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

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24 views

riammanam manifold [on hold]

Lemma 4.6. Let ∇ be a linear connection on M . There is a unique con- nection in each tensor bundle T k l M , also denoted ∇ , such that the following conditions are satisfied. ( a ) On TM , ∇ agrees ...
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1answer
12 views

Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
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1answer
25 views

Circle problem and area [on hold]

What is the area of the circle centered at the origin with radius $5$, restricted to the domain where $x>0$ and $y>0$.
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1answer
18 views

Are congruent shapes similar, too?

I know that congruent shapes have the same size and the rotations don't matter. I just want to know if congruent shapes can also be similar at the same time. I've seen that similar shapes have one ...
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0answers
22 views

Compare time it takes to travel a curve and a line

Suppose you have a right triangle ABC with hypotenuse AB, AC is along gravity direction, C is the right angle. c1 is AB, c2 is a smooth and convex curve within the triangle connecting A and B. You ...
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0answers
24 views

How to check if a set of coordinates creates a polygon?

Well, hello. I've got a set of coordinates and i want to check if it creates one polygon or 2 (or more) polygons. Coordinates are being read from input stream, one after another, for example: ...
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2answers
28 views

A tangent circle element between 2 intersecting vectors

2 Vectors, which are originating from one point I. I want to the replace the sharp corner (I) with an arc (circle element) with a radius of r. The arc touches the vectors at T1 & T2. What is the ...
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0answers
9 views

verifying the geometric relationship in the Blinn-Phong reflection model.

I would like to prove to myself that the angle between the normal and the half vector is twice the angle between the reflection vector and the eye vector, as shown in this image taken from the ...
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0answers
14 views

Equal time path

Given two points A and B, where A is at a higher position than B. It's easy to find the time it takes a mass point to travel from A and B over a straight slope under gravity. Now can you find a smooth ...
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1answer
27 views

Time it takes a mass point to go down a curve under gravity

An age old question. How to calculate the time it takes a mass point to go down a frictionless curve under gravity? P.S. The curve is convex and smooth and can be of any kind of shape.
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1answer
23 views

Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...
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1answer
16 views

Two circles externally tangent find distance from the center of smaller circle to the point of tangency.

The radius of two circles that are tangent externally to each other is $r$ and $s$.Suppose $r>s$ and two outer tangent of circles intersect at point $P$.Denote the center of smaller circle by ...
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0answers
18 views

Affine Buildings

I am trying to study affine buildings. So far I learn a lot of theoretical properties and definitions, but it was hard for me to find a good example of this object to "visualize" the theory. (Yes, I ...
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1answer
38 views

To calculate side of the Equilateral triangle

The figure is an equilateral triangle. 3 line segments , which meet at a(any) point in the triangle , are of the length 5cm, 4cm, and 3 cm as shown in the figure. Find the side of the equilateral ...
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2answers
43 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 ...
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2answers
51 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 $B_2$ ...
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0answers
9 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 ...
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1answer
16 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 ...
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1answer
16 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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1answer
13 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 ...
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0answers
31 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 ...
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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 ...
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2answers
47 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 $-\nabla_{[u,v]}w$ ...
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1answer
44 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?
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1answer
17 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 ...
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5answers
233 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°. ...
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0answers
14 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 ...
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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.) ...
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0answers
21 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 ...
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1answer
21 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 ...
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1answer
13 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
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1answer
63 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
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0answers
35 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' ...
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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 ...
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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 ...
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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 ?
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2answers
43 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
53 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 ...
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2answers
18 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
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0answers
58 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, ...
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2answers
31 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 ?
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0answers
24 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 ...
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0answers
12 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 ...
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1answer
29 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 ...
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1answer
19 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 ...
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1answer
29 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
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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
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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
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1answer
19 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 ...
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0answers
26 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 ...