so? How would I do this for $\mathbb{P}^{1}(\mathbb{F}_{4})$? What about general ...
2
votes
3answers
244 views

Vector path length of a hypotenuse

Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to: $ P_n = P_x + P_y = \frac{n}{2}\times w + ...
0
votes
1answer
162 views

Chain Rule and Homogenous Coordinates

I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$. Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of ...
0
votes
2answers
31 views

Creating a 'virtual' point

I have co-ordinates (x,y,z) for three markers placed on the leg, and I need to calculate the position of a 'fourth' marker which appears in the first frame of my recording and then disappears. ...
0
votes
1answer
267 views

Radius of incircle and Pythagorean triangle

How is the following statement true? If $m$ and $n$ are positive integers with $m > n$. Let $a = 2mn, b = m^2 - n^2$ and $c = m^2 + n^2$ be the sides of a Pythagorean triangle. Then the radius ...
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?
1
vote
1answer
75 views

Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...