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
16 views

Quaternion - Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
3
votes
0answers
19 views

Optimal Box-in-a-Box-in-a-Boxing

As inspired by this closely related problem, suppose I have $n$ cuboid boxes, all with arbitrary (possibly random) finite dimensions. For any two boxes, $B_1$ with dimensions $w_1,h_1,d_1$, and $B_2$ ...
1
vote
0answers
8 views

How is distance between two points defined in barycentric coordinates?

Hope someone can help. I have this 3-d simplex (a tetrahedron) and its vertexes have barycentric coordinates defined as follow: $A_1=(1,0,0,0), A_2=(0,1,0,0), A_3=(0,0,1,0), A_4=(0,0,0,1)$. I am ...
3
votes
1answer
14 views

Compute direction of a cylinder by using 10 coefficients

I am wondering if anyone knows how to compute the direction of a cylinder by using the 10 coefficients. For example, we have the equation of a cylinder as ...
0
votes
1answer
15 views

Create a Ray from two points

I know it's too easy for this website but I couldn't think of it myself. I have a point A(x1,y1,z1) and another point B(x2,y2,z2). And I represent a ray like this : r(t) = o + t *d. Using the given ...
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votes
1answer
12 views

Gauss Curvature…Product of Minimum and maximum values

The function g(ϑ ) = cos2 (ϑ ) fxx (x0 , y0 ) + 2 cos(ϑ )sin(ϑ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in ...
3
votes
0answers
23 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
-1
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1answer
13 views

Trapezoids and Bases [on hold]

A trapezoid has bases of length $x$ and $4$. Let $P$ and $H$ be points on opposite legs of the trapezoid. $PH$ is parallel to the bases and divides the trapezoid into two quadrilaterals of the same ...
3
votes
1answer
29 views

About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
0
votes
1answer
43 views

Square is a parallelogram?

I remember, in the geometry class, our teacher used to tell us some definitions or something that i don't really know about. Why is square a parallelogram?
0
votes
0answers
14 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
2
votes
5answers
227 views

How can I find the radius of a circle from a chord and a section of the radius?

Draw a circle with center O. Line AD is a chord that is 8cm long. The arc above is smaller than the one below. B is the center of AD. Line CB is a line that is 2cm long. It meets AD at 90°. ...
0
votes
0answers
12 views

Problem with an inclined cone and planes

From the image given below, I want to prove that there exists a unique plane $p \neq P$ s.t. $p \cap$ inclined cone $=$ circle centered at $O_{2}$. I also want to prove that if ray $SO_{1}$ (where ...
1
vote
2answers
20 views

Creating a parametric Equation when given the points of a collinear line?

$(-70, 3)$, $(88, 81)$, and $(246, 159)$ are three collinear points. Write parametric equations for $x$ and $y$. (In other words, write equations that produce points when $t$-values are assigned.) ...
0
votes
0answers
18 views

Find planes of a solid with all lines

I'm searching to obtain all planes of a solid out of its corresponding lines. The lines are composed of two connecting points and this is all the information that I've got. What is the best way to ...
1
vote
1answer
18 views

Prove that the locus of a point P is a circle

I'm struggling with this geometry question: The fixed points A and B have coordinates $(-3a,0)$ and $(a,0)$ respectively. Find the equation of the locus of a point P which moves in the coordinate ...
1
vote
1answer
12 views

Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise. Given $|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$ , Find $|AC|$

Let $A,B,C,D$ be the vertices of a four sided polygon taken in anti clockwise. Given $$|AB|=|BC|=3,|AD|=|CD|=4,|BD|=5$$ Find $|AC|$ My try:I have noticed trangles $ABD$ and $BCD$ are right ...
3
votes
1answer
60 views

Parametrising the unit circle without sine and cosine

Is there a nice way to make a smooth and periodic parametrisation $\gamma\colon\mathbb R\to S^1$ of the unit circle $S^1$ in $\mathbb R^2$ that does not somehow involve sine/cosine or (what I find to ...
2
votes
0answers
33 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...
2
votes
3answers
19 views

Equation for a plane perpendicular to a line through two given points

The following type of question is quite popular with examiners at the institution where I study. Find an equation of the plane containing the point $(0, 1, 1)$ and perpendicular to the line passing ...
1
vote
1answer
21 views

Assistance in drawing an Obtuse angled triangle $ABC$ with altitudes

In obtuse angle $ABC$, with the obtuse angle at $A$, let $D$,$E$,$F$ be the feet of the altitudes through $A$,$B$,$C$. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle ...
2
votes
1answer
41 views

Is this a proper way to prove simple geometrical result?

I found this on Quora : Is there anything wrong in the steps illustrated ?
1
vote
2answers
42 views

Fuel level in horizontal cylindrical tank [on hold]

how many meters should the fuel level drop down to in a horizontal cylindrical tank full of fuel with a 2.5 m diameter and 6.25 m length, if i want to take 3 cubic meters of fuel from the tank?
3
votes
3answers
47 views

High school geometry proofs and first order logic?

I am a student of logic who recently came across two column geometry proofs which seem to be the bane of many a high-school student. My main question though, is that is there any way of doing these ...
0
votes
2answers
17 views

Measurement and computation

A large wall in the shape of a parallelogram is to be painted at a cost of $\$ 20$ per litre. Each litre covers 5m$^2$. The wall has a base length of 30m and height of 10m. Find the cost of painting ...
2
votes
0answers
53 views

Shortest distance from a point to vertices of a cube

A $d$ dimension cube has vertices $P_1,...,P_{2^d}$, where the coordinates of each vertex are either $0$ or $1$. To find which vertex of $P_1,...P_{2^d}$ is closest to a given point $P=(p_1, ...
-2
votes
2answers
29 views

$ABC$ is a right angled triangle,B being the right angle.Midpoints of BC and AC are B' and A' .area of triangle $A'B'C$ is? [on hold]

$ ABC$ is a right angled triangle,B being the right angle.Midpoints of BC and AC are B' and A' .Area of triangle $A'B'C$ is ?
0
votes
0answers
23 views

Geometry problem on ratios of areas.

The vertices of a smaller square are the trisection points of a larger square. What is the ratio of the area of the smaller square to the area of the larger square. Given that the smaller square is ...
0
votes
0answers
10 views

Triangles form a harmonic set with their medians and altitudes

In a triangle $\triangle ABC$, let $AD,BE,CF$ be its altitudes and $AK,BL,CM$ their medians. Show that $D\{EF,AB\} = -1$ and $K\{LM,AB\} = -1$ I don't get any of the problems here. Not any of these ...
2
votes
1answer
28 views

Locus if orthocenter

To a circle of radius $1$, two tangents are drawn from any point $P$ on a line $3$ units away from its center. They touch the circle in $A$ and $B$. Find the locus of the orthocenter of ...
0
votes
1answer
18 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
0
votes
1answer
28 views

How to find the length of a line segment in a rectangle

there is a rectangle abcd (vertexs) and there is point labeled P inside the rectangle. AP=55 PD=60 PC =33 what is PB
4
votes
1answer
26 views

Geometric interpretation of linear forms in the sum of four (or eight) squares identity

There is a well-known sum-of-squares identity $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2. \tag{1}$$ Moreover, letting $\vec{u}=[\begin{smallmatrix}a\\b\end{smallmatrix}]$, ...
0
votes
1answer
31 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
2
votes
1answer
18 views

Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
-1
votes
0answers
24 views

What's area of a triangle described below.

A scalene triangle has one point on each side that divides the side so the two part's that make up the side form a ratio k. What's the area of the triangle formed by connecting those points, if the ...
0
votes
1answer
19 views

How do I find the relative coordinates of a picture of a plane in 3d space.

I have a box, with corners $A$ through $H$, as depicted above. I'll consider $B$ the origin of a coordinate system, with the $x$ axis in the direction through $C$, the $y$ axis through $A$ and the ...
1
vote
1answer
62 views

Mathematical aspects of General Relativity

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also ...
0
votes
0answers
31 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
1
vote
0answers
36 views

Algorithm to compute wether a stabbing line exists for a set of line segments

Let $S$ be a set of n segments in the plane. A line $L$ that intersects all segments of $S$ is called a traversal or stabber of $S$. Give an $\mathcal{O}(n^2)$ algorithm to decide if a stabber for $S$ ...
1
vote
1answer
20 views

Computing overlapping circle positions, equidistant from each other.

Hello, I am a programmer and I wanted to develop an application that would have several overlapping circles in the same location, where each circle can be selectable. Is there a mathematical way of ...
-2
votes
0answers
28 views

Area of a circle shall equal the area of a square [on hold]

How can I, using bolzanos theorem, discuss the equal areas of a circle and a square? How can this be shown in a graph? Would be really grateful if any could help me! :)
0
votes
1answer
16 views

Measure of angle formed by chords and two circles

The following is a question from a practice GRE Math Subject Test: In the Euclidean plane, point A is on a circle centered at point O, and O is on a circle centered at A. The circles intersect at ...
3
votes
3answers
59 views

How to prove that tg 55º<$\pi/2$

How to prove that tg 55º<$\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
0
votes
1answer
16 views

Determine if this interpretation satisfies axiom Congruent axiom 1. If so, prove it. If not, find specific

Recall the interpretation of the rational plane: points are ordered pairs $(x, y)$ with $x, y \in \mathbb{Q}$; lines are solution sets of equations $ax + by + c = 0$ with $a, b, c \in \mathbb{Q}$ and ...
1
vote
1answer
27 views

How can I prove the existence of an octagon/decagon/dodecagon?

I've had this question in my head recently due to my math teacher giving me this problem as a bonus on a test. So I have two regular hexagons inscribed in two distinct circles, of radius n. The two ...
0
votes
1answer
16 views

How to identify the surface of a cylindrical patch?

Consider that I have a 3D facet/patch that lies on the surface of a sphere. Taking four non-collinear, non-coplanar points that lie on the facet/patch I can find the patch's underlying sphere/surface. ...
0
votes
1answer
14 views

Proof: Minkowski sum polytope implies A and B polytopes

Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. How do you prove that $A$ and $B$ are polytopes as well?
1
vote
1answer
22 views

Triangulation of 4 points why Delaunay maximizes the minimum?

I have been going through the chapter on Delaunay triangulations from the book by DeBerg (http://www.cs.uu.nl/geobook/interpolation.pdf). In lemma 9.4, he simply says that "from Thales theorem" we can ...
0
votes
0answers
10 views

Inverse of Pascal theorem

I need a simple proof of Braikenridge-Maclaurin theorem, which is also known as the inverse of Pascal theorem (for conics). Do you know any article or book that contains this theorem's proof? Thank ...