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
4 views

Identical Geodesics implies scalar multiple of metric?

Suppose $(M,g^1)$ and $(M,g^2)$ are two intrinsic metric spaces with the same underlying set $M$. Assume that for every $p,q\in M$, the geodesics $\gamma^1_{[p,q]}$ and $\gamma^2_{[p,q]}$ ...
1
vote
2answers
16 views

Place x number of points equidistant from each other on a circle segment circumference

So I have a circle segment, a 90 degrees part of a circle, so 1/4 of the circle. I know the radius and I want to place an x number of points spread out evenly on the circumference. How can I achieve ...
0
votes
2answers
531 views

deriving formula for reflection over y=mx+b using dot product

So, I know that the formula for a generic point is $$\left(\frac{1-m^2}{1+m^2}x + \frac{2m}{1+m^2}(y-b), \left(\frac{2m}{1+m^2}\right)x - \left(\frac{1-m^2}{1+m^2}\right)(y-b)+b\right)$$ when you ...
1
vote
2answers
49 views

Geometry proof given diagram

$ABCD$ forms a square. $CDE$ forms a triangle. Given $\measuredangle AED=15^{\circ}$ and $DE=CE$, prove $\triangle CDE$ is equilateral. The question is surprising hard, the problem is basically ...
0
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1answer
54 views

Is it possible to solve this or it has infinite solutions? [on hold]

How do you calculate the perimeter of the blue square? Sorry, initially I misread the question. It asks for the smallest possible perimeter. Therefore solved with optimisation.
3
votes
1answer
23 views

Relationship between circumscribed sphere radius and edge length of a dodecahedron?

Hello and I'm a secondary student doing a math exploration, but I'm currently stuck with this problem... Can anyone kind enough to show me the derivation of the relationship between the circumscribed ...
5
votes
1answer
42 views

A question on dissecting quadrilaterals

How to prove that if $n\geq 4$, every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals?
-3
votes
1answer
33 views

Clock angles question. [on hold]

At 4:00 oclock, what will be the angle between the two pointers ? Adding a picture :
0
votes
1answer
31 views

Transformations between coordinate frames

Suppose I have three coordinate frames: $A$, $B$ and $C$, all in 2D space. In homogeneous coordinates, I deduce, by inspection, the transformation matrices between each of these ($T_{AB}$, $T_{BC}$ ...
0
votes
2answers
45 views

How can I solve this trigonometry question? [on hold]

$ABC$ is a triangle $m( A \widehat BC) = m( A \widehat CB) + 90^\circ$ $3 \lvert AC \rvert = 7 \lvert AB \rvert$ area of the $ABC$ triangle is $4{,}2$ cm$^2$ $\lvert BC \rvert =$ ?
2
votes
1answer
516 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
2
votes
2answers
749 views

calculate centroid of triangle on a graph

Given ANY three points on a graph that form a triangle, how do you find the centroid using geometry? So basically I have three points (X1, Y1), (X2, Y2), and (X3, Y3). I am trying to use the slopes ...
1
vote
3answers
116 views

Finding sides of a trapezium

In a trapezium $ABCD$, $AB$ is parallel to $CD$ & $\angle D$ = $2\angle B$. If $DC =p $ and $ AD = q$ , then $AB = ?$ My attempt to solution is shown in the picture attached
0
votes
0answers
9 views

Mahalanobis distance between two orientations

I am performing bundle adjustment and besides minimizing the reprojection errors I have a GPS/INS system providing me input which I want to use as constrains. So far I have added the positional ...
0
votes
0answers
5 views

finite partitions of the square that separate all equipotent sets of points

This question asked whether there exists a finite partition of $[0, 1]^2$ and a finite set of points in $[0, 1]^2$ that can't be affinely transformed to fall into one part of the partition. I would ...
0
votes
0answers
24 views

What is the minimum number of vertices needed to create n non-overlapping triangles

How can I calculate the minimum number of vertices needed to draw $n$ non-overlapping triangles? Details There is no restraint on collinearity of the vertices. I don't think $^nC_3$ is correct (...
1
vote
1answer
522 views

definition of thickness of a shape (ring)

I want to know how thickness is defined. Let me start with the simplest shape due to its symmetry. This is a circle. Assume a circle with radius of 1. If we draw a circle with the same center and ...
0
votes
1answer
12 views

Line spacing for scaffold porosity calculations

I'm trying to calculate the correct distance to achieve a certain porosity in a 3D printed scaffold. The scaffold is circular with fixed distances between lines of printed material, with alternating ...
1
vote
1answer
44 views

Volume of solid lies under $z=x^2+y^2$ [on hold]

Find the volume of solid lies under $z=x^2+y^2$ above $x$-$y$ plane and inside the cylinder $x^2+y^2=2x$. I know, for volume we have to us $V=\iiint { \mathrm dx\mathrm dy\mathrm dz}$ but i was not ...
0
votes
3answers
49 views

A geometric approach to this problem?

Question: A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane ...
5
votes
1answer
30 views

Parallel planes and existence of a regular tetrahedron

Could somebody please guide with an appropriate approach for the following problem? Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
0
votes
1answer
403 views

Minkowski difference of two convex polygons

I just want to make sure that the following algorithm is correct for computing the Minkowski difference of two shapes $A,B$: $\text{Minkowski}(A,B) = \text{ CH } \{x: x = a - b \text{ for } a \in ...
3
votes
3answers
355 views

Circle Geometry Questions

In rectangle $ABCD$, we have $AD = 3$ and $AB = 4$. Let $M$ be the midpoint of $\overline{AB}$, and let $X$ be the point such that $MD = MX$, $\angle MDX = 77^\circ$, and $A$ and $X$ lie on opposite ...
4
votes
3answers
123 views

What is the geometric interpretation of $|z-1|^2+|z+1|^2=4$ for all $z$ such that $|z|=1$?

Show that $|z-1|^2+|z+1|^2=4$ for all z such that $|z|=1$. [Note that $|z|$ refers to the magnitude of z where $z=a+bi$]. I was able to 'prove' the question; however, I cannot think of a geometric ...
0
votes
1answer
21 views

What is a geometric interpretation of multiplication/division in the complex plane? [duplicate]

How can one visualize the multiplication/division of a complex number, z, by a real number, an imaginary number, or another complex number?
0
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2answers
34 views

A theorem about one-dimensional convex sets

Suppose we have a non-empty convex set which does not consist of only one point such that it belongs to the same line, then this set is either a line segment(closed, half-open or open),a ray(closed or ...
0
votes
4answers
111 views

Does a right circular cone only consists of pair of straight lines, hyperbolas, parabola, circles and ellipses? [closed]

I was reading about the conic sections and that a conic section includes pair of straight lines, ellipses, hyperbola, circles and parabola. Are all these 5 components enough to form a right circular ...
5
votes
3answers
205 views

Find the red coloured area

A circle is in a square of side 10 and a quadrant circle with radius 10 overlaps as shown in the figure. Find the red coloured area. $\hskip2.4in$ I guess I could find the value by subtracting the ...
9
votes
1answer
125 views

Dividing an equilateral triangle into N equal (possibly non-connected) parts

It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an ...
1
vote
1answer
30 views

Pentomino Tessellation Explanation

I need to explain why this pentomino tessellates in a mathematically coherent way. Here is the pentomino and the tessellation I have made. This pentomino can be translated to form a diagonal ...
0
votes
3answers
21 views

How do I find the the third vertex of an isosceles triangle, if the vertex is on the y-axis?

$ABC$ is an isosceles triangle ($AB=AC$). We know that $A(5;9)$, $B(4;2)$ and $C$ lies on the $y$-axis. How to find $C$?
0
votes
2answers
49 views

What is embedding?

I am new to this so do I need to learn topology in order to understand this? Cause I come across this which says that unlike the 2D sphere, 2d saddle surface cannot be embedded in 3D Euclidean space(...
8
votes
3answers
72 views

Does a set of $n+1$ points that affinely span $\mathbb{R}^n$ lie on a unique $(n-1)$-sphere?

In $\mathbb{R}^2$ every three points that are not colinear lie on a unique circle. Does this generalize to higher dimensions in the following way: If $n+1$ element subset $S$ of $\mathbb{R}^n$ does ...
0
votes
0answers
11 views

Determine SE(3) transform between pair of sensors producing 2d line segments

Say we have sensorA and sensorB with an unknown transform between them, and that we have n ...
0
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0answers
21 views

geometry surface area and cubes

a cake is formed in the form of a cube of side 20 cm, coated from all the sides with chocolate. If one corner of the cake is cut by a plane through midpoints of the edges meeting at the corner find ...
0
votes
1answer
20 views

How to derive the five-segment axiom of Tarski's geometry from Hilbert's axioms?

We are trying to prove that in an arbitrary Hilbert plane (assuming Hilbert's axioms of Group I:Incidence, II:Order and III:Congruence https://en.wikipedia.org/wiki/Hilbert%27s_axioms), Tarski's ...
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0answers
21 views

Plane containing two points $a,b \in \mathbb{S}^3$ but avoiding two other antipodal points

Let $\mathbb{S}^3$ be the three-dimensional subset of $\mathbb{R}^4$ given by $$ \mathbb{S}^3 = \{ (x_1,x_2,x_3,x_4) | x_1^2 + x_2^2 +x_3^2 +x_4^2 = 1 \}.$$ I want to construct a plane through the ...
2
votes
2answers
26 views

Finding axis of a cylinder

I have to find axis of a cylinder that has the top in the origin and the points $A(-5,6,-4),B(-4,-1,2),C(-1,2,4)$ lie on its lateral area. Now I know that points A,B,C have the same distance to the ...
-3
votes
0answers
23 views

Sum of Area of Circles. [duplicate]

A circle of radius x cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to ...
1
vote
1answer
27 views

Expression for nano scale crack propogation

The following image is of a nano-scale self replicating crack on a thin film/substrate pair. I want to construct a "position" function for the crack front's propagation. So far the only way I can ...
2
votes
1answer
23 views

Dissecting a circle with an irregular rectangular grid

Can a circular disc be 'dissected' by a rectangular grid into a finite number of pieces in such a way that the individual pieces of the circle can be grouped into regions of equal area? Clearly ...
1
vote
1answer
28 views

What is special about glide reflection?

In wikipedia article https://en.wikipedia.org/wiki/Translation_(geometry) it is written: A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and ...
0
votes
1answer
19 views

Prove using dot product that the angle bisectors of two adjacent angles are perpendicular

I am asked the following problem: Prove using scalar product that the angle bisectors of two adjacent angles are perpendicular. I am not sure if the following solution satisfies the problem: $...
0
votes
0answers
8 views

Relation between angle and curvature in sphere

Consider a canonical 2-dimensional sphere of radius $\frac{1}{\sqrt{k}}$ where $k>0$ is a sectional curvature Consider a geodesic triangle $ABC$ Prove that $$ \frac{d}{dk}\angle CAB > 0 $$ ...
2
votes
2answers
58 views

Finding an angle of a triangle

Suppose $ BD=AC=4$, $\angle ABC=50$, $\angle ACB=100$, and $\angle BCD=130$. Find degree of $\angle BDC$. For this I could not find the measure of the angle
5
votes
1answer
1k 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 ...
2
votes
6answers
47 views

Choosing a value so a line and circle intersect a one, two, and no points

Let l be a line and C be a circle. $y=x+d$, where $d$ is to be determined. $C=x^2+y^2=4$ Pick a value for $d$ so that l and C intersect at one point. Pick a value for $d$ so that l and C ...
0
votes
1answer
17 views

Unit sphere intersected with hyperplanes, largest chamber?

I have a question which seems not that hard and maybe I can figure it out myself, but probably there is already a known theorem about it. Here it is: Imagine the unit $n$-dimensional unit sphere is ...
1
vote
2answers
30 views

In the figure below,three congurent semicircles with centres P,RQ,R are drawn on each side of three equilateral triangle.Find shaded part's area?

In the figure below,three congurent semicircles with centres P,RQ,R are drawn on each side of three equilateral triangle.Find shaded part's area?