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

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3
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0answers
17 views

Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
0
votes
0answers
4 views

How to calculate lengths from perspective projection?

Suppose that I have a single point perspective drawing like . Suppose that I know some of the real horizontal distances and distances along lines converging to vanishing point. for e.g if i know the ...
1
vote
2answers
22 views

Determining the 3rd vertex of an Equilateral and Right angled isoceles triangle.

I am really having problems solving the following problems: If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two vertices on the hypotenuse of a right angled isosceles triangle then the ...
0
votes
0answers
34 views

A geometric question, no idea how to do it

Let us have a cube with the middle point: $M = (4,3,1)$. We know that $6(x+10) = 7(y+20) = 7z$ is one of the edge of the cube. How to find the vertices of the cube? I have no idea how to do this, ...
0
votes
0answers
20 views

Triangles with vertices in lattice points

I'm trying to solve a problem here. Let's have a lattice Z*Z, where Z is the set of integers. We form a triangle on this lattice with vertices in lattice points(all the vertices have integer ...
2
votes
3answers
72 views

Why can we always take the zero section of a vector bundle?

$\require{AMScd}$ As I understand it, a rank $k$ vector bundle is a pair of topological spaces with a map between them $$ E\xrightarrow{p}B $$ such that there exists an open cover $(U_\alpha)$ of $B$ ...
2
votes
0answers
20 views

In convex pentagon $ABCDE$ we have $BC=DE,\angle ABE=\angle CAB=\angle AED-\frac{\pi}{2},\angle ACB=\angle ADE$. Prove that $BCDE$ is parallelogram.

In convex pentagon $ABCDE$ we have $BC=DE$, $\angle ABE=\angle CAB=\angle AED-\frac{\pi}{2}$,$\angle ACB=\angle ADE$. Prove that $BCDE$ is parallelogram. I proved it by using law of sines in ...
1
vote
0answers
30 views

Is projective space really a moduli space for lines through the origin?

The Wikipedia page for Moduli spaces states that real projective space $\mathbb{RP}^n$ is a moduli space which parametrizes the space of lines in $\mathbb R^{n+1}$ passing through the origin. ...
2
votes
1answer
15 views

Determining whether two 2D polynomial curves are everywhere close to each other

Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials ...
0
votes
0answers
18 views

Finding angle in heights and distances problem

AB is a vertical pole and C is its mid point. The end A is on the level ground and P is any point on the ground other than A. The portion CB subtends an angle beta at P. If AP:AB = 2:1, then find ...
1
vote
1answer
33 views

FLATLAND's sphere intersection scenario, explored for four dimmensions

I recently finished this wonderful new vintage edition of FLATLAND. http://amzn.com/918775116X In 1884, Edwin Abbott wrote this strange and enchanting novella called FLATLAND, in which a square who ...
0
votes
1answer
39 views

Is a heart shape convex or concave?

Is a heart shape convex or concave? To me the inward points tells me that it is concave, but according to this answer, it says the inwards point makes it convex?
0
votes
0answers
15 views

Construction based on circumcenter and incenter

Construct a triangle given the exact location of its circumcenter and its incenter, and the position of its angle bisector (including its direction), but not its length. I tried to consider the ...
4
votes
1answer
48 views

Visual understanding for “the genus” of a plane algebraic curve

I am trying to understand the genus of an algebraic curve in the complex plane $\mathbb{C}P2$. I am looking for a visual or intuitive understanding. The difference between a sphere and a torus as a ...
0
votes
0answers
21 views

Find the minimum distance between the curves (Y - 2)² = 8(X - 4.5) and (X - 4)² = 4(Y - 6)

Find the minimum distance between the curves (Y - 2)² = 8(X - 4.5) and (X - 4)² = 4(Y - 6) Attempt:- The minimum distance between these two parabolas lies along their common normal. Relevant ...
0
votes
1answer
26 views

The bisector of $\angle BAC$ of triangle $\Delta ABC$ cuts $BC$ at $D$

The bisector of $\angle BAC$ oF triangle $\Delta ABC$ cuts $BC$ at $D$ and circumcircle of triangle at $E$. if $$AD=5 \text{ cm} ,\ DE=3 \text{ cm},\ AC=4 \text{ cm}, $$ then what is the length of ...
2
votes
1answer
30 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
7
votes
2answers
86 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
0
votes
1answer
28 views

Prove all right angles are congruent?

Prove all right angles are congruent. I only have to prove one side to this argument, so I just need to the the other argument. So basically, if two angles are right, then they must be congruent is ...
0
votes
1answer
14 views

Prove the distributibe property of the dot product using its geometric definition?

The geometric definition of the dot product: $$\mathbf{a} , \mathbf{b} \in \mathbb{R} ^n$$ $$\mathbf{a} \cdot \mathbf{b} = a \cdot b \cos \theta$$ The distributive property of the dot product: ...
0
votes
2answers
37 views

In the figure ,Prove that $AG=GH=CH$.

$ABCD$ is a parallelogram.$E$ and $F$ are midpoints of $AD$ and $BC$. Prove that diagonal $AC$ is trisected. Till now i have proved that $\triangle BAE\cong \triangle DCF\\ \implies \angle ...
4
votes
0answers
18 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
0
votes
0answers
16 views

Why does the Schwarz Christoffel Transform have three independent parameters?

Why is the Schwarz Christoffel Transformation for the half plane uniquely determined up to an arbitrary choice of three prevertices? Is there a simple intuitive way to motivate this, rather than ...
0
votes
2answers
12 views

Orthogonal vectors in case of Lorentz Metric

Let us consider $\mathbb{R}^4$ equipped with the Lorentz metric $$\eta(X,Y)=x^0y^0-(x^1y^1+x^2y^2+x^3y^3)$$ Let $X\in\mathbb{R}^4$ a time-like vector, that is $\eta(X,X)>0$. I want to show that all ...
1
vote
2answers
21 views

Determining whether images of two curves are close to each other

I am coding an implementation of Boolean operations on SVG paths, and need to solve the following problem: Given two sequences of curves, determine whether the distance between their images never ...
-1
votes
0answers
27 views

External Cirlces [duplicate]

For two circles (with centres $(a_1,b_1),(a_2,b_2)$ and radius $r_1,r_2$ respectively) touching externally and the tangent at their common point passing through the origin, I have shown that $(a_1^2 ...
48
votes
16answers
4k views

Fastest way to meet, without communication, on a sphere?

I was puzzled by a question my colleague asked me, and now seeking your help. Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and ...
1
vote
1answer
18 views

Calculating the length of $DP$ in a rectangle.

$P$ is a point in rectangle $ABCD$. Calculate the length of $DP$ if $AP = 3$, $PC = 5$ and $BP = 4$. How do I go about by doing this? I clearly can't use Pythagoras because the hypotenuse is not a ...
0
votes
1answer
7 views

Interpolate/Increment Point on Circle Circumfrence

For my 2D physics engine, I'm using the unit vectors of the direction an object is facing to represent its orientation; essentially, [Cos(theta),Sin(theta)] where theta is the object's rotation in ...
0
votes
0answers
19 views

Why use eccentricity instead of ratio of major and minor axes of an ellipse?

I am researching different features that might be useful for object detection algorithms. For ellipses, it seems that eccentricity is a common choice because it describes the "elongation" of an ...
1
vote
2answers
109 views

Triangle containing most points from a set

Given a point set in $\mathbb{R}^2$, I need to find a triangle connecting three points of the set that contains the most points of the set. Points that lie on the connecting lines don't count. The ...
0
votes
1answer
17 views

Central angle of a circular sector from area and arc length

I've been doing a task which says the following: Area of a circular sector is $3.2\pi cm^2$, arc length is $0.8\pi cm$. What is the central angle? I've been attacking this from several angles ...
3
votes
0answers
27 views

Proof in Neutral Geometry

I need to prove the following proposition: Given $\triangle ABC$ and $\triangle DEF$, if we have $AB \equiv DE$ and $BC \equiv EF$, then $\angle B < \angle E$ if and only if $AC < DF$. I began ...
2
votes
0answers
30 views

A (analytical) Geometric Way to solve this complex number problem?

Problem Let $a_0,a_1,a_2$ be three complex numbers that lie on the circle $C$ with center $K(2,0)$ and radius $r=1$. Let $v \in \mathbb C$ such that $$v^3 + a_2v^2 + a_1v + a_0 = 0$$ ...
0
votes
3answers
33 views

What will be the other vertex of the triangle?

Two vertices of a triangles are $(5,-1)$ and $(-2,3)$. If the orthocenter of the triangle is the origin, what is the other vertex ? My approach was that since the three vertices and the orthocenter ...
1
vote
1answer
29 views

Reflection Of Conic Section About A Line

If a certain conic section $$ ax^2+2hxy+by^2+2gx+2fy+c=0 $$ is reflected about any line $y=mx+n$ what will be its new equation?
0
votes
0answers
25 views

Angle condition for $a^2+c^2=nb^2$

Find a necessary and sufficient angle condition (independent of $a,b,c$ -- see under "what I have got so far" for examples) such that $a^2+c^2=nb^2$ where $n$ is a positive integer. Note: As usual ...
0
votes
0answers
17 views

Similarity transformation versus Möbius transformation

What is the relationship between the Möbius transformations of $\mathbb{R}^n\cup \{\infty \}$ and the similarity transformations of $\mathbb{R}^n$? The möbius group is generated by inversions and ...
1
vote
0answers
23 views

Calculating the ratio of $CD:DF$

In rectangle $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $BD$. If the area of the rectangle is 1, then calculate the ratio of $CD:DF$. My solution: $G$ is the centroid of ...
2
votes
0answers
11 views

Finding stabilizer under group action (Erlangen)

I want to do an Erlangen approach to classical Geometry and below I discuss how $PO(n+1,1)$ acts transitively on a model for hyperbolic space. I want help finding the stabilizer under this actions and ...
0
votes
1answer
25 views

Is $\measuredangle ABF=\measuredangle DAE$ in a quadrilateral?

Is $\measuredangle ABF=\measuredangle DAE$ in quadrilateral $ABCD$? I cannot find a way to answer this question. All I know is that $ABCD$ is a cyclic quadrilateral. I mused that $\measuredangle ...
2
votes
2answers
21 views

possible polyhedra from euler's formula

I'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra all ...
3
votes
2answers
61 views

In triangle $ABC$, $a^2+c^2=3b^2$

In triangle $ABC$, we have $a=BC$, $b=CA$ and $c=AB$ as usual. What is a necessary and sufficient condition for $a^2+c^2=3b^2$ to hold? I created this problem as a generalization of $a^2+c^2=2b^2$ ...
0
votes
0answers
53 views

A parabola and circle [duplicate]

A circle touches the parabola $y^2=4ax$ at $P$. It also passes through the focus $S$ of the parabola and intersects the axis of the parabola at $Q$. If $\angle SPQ$ is $90^\circ$, what is the equation ...
0
votes
1answer
13 views

Conversion from Cartesian to Parametric function for a plane

I am given a plane in $\mathbb{R}^3$ with Cartesian equation $$ -5 x_1 - 2 x_2 + 2 x_3 = -15 $$ and I would like to find parametric equations $$ \mathbf{x} = \mathbf{x}_0 + t_1 \mathbf{v}_1 + t_2 ...
0
votes
2answers
42 views

Bijection from $\{ (x,y) : 0 \leq x ,y \leq 1\}$ to $\left\lbrace (u,v) : u,v \geq 0, u+v \leq \frac{\pi}{2} \right\rbrace$

Suppose the set $M :=\{ (x,y) : 0 \leq x ,y \leq 1\}$. Now we define $$ u := \arccos \sqrt{\frac{1-x^2}{1-x^2y^2}} \ \ \ \ \ \text{and} \ \ \ \ \ v:= \sqrt{\frac{1-y^2}{1-x^2 y^2}}.$$ How can I ...
0
votes
2answers
47 views

Convert a 2D point to 3D on a plane

I have a 2D point and a 3D infinite plane(defined by a 3D point and its normal), I want to convert 2D point to 3D point by projected 2D point onto 3D plane surface. I'm weak in math, I need a method ...
9
votes
2answers
142 views

Is there a term for two polygons with the same angles but different side lengths?

Suppose polygons $A$ and $B$ have the same number of sides, and there is a correspondence between the vertices of $A$ and $B$, in consecutive order around both polygons, so that the angles at ...
0
votes
0answers
28 views

Construct triangle from three points on base and difference in distances to third vertex

Imagine such a triangle: We know the differences in distances: $\overline{OA} - \overline{BO}$ and $\overline{CO} - \overline{BO}$, as well as the distances between the points on the base: ...
2
votes
1answer
28 views

Similar triangle side lengths given its area and similar triangle side lengths

I've been working through this task in an old textbook and can't figure out where I'm wrong. I suspect my whole approach is wrong. Task says: Given the side lengths of a triangle that are equal to ...