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

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1answer
155 views

Isometries to prove rhombus?

Suppose that the diagonals of a quadrilateral are perpendicular bisectors of each other. Use isometries to prove that the quadrilateral must be a rhombus. Im unsure how to use isometries to prove this ...
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1answer
685 views

2D triangulation

I understood what it is from the following link: http://electronics.howstuffworks.com/gadgets/travel/gps1.htm But I want to know : In a 2D plane, if we know the (x, y) positions of three “guard” ...
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2answers
178 views

split irregular line in equal parts

I have this irregular line and I want to split it in, for example, ten equal parts. How can I do that? Thank you!
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1answer
32 views

Let $f$ be glide reflection, prove $\frac{1}{2}(\vec x + f(\vec x)) \in g$

Let $f$ be a glide reflection with $\sigma_{\lambda,g}=\vec x + 2 (d-\vec n \bullet \vec x) \vec n+\lambda \vec{n_r}=A\vec x + 2 d\vec n + \lambda \vec{n_r}$ with $g=\lbrace \vec x:\vec n \bullet \vec ...
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1answer
148 views

Two sticks between two concentric circles

Let's start with two concentric circles of radii $r<R$. Then we put two sticks inside the outer circle while avoiding the inner circle, say $AB$ and $CD$. Then we compare the length of inner ...
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1answer
1k views

Finding point in two parallel lines in 3d?

The line $L_1$ that goes through the point $A(4,3,-2)$ and its parallel to the line $(x=1+3t, y=2-4t, z= 3-t)$, if $P(m,n,-5)$ belongs to $L_1$, determine the values for $m$ and $n$ I really don't ...
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1answer
2k views

Meaning and types of geometry

I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you ...
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1answer
45 views

Isometries of two dimensional space

I know that isometries of R^n are composed of orthogonal transformations followed by translation. My questions are: In 2-D space, there are glide reflections, but why must the glide be according to ...
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1answer
102 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
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0answers
63 views

What's the difference between a $2$-sided and $2$-sided strip polytan

There are $14$ $2$-sided tetratans and $13$ $2$-sided strip tetratans. The sets are identical, except the square is missing in the strip version. My best guess is that for strips, no vertex can have ...
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1answer
94 views

finding Length of a diagonal

Given Quadrilateral ABCD in such that $AB<BC<CD$ creating increasing arithmetic progression with sum of $27$ cm. $\measuredangle BCD=60^{0}$. the diagonal $BD=\sqrt{133}$ cm, and it divided ...
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3answers
4k views

Calculating circle radius from two points and arc length

For a simulation I want to convert between different kind of set point profiles with one being set points based on steering angles and one being based on circle radius. I have 2 way points the ...
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1answer
202 views

Angular alignment of points on two concentric circles

I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
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2answers
138 views

sliding a shape with area= 5 on a grid so it covers at least 6 of it's points - riddle

place a shape on an integer coordinates grid, which is continuous without holes, and that its area is 5. Explain why you can slide it (without twisting or warping it), so that it will cover at least ...
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2answers
1k views

What is the initial reason to define the evolute of a curve?

The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle. What is exactly "the envelope of the normals" ? What is the reason to ...
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2answers
195 views

Cheapest can problem

A cylindrical can which must hold 1000 mL is set to be designed so the least amount of material is necessary to make the can. What should the radius be? What is the height of the can? What is the ...
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1answer
39 views

similarity : $z'=(1-i)z+1+i$ with the curve of $e^x-1-x$.

Let $S$ be the similarity defined by : $S(z)=(1-i)z+1+i$, for a complex number $z$ in the complex plane. What is the image of the curve : $y=e^x-x-1$ by the similarity $S$. My work : Let $z=x+iy$ ...
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2answers
155 views

Quick question, does $\sin^2(4x) + \cos^2 (4x)$ equal 1?

Does $\sin^2(4x) + \cos^2 (4x)=1$? So even $\sin^2 (249023049x) + \cos^2 (249023049x) = 1$?
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1answer
105 views

Largest Convex polygon consisting of k points

The problem is Given a set of points, determinate the Largest (in terms of area) Polygon consisting of at most $k$ points. In a shape like The one below: $k = 3,polygon =A,F,G $ I would like to ...
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1answer
249 views

Rational and Irrational Angles

What are rational and irrational angles? Are they just angles, the radian measure of which is respectively rational or irrational? They came up in conversation, and I tried researching them, but ...
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2answers
218 views

Determine number of squares in progressively decreasing size that can be carved out of a rectangle

How many squares in progressively decreasing size can be created from a rectangle of dimension $a\;X\;b$ For example, consider a rectangle of dimension $3\;X\;8$ As you can see, the biggest square ...
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1answer
82 views

Find Line with specific Angle to another Line

Given any line in 3 dimensional space $$A: \vec{X} = \vec{O} + \lambda \vec{D}$$ and any angle $\phi$, I want to find another line $B$ which fullfills the following criteria: it ...
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1answer
2k views

Circle in square, calculate distance from square's corner to circle's perimeter?

I have a square that is $33\times33$ cm. I will put a circle in it that has a diameter of $33$ cm. How do I calculate the distance from the square's corner to the circle's closest perimeter in a ...
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3answers
2k views

Analytic geometry straight line problem

Prove that two straight lines represented by the equation $x^3+y^3+bx^2y+cxy^2=0$ will be at right angles if $b+c=-2$ I didn't know that even straight lines like planes can be represented by a ...
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2answers
615 views

Analytic geometry section of cone and sphere

How to show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circle ? I understand that the two equations taken together represent the circle. but how to go about finding the ...
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1answer
21 views

What is Angle(A,b) about something.

I was reading a paper and came through a notation saying .... Angle = Angle(A,B) about C. Can anybody tell me what exactly it means. Thnaks, Harsha
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0answers
136 views

usable rectangles

This question deals with efficient division of land into land-plots. For the sake of this question, assume that a land-plot is usable only if it is a rectangle whose width/height ratio is between ...
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1answer
246 views

Need “up” vector to calculate distance from a focal plane given world coordinates (SOLVED)

I have a RGB image, and for each pixel in the image I also have its real world coordinate. I also have the location (real world coordinate) yaw, pitch and roll of the camera. I am trying to produce ...
3
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1answer
430 views

Is this union of line segments a connected set?

Let $S$ be a subset of $\mathbb{R}^2$ that is a union of two horizontal line segments $(0,0)-(1,0)$, $(0,1)-(1,1)$ and a union of a countable set of vertical line segments ...
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1answer
65 views

Property about revolution surfaces

"Consider $f:[a,b]\rightarrow\mathbb{R}$ of class $C^1$, limited, such that $f(x)\neq 0$ for all $a\leq x\leq b$. After this, consider the revolution surface by turning graphic of $f$ around the $x$ ...
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2answers
69 views

Help with unknown notation

In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere): "writing $z=x+iy$, we can verify that: $$x:y:-1=x_1:x_2:x_3-1, $$ and this means that the points ...
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1answer
78 views

Angle sum of a triangle.

Can you please describe the geometry in which the sum of the angles of the triangle can be less than 180 degrees?
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2answers
23 views

Triangle inequlity improvment with the angle conditions

I was working on how to proof $a+b \leq x+y+z$? Apply triangle inequity to the triangle ADC, $x+z \geq a$ Apply triangle inequity to the triangle DCB, $y+b \geq z$ Adding above inequities, ...
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5answers
189 views

Proof: $\vec x,\vec y \perp \vec z \Rightarrow \vec x || \vec y$

I have to prove $\vec x \perp \vec z$ and $\vec y \perp \vec z$ imply $\vec x || \vec y$ where $\vec x,\vec y,\vec z \in \mathbb{R}^2$ and $z$ nonzero. I know $x \perp z \Leftrightarrow ...
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1answer
95 views

Function to create long rounded rectangles

Is it possible to describe with a function the following shape or would that result in a too cumbersome approach? I am looking for something like this, because I would like to keep the length, width, ...
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1answer
97 views

Question on the perimeter of any quadrilateral

Is it true that the perimeter of any convex quadrilateral inside a unit circle is no more than $4\sqrt{2}$?
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1answer
126 views

Rotate 3d plane

I have a plane in 3D space that formed from 3 poin $P_1=(x_1, y_1, z_1)$, $P_2=(x_2, y_2, z_2)$, $P_3=(x_3, y_3, z_3)$ I want to rotate and transform this points (equally related plane) into 2D space ...
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1answer
76 views

why are these integrals equal?

I've been trying to analyse Steiner symmetrization. I've asked before about the symmetrization preserving $\textit{volume}$, I'm still going over the same proof. I think I understand a bit what is ...
2
votes
1answer
510 views

Mean displacement for a random walk on a $d$-dimensional lattice

How does the mean displacement of a random walk on a $d$-dimensional integer lattice (or $d$-dimensional hexagonal lattice) scale with the number of steps $N$ in the walk? Is the displacement always ...
0
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1answer
67 views

Proof dividing ratio of points on a line

I'm thinking about following situation: Draw a line with two points $B,C$. Let be $X$ a point between $B$ and $C$. I can write $X$ as the following: $x=\lambda_1 \vec b + \lambda_2 \vec c$. Now ...
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1answer
52 views

Explain about convexity in geometry and in optimization.

My question is 'what is a difference between convexity in geometry and optimization?'
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2answers
111 views

What does $\{(x_1, x_2, x_3) \in\mathbb R^3: x_3 \leq x_2 \leq x_1 \}$ look like?

What does $\{(x_1, x_2, x_3) \in\mathbb R^3: x_3 \leq x_2 \leq x_1 \}$ look like? It seems to be a linear convex cone with vertex at the origin. I am trying to visualize it but cannot. Thanks!
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3answers
303 views

Pythagorean theorem and its cause

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
5
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0answers
506 views

Second derivative of a metric in terms of the Riemann curvature tensor.

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial ...
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1answer
93 views

P is a point in triangle $ABC$, what is $[APC]$?

Moderator Note: This question is part of an ongoing contest on Brilliant.org, and will be unlocked in 1 week. P is a point in triangle $ABC$. The lines $AP$,$BP$, and $CP$ intersect the sides ...
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1answer
216 views

Similarity of triangles in a circle

The problem: c is a circle with a diameter AB. t is the tangent at the point B. Now C and D are two points on t and at different sides of B. I draw the line segments AC and AD, the point where AC ...
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1answer
74 views

Can more than one hamiltonian graph have the same set of hamiltonian paths?

For some pair of non-isomorphic hamiltonian graphs, can there be a chance that it be shown to have the same set of all hamiltonian paths in each graph? we get the set of all hamiltonian paths in each ...
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2answers
757 views

what will be the parameterization of cone

I have question, I need more idea. can any one answer my question I have tried but i didnot get full idea I know this question we have to use parameterization of cone which i donot know in this case ...
2
votes
3answers
1k views

given coordinates of beginning and end of two intersecting line segments how do I find coordinates of their intersection?

There are two line segments. I know for sure they intersect (so I don't have to check it). For both line segment I know coordinates of its both ends. With what formula can I find coordinates of their ...
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2answers
159 views

Distance from the midpoint of a radius to another point on the same radius

Here is a picture of the problem. Note that $M$ is the midpoint of $OB$. How do I figure out what $MH$ is?