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
0answers
102 views

How to model bending, folding of 2D figures?

2D shape can be folded in various ways. For example, trapezoid can have its sides (with the acute angles) folded, so that it will effectively become rectangle – with the difference, that part of the ...
2
votes
1answer
166 views

I have a vector of length 1, and two angles (in $xz$ plane and $yz$ plane), how do i get the $x,y,z$ compontents of this vector?

I just can't see what's wrong, this should be relatively simple... I have a vector which has length 1. Then, I have two angles. The first ($\gamma$) is the angle between the projection of the vector ...
1
vote
2answers
350 views

Triangle in hexagon

In a regular hexagon ABCDEF is the midpoint (G)of the sides FE and S intersection of lines AC and GB. (a) What is the relationship shared point of straight ...
0
votes
2answers
105 views

Finding An Equation For A Parabola

The information given in this particular problem: Axis is parallel to y-axis; graph passes through and $(4,11)$.$(3, 4)$ $(0,3)$ From this information, I know that it opens either upwards or ...
-2
votes
2answers
180 views

Can it be argued that this question only has 1 answer ultimately?

I came across this quiz question in a forum today. I would like to ask for your opinion of this notorious mathematics question, and also, to share. This question came out in Singapore's PSLE ...
3
votes
0answers
69 views

Is there a way to derive the surface of a ball without integral? [duplicate]

Possible Duplicate: Can the Surface Area of a Sphere be found without using Integration? A ball is effectively a pyramid with "curved" based. If we know the surface, which is $O=4 \pi r^2$, ...
0
votes
1answer
381 views

What are spatial Transformations?

What are spatial Transformations? Are Affine transformations also part of spatial transformations?
1
vote
3answers
807 views

What is the maximum number of regions produced, i.e. $f(n)$, by joining all vertexes with line segments of a convex polygon with $n$ sides?

What is the maximum number of regions produced, i.e. $f(n)$, by joining all vertexes with line segments of a convex polygon with $n$ sides? For example, for the hexagon on the left, number of ...
1
vote
2answers
63 views

Determine whether the triangles $ABC$ and $DEF$ are rectangles

How can we determine whether the triangles $ABC$ and $DEF$ are rectangles? We have $A(-6,5),B(-3,3),C(1,9),D(1,3),E(5,1),F(11,10)$.
1
vote
2answers
140 views

finding area of the fourth circle

Three circles of the same radius are arranged in such way that one circle is tangent to the other two. A fourth circle is drawn so that it will contain three circles and be tangent to the other ...
1
vote
1answer
43 views

Why does this system of equations result in an always-positive output?

I have two variables, M and P, whose relationship is described by the following two equations: [1] P = 50.5 * M / ( M - 50) [2] M = C * P where C is a positive ...
4
votes
2answers
587 views

Construct a triangle given one side, its height and inradius

I've been scratching my head with this problem: "Draw a triangle given one of its sides, the height of that side and the inradius." Now, I can calculate the area and obtain the semiperimeter. From ...
2
votes
4answers
2k views

what's the name of the theorem:median of right-triangle hypotenuse is always half of it

This question is related to one of my previous questions. The answer to that question included a theorem: "The median on the hypotenuse of a right triangle equals one-half the hypotenuse". When I ...
11
votes
2answers
1k views

Relationship between diameter and radius of a point set

Consider a set of $n$ points in $\mathbb{R}^k$. The diameter of this set is the maximum distance between two of its points; its radius is the radius of the smallest (closed) k-ball that contains all ...
0
votes
1answer
186 views

Is the set of points of equal distance to the surface of an ellipsoid again an ellipsoid?

Consider the hyperellipsoid $A$ in $\mathbb{R}^d$ given by the semi-major axes $a_1,a_2,\ldots,a_d$. Do points on the surface of the hyperellipsoid $A'$ with semi-major axes $a_1-\varepsilon, ...
8
votes
3answers
451 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
4
votes
2answers
705 views

Resource for learning straightedge and compass constructions

Does anyone know a good resource for learning about straightedge and compass constructions besides "The Elements?" I tutor geometry to middle-schoolers and high-schoolers and thought that including ...
2
votes
3answers
7k views

Optimization of the Area of a rectangle with regards to an Ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I got as far as coming up with the equation for the ...
2
votes
3answers
3k views

How to find the other vertices of an equilateral triangle given one vertex and centroid

If I know the coordinates of the center and one vertex of an equilateral triangle, how do I find the coordinates of the other vertices? I'm thinking I need to find (x,y) such that the distance to the ...
5
votes
1answer
149 views

Probability of seeing to a certain distance in a forest and related problems

I was walking in a forest one day and saw trees all around me. I begun wondering about how far do I see in the forest on average. I was also reminded to the "proof" that the age of the universe is ...
1
vote
2answers
127 views

Can an ellipse with fixed semi-axis have different values of eccentricity?

Warning: this is probably a ridiculous question but here goes... Can an ellipse with a semi-major axis $a$ take on different values of eccentricity $e$? I have seen various places where it seems to ...
1
vote
1answer
285 views

Straightening the boundary in concrete examples

Let $\Omega \subset \mathbb{R}^d$ be open and with $C^1$ boundary $\Gamma$. For any given point $x_0 \in \Gamma$ we know there's a neighborhood where $\Gamma$ is the graph of some $C^1$ function ...
1
vote
0answers
29 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
1
vote
1answer
215 views

Analytically derive n-spherical coordinates conversions from cartesian coordinates

I'm finding it difficult to find any non-geometrical derivation of coordinate conversions from cartisan to spherical. I can understand the derivations geometrically, because I can visualize the ...
3
votes
2answers
84 views

What's wrong with an irregular digon?

I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a ...
1
vote
3answers
215 views

Prove that point M is on circle c

It's hard to create question names that make sense. Anyhow, the following is another question from my math assignment. Line-segment AB has a fixed length of 10 units. point A moves on the positive ...
2
votes
1answer
522 views

Puzzle on the triangle.

In triangle top four figures that have to be repositioned to form the "triangle" without a unit square. How to explain this? Thank's.
1
vote
1answer
7k views

How to calculate radius when I know the tangent line length?

For my math homework, I was asked this question: The tangent lines from O hit a circle with center M and radius r in R and S. Calculate r. -The length of OR and OS is 4 How do I calculate the ...
1
vote
0answers
147 views

Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
2
votes
2answers
323 views

Möbius map from circles to lines

I want to find a Möbius transformation that takes the two circles $C(i,3), C(-i,1)$ to the parallel lines $Re(z)=0, Re(z)=1$. I know that they intersect at -2i, which means I have to map $2i \mapsto ...
1
vote
2answers
676 views

Dividing a Triangle into Two Parts of Equal Area with Constraints

Given a triangle $ABC$ and a line labelled 'n' that passes through the triangle but its not parallel to any of the sides, how do we construct a line parallel to 'n' that will divide $A$$B$$C$ into two ...
3
votes
1answer
55 views

What is inversion and how does it act on figure inscribed in a circle?

Trying to wrap my head around inversions. I understand it takes things from inside to outside, such that the distance from some point inside circle + the distance to new point is r^2 where r is radius ...
3
votes
3answers
148 views

Ratio of lengths in isosceles triangle

In $\triangle ABC$ , $BC = AC$. Also $D$ is a point on side $AC$ such that $BD = AB$. Find the ratio $\frac{AB}{AD}$. Justify your answer. The answer is supposed to be $\frac1 {cosA}$ where $A = ...
4
votes
0answers
157 views

How many points does one need for an epsilon-net

Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
1
vote
3answers
82 views

Prove that altitude² = pq?

The following is a question for my math class. I just cannot figure it out. Given is that: h is the altitude that divides the longest side of this right triangle into p and q. Question: Prove that ...
8
votes
3answers
208 views

Other ways of solving $\cot^{-1}(x)=\sin^{-1}(x)$

Real solutions to $$\cot^{-1}(x)=\sin^{-1}(x)$$ I found this problem in an exam years ago and I solved it using geometry. The first mistake I made was assuming ...
6
votes
2answers
151 views

Voronoi Diagrams Proof

I am having a real problem with this proof about voronoi diagrams: Prove that $V(p_i)$ (i.e., the cell of $\operatorname{Vor}(P)$ which corresponds to $p_i$) is unbounded if and only if $p_i$ is on ...
6
votes
1answer
885 views

Find volume of crossed cylinders without calculus.

I found this puzzle here. (It's labeled "crossed cylinders".) Here's the description: Two cylinders of equal radius are intersected at right angles as shown at left. Find the volume of the ...
1
vote
2answers
551 views

4 Points on Circumference of Circle and center

This is actually a computer science question in that I need to create a program that will determine the center of a circle given $4$ points on its circumference. Does anyone know the algorithm, ...
1
vote
1answer
240 views

A proof that the hexagonal lattice describes the optimal sphere packing in two dimensions

In an effort to understand sphere packing in 24 dimensions, I figured that I should start with 1 and 2 first. The proof of 1 dimension is obvious, since we can achieve 100% density, and you cannot do ...
1
vote
4answers
573 views

Geometry Prove - two perpendicular lines in a circle

In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X. Show that:
0
votes
1answer
94 views

Bounding L2 distance with mean and standard deviation

Let $\mathbf{x}=[x_i]_{i=1}^d, \mathbf{y}=[y_i]_{i=1}^d$ be two vectors in $R^d$. Is it possible to find a lower bound $l\leq \|x-y\|$ and an upper bound $u\geq\|x-y\|$ as a function of ...
3
votes
1answer
441 views

Unfolding Polyhedra

I'm interested in learning more on unfolding polyhedra. Are there any known algorithms that unfold polyhedra into nets? I'm interested in writing code on this in either MATLAB, Python, or C#. On ...
2
votes
4answers
91 views

Are the areas of the same size?

The colored areas, are they of the same size? Regards
2
votes
2answers
83 views

Points placing in plane

This is famous puzzle that I can came across recently. Place six points on a plane so that distance between any two points is integer such that no three points are collinear. In 3D it is easy( ...
0
votes
1answer
124 views

Geometry problems: Area problem…[Homework]

1) $\triangle ABC: D \in AB; E\in BC$ such that BD=3AD, BE = 4EC. F is the intersection of AE and CD. Prove that FD = FC (I think we should prove $S_{ACE} = S_{ADE} = \frac{S_{ABE}}{4}$) 2)$\triangle ...
2
votes
0answers
336 views

Find Orthonormal Basis for Image Plane Given Camera Matrix

Given a projection matrix $P = [M | p_4]$, ($M \in 3 \times 3$, $p_4 \in 3 \times 1 $), the principal axis (the vector that passes through the center of projection and is perpendicular to the image ...
2
votes
4answers
926 views

Calculate a point on the line at a specific distance .

I have two points which make a line $l$ , lets say $(x_1,y_1) , (x_2,y_2)$ . I want a new point $(x_3,y_3)$ on the line $l$ at a distance $d$ from $(x_2,y_2)$ in the direction away from $(x_1,y_1)$ . ...
1
vote
1answer
155 views

Finding the maximum area

If $P$ is a point inside quadrilateral $ABCD$ with $P A = 2$, $P B = 3, P C = 5$ and $P D = 6$, find the maximum possible area of $ABCD$.
0
votes
1answer
67 views

Parametric Equation of a Surface

If $\left\{P_{k,l}^{0}\right\}_{k,l=0}^{n}$ is a set of $\left(n+1\right)^2$ three-dimensional points, and $$ ...