shape, congruence, similarity, transformations, properties of classes of figures, points, lines, angles
1
vote
1answer
28 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 ...
0
votes
0answers
10 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
0answers
11 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
27 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
43 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
1answer
17 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
39 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
26 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 ...
3
votes
2answers
57 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
20 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
1answer
31 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
31 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 $D$ ...
11
votes
0answers
74 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
44 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
51 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 ...
2
votes
2answers
70 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
52 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
35 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
17 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
53 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
25 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
31 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
24 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
20 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:
1
vote
1answer
34 views
Using Semi-circle find side of triangle
The figure below above shown a bicycle path. If semicircular portion $ABC$ is $100$ $\pi$ and $CD$ is $100$$ft$ then what is $AD$?
I have tried to find the diamenter of the circle and the ...
4
votes
1answer
41 views
How to find area of triangle from its medians
The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is
a) $48$
b) $144$
c) $24$
d) $72$
I don't want whole solution just give me the hint how ...
3
votes
3answers
34 views
Finding side of rectangle using given information
Really simple question but I am stuck. The following information is given:
$$BD=8,\quad AB = 6,\quad ED =5,\quad EF = EC$$
and we want to find $AF$.
If we have three $90^\circ$, what does that ...
1
vote
1answer
30 views
Packing circles on a line
On today's TopCoder Single-Round Match, the following question was posed (the post-contest write-up hasn't arrived yet, and their explanations often leave much to be desired anyway, so I thought I'd ...
2
votes
3answers
32 views
Right-angled isosceles triangles
If a right-angled triangle is isosceles then the other two angles must be equal to $45^\circ$ ?
Is this always the case or are there other possible right-angled isosceles triangles?
1
vote
1answer
39 views
Find next point in ellipse given the chord length
I would like to draw a cloud programmatically. For this reason I need to know where to draw the next circle around the ellipse.
Given the chord (circle radius), how can I calculate the next point in ...
0
votes
1answer
32 views
triangle, vectors, proving an identity.
I'm trying to prove something but unfortunately I can't.
Let $ABC$ be a triangle and $M$ a point in $[AB]$ where $d(A,M)=d(B,M)$.Let also be
$N$ be a point in $[AC]$ where $d(A,N)=d(B,N)$.
Prove ...
0
votes
1answer
25 views
Expressing a point in two coordinate systems
Let $(O,e_1,e_2,e_3)$ and $(O',e_1',e_2',e_3')$ be two coordinate systems. Let $\overline{OO'}=2e_1-e_2+3e_3$, $e'_1=e_1-e_2+3e_3$, $e'_2=e_1+e_2+e_3$ and $e'_3=e_1-e_2-e_3$.
a) Find the coordinates ...
2
votes
0answers
38 views
Puiseux series and Resolution of Singularities
I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities.
So the Newton's method gives us a Puiseux series parametrizing the branches of ...
2
votes
1answer
79 views
How to prove this inequality $xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$
Let $x,y,z$ is real numbers,and such that $x+y+z=1$,and in $\Delta ABC$,prove that
$$xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$$
I think this inequality maybe use $x^2+y^2+z^2\ge ...
0
votes
1answer
12 views
Ray Disk intersection
So if I have a ray parameterized as $O + tD$ where $O$ is the origin, $D$ is the direction and $t$ is the parameter variable and a flat circular disk with a center point $P$ in 3D space and a radius ...
2
votes
1answer
21 views
Scale rectangles so they have same height and don't exceed a total width?
I have three rectangles of different sizes side by side.
I want to scale them all (maintaining their aspect ratio) so they have the same height and don't exceed a total width.
I know I could find ...
5
votes
0answers
54 views
square cake with raisins
Alice bakes a square cake, with $n$ raisins (= points).
Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins.
Note that a single ...
4
votes
1answer
36 views
Book on quadric surfaces with linear algebra
Most information that I can find about quadric surfaces is written from a calculus perspective - without using any matrices or vectors. However, I would like to have a reference that tells me the ...
0
votes
0answers
28 views
How do you calculate the angle of deflection of a plumb line towards a mountain?
How do you calculate the angle of deflection of plumb line being pulled down by the entire mass of earth, 5.89 x 10^24 kg and being pulled horizontally by the entire mass of mount everest, 6.399 x ...
2
votes
4answers
43 views
Relation between chords length and radius of circle
Two chords of a circle, of lengths $2a$ and $2b$ are mutually perpendicular. If the distance of the point at which the chords intersect,from the centre of the circle is $c$($c<$radius of the ...
