shape, congruence, similarity, transformations, properties of classes of figures, points, lines, angles
1
vote
2answers
23 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
vote
2answers
33 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 ...
4
votes
0answers
30 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 at the shortest possible distance such that every node is placed on the radius $R$ from at least one other node.
I ...
-2
votes
0answers
27 views
similarity of triangles and tangents
A triangle has sides of length 2, 3, 4. A tangent is drawn to the incircle (inscribed circle) parallel to side 2, intersecting the other two sides at points x, y. Then length of xy =
A. 5/3
B. 10/9
C. ...
5
votes
1answer
54 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
vote
1answer
24 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 ...
0
votes
1answer
22 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
55 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 ...
0
votes
0answers
36 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
45 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
37 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
29 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
69 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 ...
2
votes
0answers
39 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
164 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
0answers
17 views
Normalizing the Second Moment of $n$ Discs
Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($||P_i - P_j||\geq d, i\neq j$).
Graham and Sloane use the second moment as a measure of compactness for ...
1
vote
1answer
32 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 ...
1
vote
0answers
11 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
16 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
32 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
votes
0answers
53 views
Multiplicity of a root
What is the multiplicity of a root $(0,0,0)$ if we have an ideal $I$ which has the next primary decomposition: $(x-y^2,z^3,y^3)$?
Thanks for answers.
0
votes
2answers
26 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 ...
1
vote
1answer
45 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
31 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
70 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 ...
1
vote
1answer
22 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
53 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
33 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 $D$ ...
13
votes
0answers
118 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
21 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
0
votes
2answers
50 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
65 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
74 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
3
votes
2answers
64 views
Find min of $IA + IB + IC +ID$ in tetrahedron $ABCD$
Let the point $I$ in tetrahedron $ABCD$. Find $\min\{IA + IB + IC + ID\}$.
I can't solve this problem, even in the case ABCD regular. Please help
1
vote
1answer
36 views
Circle Packing: Unsolved Problem in Geometry?
Graham and Sloane minimize the second moment of the centres of a number discs in order to maximize their compactness. They use computational geometry techniques to find the optimal packings for ...
3
votes
2answers
28 views
Rotation of a point in 3d space
I'm trying to rotate a point around a single axis of a 3D system.
Given $P=\begin{pmatrix}
101 \\
102 \\
103
\end{pmatrix}
$,
And the rotation matrix formula for rotation around the X axis only, I ...
0
votes
1answer
23 views
Find the bases of a Non Isosceles Trapezoid by the median and diagonal [closed]
One of the diagonals of a non-isosceles trapezoid splits its median into two parts of $3$, $5$ and $2$ cm. Find the length of the trapezoid's bases.
3
votes
2answers
37 views
Physical representation of volume to surface area
I was looking at this XKCD what-if question (the gas mileage part), and started to wonder about the concept of unit cancellation. If we have a shape and try to figure out the ratio between the volume ...
2
votes
0answers
19 views
Determine direction of minimum overlap of convex polygons
Given two convex polygons $P$ and $Q$ what is the minimum intersection polygon $A=P\cap Q'$ where $Q'$ is the polygon $Q$ offset by a vector $\overline r$ of fixed length?
Put another way, what is ...
-1
votes
0answers
39 views
Distance on the surface of a sphere
Given a sphere which radius is r.
There are two red points on the sphere. Given the location of the two points in spherical coordinate system.
If the surface distance between a point and a red point ...
1
vote
1answer
54 views
Property of bisectors of right triangle
In triangle $ABC$ $\angle C=90^\circ$, $AA'$ and $BB'$ are angle bisectors intersecting at $I$ ($A'\in BC$, $B'\in AC$). What would be the easiest way to prove that projection of $I$ onto $AB$ lies in ...
2
votes
1answer
26 views
How to Find the Center of a Parallelogram
I want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It seems ...
0
votes
2answers
32 views
Ray-Lens Intersection
So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector.
I also have a spherical lens with aperture ...
0
votes
0answers
25 views
How to introduce perpendicular or congruence of angles in affine space
$n$-dimensional affine point-vector space is a pair $\mathbb A^n = \langle \mathbb A, V^n \rangle$, where $\mathbb A$ is an arbitrary set, which elements are called points of affine space, $V^n$ is an ...
3
votes
3answers
65 views
Right triangles with integer sides
Most of you know these triples:
$3: 4 :5$
$5: 12 :13$
$8: 15 :17$
$7: 24 :25$
$9: 40 :41$
More generally we can construct such triangles such as
$$2x:x^2-1:x^2+1$$
My question is why one of ...
-4
votes
0answers
40 views
Find ebook A.V. Pogorelov, “Foundations of geometry”. [closed]
Can you help me find ebook : A.V. Pogorelov, "Foundations of geometry" , Noordhoff (1966).
Or book write about axoxiom systems Pogorelov in Euclidean geometry.
0
votes
0answers
40 views
How can I eliminate duplicate set elements?
Given the set of eight angles A={0,45,90,135,180,225,270,315}, if we want to draw all possible graphs that have k vertices, where each vertex must have an exterior angle chosen from A, we need to draw ...
2
votes
0answers
21 views
Minimal surface representation from a 3D contour
I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane and I want to obtain a 3D triangular mesh representation of the surface inside ...
0
votes
1answer
19 views
Maximal square covering
Let X be a shape in 2-dimensional space.
Define a square covering of X as a set of axis-aligned squares, whose union exactly equals X.
Note that some shapes don't have a finite square covering, for ...
1
vote
4answers
30 views
Simple geometry/trigonometry question
How to find the X coordinate of the red point if i know it's Y coordinate and the angle? Let's say the Y is 40 and the angle is 30 degrees:
