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

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0
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2answers
143 views

How many times can quadric kiss cosine at given point?

Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of ...
1
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1answer
80 views

How can we draw a line parallel $l_1$ and $l_2 $ to that passes of $p$

How can we draw a line parallel $l_1$ and $l_2 $ to that passes of $p$ ? Please help me. Thank you.$\Large\color{red}{☺}$
6
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2answers
226 views

Is There A Function Of Constant Area?

If I take a point $(x,y)$ and multiply the coordinates $x\times y$ to find the area $(A)$ defined by the rectangle formed with the axes, then is there a function $f(x)$ so that $xy = A$, regardless of ...
1
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1answer
196 views

Why is the volume of a parallelepiped equal to the area of its base times its height?

Is there a formal proof? or is it by definition?
-1
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1answer
130 views

Probability that points are on a straight line

I am looking at a formula to calculate the probability that $n$ points are on a straight line between point $1$ and point $n$ in 2d Euclidean space. If the points are exactly on the line, the ...
2
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2answers
140 views

Universal property of universal bundles.

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the ...
0
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1answer
119 views

Prove that the tetragon ABCD also has an inscribed circle

There is given convex tetragon ABCD and some points J,K,L,M on the line segments AB,BC,CD,DA respectively. The intersection of JL and KM is T. Each of the tetragons AJTM,BKTJ,CLTK, and DMTL have an ...
1
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2answers
297 views

Show that the area vectors for a general $n$-sided closed shape sum to zero

It is possible to show that the sum of the area vectors for a general, closed, $n$-sided figure in $\mathbb{R}^3$ (3-space) is zero. Hint: it may be easiest to consider orientable and non-orientable ...
3
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3answers
2k views

Show that the area vectors for a general tetrahedron sum to zero

Using vector addition and multiplication, it is possible to show that the sum of the area vectors for a general closed tetrahedron in $\mathbb{R}^3$ (3-space) is zero. Hint: start by writing down ...
0
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2answers
40 views

Geometric Maximization Exercise

What is the height of a circular cone with surface line ("Mantellinie") s, which has the maximal volumina? My problem here is, that I am not really sure what surface line is..is this the same thing as ...
4
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1answer
192 views

How to slice an area in rectangles optimally? [duplicate]

Given a contiguous subset of a chessboard (or, more general, a 2d rectangular grid), how can I algorithmically determine a minimal set of rectangles covering the area? In this example, the ...
2
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3answers
703 views

What is the relation between vectors in physics and algebra?

Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. ...
2
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1answer
1k views

A unique circle with 3 points proof

I have prove the theorem: There is only one circle passing through three given non-collinear points in both geometrical and algebraic ways. THere is one question that I just have no idea with. 'the ...
12
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1answer
2k views

A hard geometry problem on circles

I found this problem on a website and I couldn't do anything. Do you have any ideas, hints? Edit: If I say $$\frac { { \partial }^{ 2 }f }{ \partial { a }^{ 2 } } +\frac { { \partial }^{ 2 }f }{ ...
0
votes
2answers
58 views

Geometric-Variational idea of Sine

I like to see sin(theta) as a property of a line making clockwise angle of theta with a horizontal axis.That property would dictate how much % the vertical component of each point in the line ...
9
votes
1answer
268 views

series for $\pi$ which correspond to apollonian gaskets or hyperbolic tilings of the unit disk

Consider the two partitions of the unit disk in $\mathbb{R}^{2}$, the first an Apollonian gasket and the second is the $\{7,3\}$ hyperbolic tiling: Since the unit disk has radius $1$, both of these ...
2
votes
1answer
108 views

comparing angle in measurement for concave mirror

From the above picture we can prove easily $ CF = XF$ because CFX is a isosceles triangle. Now if we move the point X more near near P, it will give the same results(I mean we will get again $ CF = ...
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3answers
115 views

Similar triangles question

If I have a right triangle with sides $a$.$b$, and $c$ with $a$ being the hypotenuse and another right triangle with sides $d$, $e$, and $f$ with $d$ being the hypotenuse and $d$ has a length $x$ ...
3
votes
1answer
126 views

Prove that $A_1,B_1,C_1$ is collinear

Let $ABC$ be a triangle inscribled inside circle $(O)$ . M is a point inside the triangle $ABC$ ($M \notin BC,CA,AB$) $AM,BM,CM$ meets $(O)$ again at $A',B',C'$ respectively. Midperpendicular of ...
2
votes
2answers
199 views

Perpendiculars on a line segment

Two points A and B are given. Find the set of feet of the perpendiculars dropped from the point A onto all possivle straight lines passing through the point B.
1
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2answers
968 views

Equation of a line passing through a point and forming a triangle with the axes

How can I find the equation of a line that; is passing through the point (8, 6) and is forming a triangle of area 12 with the axes ? So I tried to start using $A = |{\frac{mn}{2}}|$ and ...
0
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2answers
79 views

Finding Area of a shape

I'm doing some revision and I'm a bit stuck. How do you find out the area of this shape? I know you have to do 10cm x 2cm = 20cm devided by 2 = 10cm - 2cm x ? (something) = area. I'm not sure what ...
4
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0answers
73 views

Spacing nodes by moving the shortest distance possible.

I have a list of N nodes with positions $(x, y)$ each. I want to move each node the shortest possible distance such that every node is placed on the radius $R$ from at least one other node, and is at ...
8
votes
3answers
152 views

One square per person

There are n persons. Each person draws k interior-disjoint squares. I want to give each person a single square out of his chosen k, so that the n squares I give are interior-disjoint. What is the ...
1
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1answer
455 views

Claim: Equilateral triangle is intersection of three equal circles while $R\rightarrow \infty $

I believe that equilateral triangle is intersection region of three equal circles with radius $R\rightarrow \infty $ , which have the same specific distance to each other. While distances changes ...
2
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2answers
1k views

Equation for a line through a plane in homogeneous coordinates.

For calculations in 2D space, there exist a few useful equations to compute general geometry with the vector dot product . and the vector cross product x when working with homogeneous coordinates ...
3
votes
3answers
900 views

Plane geometry tough question

$\triangle ABC$ is right angled at $A$. $AB=20, CA= 80/3, BC=100/3$ units. $D$ is a point between $B$ and $C$ such that the $\triangle ADB$ and $\triangle ADC$ have equal perimeters. Determine the ...
1
vote
1answer
353 views

Quaternion exponential map, rotations and interpolation

A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
0
votes
1answer
61 views

A simple geometry problem with points

Given the points $M(3,4)$ and $N(1,2)$, find $x$ in the point $P(x,0)$ so that $PM + PN$ is a minimum.
2
votes
1answer
47 views

Total area of squares.

We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
1
vote
1answer
129 views

Is this enough info to solve this time dilation problem

There are two clocks. One is a regular clock measuring regular time $\tau$. The other is a clock measuring time $t$ which also advances clockwise, but does not advance uniformly--it accelerates ...
6
votes
1answer
92 views

Area of circles: represent $x$ in terms of $r_1$ and $r_2$

See the image. Area of green and red regions are equal. Can you represent $x=|O_2D|$ in terms of $r_1$ and $r_2$ for $r_1> r_2$ ? Edit: The point $O_1$ does not enter in the region of small ...
4
votes
1answer
149 views

Count the number of possible drawings

Take a square, ABCD. Add two points, E and F on AB such that AE=EF=FB. Now, add G, H on BC, I and J on CD, K and L on AD. Now pick four pairs of the eight points, E,F,G,H,I,J,K,L. Draw the line ...
8
votes
3answers
512 views

What does area represent?

Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of ...
1
vote
1answer
116 views

What is orthogonal projection of zero to triangle generated by three points (0,1) (5,0) and (2,4)

What is orthogonal projection of zero to triangle generated by three points (0,1) (5,0) and (2,4) Well, in my opinion, there is none. However, my teacher think that it has but he don't know how ...
3
votes
1answer
243 views

Estimate for a rigid transform given a set of noisy measurements

I have a set of rigid transforms $\in \mathbb{R}^{4x4}$, where each transform is an approximation to some unknown, "correct" transform. I'm looking for an algorithm to estimate the correct transform ...
0
votes
1answer
179 views

Problem on hyperbolic hyperboloid generated by a rotation

This is the problem: In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
0
votes
1answer
118 views

Why in the affine space can not we use Grassmann formula?

For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar. For this reason it is not worth the Grassmann ...
0
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2answers
167 views

Which polygon tile grids allow convex polygons to be formed from multiple tiles?

If I have a grid made of equilateral triangles, I can easily form larger convex polygons as a set of tiles in that grid. I believe this holds for some (but not all) tilings of non-equilateral ...
2
votes
1answer
386 views

Pullback Calculation

If we define the 2-form $\omega=\frac{1}{r^3}(x_1dx_2\wedge dx_3+x_2dx_3\wedge dx_1+x_3dx_1\wedge dx_2)$ with $r=\sqrt{x_1^2+x_2^2+x_3^2}$ If we now define ...
0
votes
2answers
134 views

Proof that the cartesian plane is an incidence geometry with only vector definition of a line.

I can get up to showing that it is an abstract geometry, but I cannot figure out how to show that for every two points, there is a unique line. The definition of a line in vector form is given $L ...
4
votes
2answers
6k views

Trigonometry Airplane question. Finding bearing and distance.

A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
2
votes
1answer
165 views

Mapping of a Lens-shaped region by a Möbius Transformation

Consider the 'lens' described by $\{z:|z-i|<\sqrt{2}\ \text{and}\ |z+i|<\sqrt{2} \}$ . We want to map this to the upper right quadrant using a Möbius transformation. The two circles meet at ...
2
votes
2answers
282 views

Constructing a conformal map from $\mathbb{D}$ to a cut plane

Source: Oxford Exam $A2 \ 1999$ We want to construct a conformal map $F$ from the unit disc $\mathbb{D}=\{z:|z|<1\}$ to $\mathbb{C} \setminus S$ where $S$ is the half-line $\{x+i:x \in (-\infty,0] ...
0
votes
1answer
59 views

analytical geometry

We have an affine coordinate system and $3$ points given: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,1)$. I have to find a linear transformation, which depicts the points $A$, $B$, $C$ and $D$ ...
18
votes
1answer
491 views

How many points can you find on $y=x^2$, for $x \geq 0$, such that each pair of points has rational distance?

Open problem in Geometry/Number Theory. The real question here is: Is there an infinite family of points on $y=x^2$, for $x \geq 0$, such that the distance between each pair is rational? The ...
0
votes
2answers
63 views

Calculate Point based on distance in 2D-Space

I have a Point P in unit circle (on or in it) with a radius of r. How can I calculate a Point Q with a fixed radius of x, which has the same angle like P
1
vote
2answers
196 views

Hydraulic radius of a complex shape

I'm working on a project involving thermoacoustics, and one of the important parameters is known as the hydraulic radius. If you have a pipe with some odd geometry, the hydraulic radius is its ...
2
votes
1answer
88 views

Computational geometry

Computational geometry? (Computational geometry) Given a set of n randomly scattered points for even n = 2,4,6,...,50 . Find the maximum number of lines between the pairs of nodes in such a way the ...
3
votes
2answers
435 views

Prove $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $

Prove that $\sin \alpha+\sin \beta+\sin \gamma \geq\sin 2\alpha+\sin 2\beta+\sin 2\gamma $ where $\alpha$ $,\beta$ $,\gamma$ are the angles of a triangle