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
votes
3answers
224 views

Visibility of the surface of a sphere

If you are $N$ radii above a sphere, what fraction of the hemisphere below you can you see? The answer is so nice that it prompted another question: is there an intuition behind it, in the sense ...
2
votes
1answer
1k views

Is it possible to divide an equilateral triangle into 12 congruent triangles?

Can you divide an equilateral triangle into exactly 12 congruent triangles? interesting question i haven't yet been able to work on. The sides can be of any length.
1
vote
1answer
1k views

rotate a roll pitch and yaw using a rotation matrix

I have a body in space, precisely a robots foot, defined by xyz and a roll pitch and yaw. The pitch is along the degree of freedom defined by the pitch joint. I want to rotate the foot using a ...
1
vote
1answer
76 views

Name of this angle?

Given a planet and a point $P$, is there an existing name for the angle $\theta$ as seen in the diagram below? If not, what would you call it? ("Angle of elevation"?) Thanks!
6
votes
1answer
644 views

How do mathematicians think about high dimensional geometry?

Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more. How do mathematicians think about higher ...
2
votes
0answers
87 views

Classical Varieties

I want to learn more about classical varieties. How do I proceed? I was looking for more information on Classical Varieties and what exactly it means. I worked through the wikipedia article - ...
-1
votes
2answers
75 views

Operator from the trace

let 's assume we have a self-adjoint operator whose trace is known $ \mathrm{Tr}(sT)= g(s) $ for a known function $ g(s) $. My quesiton is, can we recover the operator simply by knowing the trace ?
5
votes
2answers
96 views

How to check that whather a Polygon is completly inside of another Polygon?

Let's say I have two polygons. I know the co-ordinates of both polygons. Now, I need to check whether the first Polygon is completely inside of second polygon? IN this figure only 1 polygon is ...
1
vote
1answer
316 views

Minimum sphere containing a tetrahedron

Is there an equation which would give me the radius of the smallest sphere containing a certain tetrahedron (no need to touch all vertices); given that I know the insphere, circumsphere radii and the ...
3
votes
1answer
204 views

Defining distance in fractal dimensions.

Is it possible define a distance measure in fractal dimensions? namely, what the generalization of $$ D(x,y)=\left(\sum_i(x_i-y_i)^2\right)^{\frac{1}{2}} $$ in fractal dimensions?
1
vote
1answer
397 views

Regular polygons that touching to a sphere surface

What is the possible number of n sided polygons(every face is the same regular polygon) that touching their corners to sphere surface and also touching each other ? I would like to know the relation ...
1
vote
1answer
491 views

Analytic proof for Circles of Apollonius

I'm looking for an analytic proof the statement for a Circle of Apollonius (I found a geometrical one already): If $\overline{AC}:\overline{BC}=s$, then $P \in k_s$. $s \in (0,1)$. $k_s$ is the ...
7
votes
0answers
375 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
4
votes
2answers
116 views

Geometry related limit

Let's consider the right triangle ABC with angle $\angle ACB = \frac{\pi}{2}$, $\angle{BAC}=\alpha$, D is a point on AB such that |AC|=|AD|=1; the point E is chosen on BC such that $\angle ...
7
votes
5answers
505 views

Partition of the plane into union of open segments

A subset of ${\mathbb R}^2$ is open in the usual topology iff it is a (not necessarily disjoint) union of open disks. It is well-known that the plane is connected, so that there is no nontrivial ...
2
votes
1answer
2k views

How to find a point after rotation?

Initially the position of the shape was in (100, 100). I am rotating (say 30 degrees) the shape as shown in the image below. I have found the starting point of the rotated object. Is there a formula ...
0
votes
1answer
95 views

What are the following shapes in 3-space?

I have these 3 equations. I tried to use wolfram alpha to graph in 3d, but did't succeed. 1) $x^2 + 2y^2 - 6x + 4y + 7 = 0$ 2) $z^2 - 4z - 6x = 2$ 3) $z = -y + 2$ I think that: 1) is a cylinder 2) ...
3
votes
2answers
188 views

high school line equation question

I hit stumbling block below. But i try everything that I can think of but i failed to find any satisfactory answer. my workout is that I find mid point between point B and C. and then using A and ...
0
votes
1answer
112 views

Height of this triangle?

Each edge of the following cube is 1 and C is a point on the edge. What would the height of triangle be in this case , how would you measure it?
2
votes
2answers
401 views

Relation between sides and angles.

Is this phrase safe to consider in general: Equal sides of a polygon have corresponding equal angles if not how would you refine or correct it Example of a corresponding angle would be ...
2
votes
1answer
500 views

In center-excenter configuration in a right angled triangle

My question is: Given triangle ABC , where angle C=90 degrees. Prove that the set { s , s-a , s-b , s-c } is identical to { r , r1 , r2 , r3 }. *s=semiperimeter , r1,r2,r3 are the ex-radii. Any ...
3
votes
1answer
300 views

Name that curve!

What curve will a kayak describe if the paddler aims her bow at an object on a distant shore ahead and keeps the bow pointing to that object as she paddles toward it with constant velocity, in the ...
11
votes
1answer
412 views

Covering points on a sphere with a disk

Suppose $m$ points ("sites") are selected on the unit sphere $S^2$. For a given radius $r < \pi$, we can define a disk around any point on the sphere as the set of points at geodesic distance at ...
7
votes
3answers
573 views

Banach-Tarski theorem without axiom of choice

Is it possible to prove the infamous Banach-Tarski theorem without using the Axiom of Choice? I have never seen a proof which refutes this claim.
3
votes
2answers
2k views

Finding the coordinates of points from distance matrix

I have a set of points (with unknow coordinates) and the distance matrix. I need to find the coordinates of these points in order to plot them and show the solution of my algorithm. I can set one of ...
0
votes
1answer
376 views

$l_1$ distance of a point to a convex polygon

Let us have a set of $n$ points, $x_1, x_2, \ldots, x_n \in \mathbb{R}^d$, that form a convex polytope. And let us have a single point $x \in \mathbb{R}^d$ that is outside of the polytope. How can I ...
3
votes
1answer
143 views

Can't understand this solution.

I came across a problem which was already present on the internet. If an arc with a length of $12\pi$ is $\frac{3}{4}$ of the circumference of the circle, what is the shortest distance between ...
3
votes
2answers
282 views

Angles of triangle inside a cricle

In the figure shown if area of circle with center o is 100pi and CA has length of 6 what is length of AB ? I looked around on the web and cant seem to get an idea of what the angles AOC ...
0
votes
1answer
244 views

How to “stretch” a procedural half-sphere texture on X and/or Y axis

I've implemented an Objective-C function to display the "height" of a half-sphere, with "1.0" being "full-height" and "0.0" being "no-height" The sphere currently has a few parameters: Center (x,y: ...
28
votes
3answers
1k views

Guaranteed Checkmate with Rooks in High-Dimensional Chess

Given an infinite (in all directions), $n$-dimensional chess board $\mathbb Z^n$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite ...
2
votes
1answer
151 views

Area of Square - Comparing squares

The question is: If the area of a parallelogram $JKLM$ is $n$ and if length of $KN$ is $n+(1/n)$, then find the length of $JM$. (The answer is $n^2 /( n^2+1 )$.) How would i go about ...
3
votes
2answers
490 views

Inscribed quadrilateral

I need a hint on this problem. ABCD is inscribed quadrilateral. Diagonals AC and BD intersect at point O. OP and OQ are the perpendiculars from O to BC and AD. M and N are the midpoints of AB and CD. ...
1
vote
0answers
63 views

Looking for a reflection of 30° at a line

I'm trying to find a matrix-expression of a 30° reflection at the line $g(x)=2x+4$ Somebody can give a hint? Greetings
3
votes
1answer
131 views

Can all convex polytopes be realized with vertices on surface of convex body?

Each convex polytope $P$ has a combinatorial type, its so-called face lattice. This lattice is just the poset of all faces of $P$ ordered by inclusion. Given one realization of such a combinatorial ...
0
votes
2answers
2k views

Triangles inside a square

I have a question with a figure of Triangle inside a square. The base of the triangle is on the base of the square and the peak of the triangle touches the top of the square.It then asks the ratio ...
3
votes
2answers
3k views

How to get Point between two points at any specific distance?

I have two points, approximately we take values for that: Point $A = (50, 150)$; Point $B = (150, 50)$; So the distance should be calculated here, $\text{distance} = \sqrt{(B_x - A_x)(B_x - A_x) + ...
1
vote
1answer
101 views

Circumcenter and incenter

I need a hint on this problem: Given a triangle ABC. CH is the altitude. CM and CN are the bisectors of $\angle ACH$ and $\angle BCH $. The circumcircle of $\triangle MNC$ and the incircle of ...
6
votes
1answer
907 views

Four turtles/bugs puzzle

I was reading about the the four turtles/bugs math puzzle Four bugs are at the four corners of a square of side length D. They start walking at constant speed in an anticlockwise direction at all ...
2
votes
4answers
128 views

A problem about rational number for right triangles.

If a right triangle has hypotenuse of length 5, and the remaining two sides $a, b$ have rational lengths, what can we say about $a$ and $b$?
3
votes
0answers
105 views

“Round” regions on surface of convex polytope

A convex $d$-polytope $P$ is the convex hull of finitely many points. Given such a polytope with $n \gg d$ vertices, I would like to prove that its surface has to be "round" in some region. Let me ...
1
vote
1answer
232 views

Triangle related question 2

In triangle $\triangle ABC$, if $AD$ is the angle bisector of angle $\angle A$ then prove that $BD=\frac{BC \times AB}{AC + AB}$. Any help/hints to solve this problem would be greatly appreciated.
3
votes
1answer
2k views

Find length of segment in triangle

In triangle $ \bigtriangleup ABC$, the known sides are: $AB=5$, $BC=6$ and $AC=7$. A circle passes through points $A$ and $C$, crosses straight lines $BA$ and $BC$ at points $K$ and $L$, which is ...
0
votes
1answer
111 views

The Nearest Points

Given a set R of N points R={(x1,y1,z1),(x2,y2,z2),.....,(xn,yn,zn)} and set S of M points S={ ((a1,b1,c1),(a2,b2,c2),...(am,bm,cm))}. for each point pi(i=1 to N) in Set R ,find the point qj(j=1 to ...
3
votes
2answers
1k views

Smallest possible perimeter of a triangle if area is 135

What is the smallest possible perimeter of a (flat, regular) triangle if the area is 135? I tried various equations but I always ended up with two unknown variables.
0
votes
3answers
468 views

Solving problem: Area of Triangle

I have this data: $a=6$ $b=3\sqrt2 -\sqrt6$ $\alpha = 120°$ How to calculate the area of this triangle? there is picture:
7
votes
4answers
4k views

Find the area of overlap of two triangles

Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
2
votes
2answers
220 views

Triangle related question

My question is: In Triangle ABC , let AE be the angle bisector of angle A. If 1/AE = 1/AC + 1/AB , then prove that angle A = 120 degrees. What i tried: I extended side AB and took a point M on it ...
2
votes
0answers
97 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
0
votes
1answer
35 views

“Lock” a direction vector by preventing motion along a second vector

I have 2 unit vectors, o and v. o is the orientation of a cylinder and v is a direction I wish to move inside this cylinder. However, I want to allow v to only move perpendicular to the cylinder, so ...
2
votes
3answers
182 views

Perimeter of a triangle

A question states: The length of each side of a certain triangle is an even number.If no two sides have the same length what is the smallest perimeter the triangle could have ? ...