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

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1answer
16 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
0
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2answers
31 views

Question about the existence of points and lines.

Say we draw a point on a graph. If the point should not take up any area than how come we could see it. Say we graph $y=x^2$, we obviously could see it. However, because $y=x^2$ is a function made up ...
-3
votes
0answers
33 views

Proving Ordering of Angles

I'm trying to prove $$\text {If}\ \angle P \lt \angle Q, \text & \ \angle Q \cong \angle R, \text{Then}\ \angle P \lt \angle R$$Which seems super basic and makes sense, but I got told that I'm ...
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votes
1answer
27 views

alternative approach than center of mass?

For my bachelor thesis i am modelling the electric output of wind-power plants with the help of a multi agent based simulation. I have the information of all wind-power plants in my country (about ...
1
vote
1answer
36 views

Fitting n number of squares into n area

I am a mobile developer and I have a problem and need to find a formula to get the dimensions of squares to fit inside a space. My problem is that I have a rectangle of dimensions lxw and need to fit ...
1
vote
1answer
32 views

What is the correct way to denote a coordinate system in writing?

I want a systematic way that is readable but also mathematically rigorous for defining a coordinate system in writing. For example for a cartesian coordinate system centered on the fixed or moving ...
2
votes
1answer
39 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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2answers
30 views

Using an ellipse, do all inscribed angles have to be congruent?

For circles, it is well known that all inscribed angles are congruent. With the definition of inscribed angles maintained to ellipses, are all inscribed angles of an ellipse congruent?
3
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1answer
24 views

Surface area of platonic solids inscribed in unit sphere.

If one inscribed each of the platonic solids in the unit sphere, what can be said about the surface areas? Is it a monotonic sequence? Does any "meaningful" (see paragraph below) continuous (or ...
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3answers
42 views

$N$ circles on a circle

Perhaps a rather elementary question, but I simply couldn't figure out the calculations on this one. Say one takes a circle centered at the origin with radius $R$. He or she then proceeds to place $N$ ...
1
vote
1answer
15 views

atanh2 / “ polar argument”

In many application you get a function of the form: $$\tan(x) = \frac{f(A)}{g(B)}$$ Where you then have to use the "polar argument" (to account for all 4 quadrants): $$\arg \left( f(A), g(B) ...
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1answer
16 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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0answers
39 views

Geometry problem – triangle [on hold]

Stiind ca $A(-4;0)$ si $B(0;-4)$ sunt doua puncte dintr un sistem de axe cu originea $O$, sa se precizeze natura triunghiului $AOB$ si sa se calculeze aria acestui triunghi. Knowing that ...
2
votes
1answer
30 views

How to calculate the surface of overlapping ellipsoids

I want to calculate the surface of a body made of at least 3 overlapping ellipsoids. Below there is a picture of the cross section of the body. I already know how to calculate the surface of single ...
0
votes
1answer
13 views

Jacobian of the Kelvin transform

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$ ...
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0answers
30 views

How to find shaded area of square? [on hold]

How to find shaded area of square if it has four circles in it each having 2 meter square area?
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2answers
54 views

All right angles are equal to each other

Why is it that All right angles are equal to each other -a postulate in Euclid's Elements (Wikipedia). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of ...
2
votes
3answers
66 views

Intersection of two moving objects

There are two objects. The first object moves with speed $U_1$ from known point $A$ to known point $B$. The second object has speed $U_2$ and starts from known point $C$. What is the direction the ...
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0answers
25 views

Prove an upper bound

Let $O$ be the origin, and $K$ be a convex polygon in $\mathbb{R}^2$ with edges $K_1, \dots K_n$. Let $\nu_i$ be the unit outer normal of each $K_i$. Suppose $O \notin K$. Prove that there exists a ...
0
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0answers
32 views

Determine the minimum and maximum angles, to the nearest tenth of a degree, that a pipe can make with the horizontal.

For residential drains, a horizontal pipe needs to have a minimum slope of 1/4 inch per foot and a maximum slope of 1/2 inch per foot for waste to drain properly. This means that for every horizontal ...
1
vote
1answer
25 views

Similar quadrilaterals in rectangle

While working on a problem on finding the shaded region in a rectangle, I realized that if a line cuts a rectangle into two quadrilaterals, then these two quadrilaterals are similar as the opposite ...
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0answers
20 views

Vector Geometry Proof with a Pentagon [on hold]

Pentagon $ABCDE$ is inscribed in a circle. For any edge of $ABCDE$, we can draw the line perpendicular through that edge that contains the centroid of the remaining three vertices. Show that these 5 ...
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1answer
39 views

Finding circumference of a circle with a hole in it [on hold]

The bigger circle has Radius R and smaller circle has radius r. I need to find circumference of circle with Radius R which has a hole cut in it of radius r(the smaller circle). I was thinking of ...
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0answers
18 views

Proof of the last part of the Reeb theorem

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. Suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both ...
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0answers
37 views

Relation between polygons

I try to solve a system but I need another equation to make it solved... So I try to find a relation that give the $\beta''$ function of $\beta'$, $\alpha'$, $N$, $D$ et $L$ as described in the ...
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votes
0answers
14 views

Calculate the Integral Points Within a Triangle

Say you have a triangle with vertices of A(-1,-1), B(1,0), C(0,1). The only integer within that triangle is (0,0) or one point. I want an algorithm that can calculate the number of points within the ...
3
votes
5answers
51 views

Showing that certain points lie on an ellipse

I have the equation $$r(\phi) = \frac{es}{1-e \cos{\phi}}$$ with $e,s>0$, $e<1$ and want to show that the points $$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = ...
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votes
1answer
31 views

Orthogonal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
-2
votes
1answer
20 views

What are the coordinates of a point given its distance from another point?

If the abscissa of a point is twice the value of the ordinate and has a distance of $2\sqrt{17}$ units from the point $(4,-5)$, what are the coordinates of the point?
-4
votes
0answers
23 views

In a trapezium $PQRS$, $QP=a,RQ=b,RS=3a$ and the diagonals intersect at $X$. Express $RP$ and $QS$ in terms of a and show [on hold]

In a trapezium $PQRS$, $QP=a,RQ=b,RS=3a$ and the diagonals intersect at $X$. Express $RP$ and $QS$ in terms of $a$ and $b$. Show that $PX:PR=QX:QS=1:4$
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vote
3answers
68 views

Peripendicular distance from a line segment

I have a line given by $Ax + By + C= 0$, and a point $x0,y0$. From that point $x0,y0$ in the direction of the line up to distance $d$, I want to find the perpendicular distance of the points from this ...
1
vote
1answer
39 views

How to find m by given information.

If the chord $y=mx+1$ of the circle $x^2+y^2=1$ subtends an angle of measure $45^o$ at the major segment of the circle then m= . How to find m.
1
vote
3answers
36 views

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$

Convert the equation to rectangular form $r = \frac {6}{1-\sinθ}$ The answer should be: $y = \frac{1}{12} x^2 -3$ But how to arrive at the answer? I tried replacing r with $\sqrt{x^2 + y^2}$, then ...
1
vote
1answer
38 views

What is the probability of two random line segments crossing in a unit square?

For the purposes of this question a random line segment is defined by connecting two random points inside the unit square, where a random point is found by generating two random numbers between 0 and ...
1
vote
0answers
43 views

How to give a rigorous proof of a fact about convex polygon?

I claim that there exists universal constants $0<\delta_1(m), \delta_2(m)<1$ such that for any convex polygon $P$ in $\mathbb{R}^n$ with $m$ faces, \begin{equation} \frac{\mathcal{H}^{n-1}(\{x ...
0
votes
0answers
9 views

What does 3D gaze direction contains? And how to convert it to yaw and pitch?

I am trying to use a dataset. But I am facing two problems or confusions in understanding it. Can anbody guide me what 3D gaze direction stands for or means (angles, (x, y, z) coordinates or what)? ...
0
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0answers
16 views

What degree do I point a 1m horizontal stick to an evelation of 1140 meters.

I have a 1 meter horizontal yagi antenna at an evelation of 685 meters. How many degrees would I need to tilt it up so it points to the mobile tower at an evelation of 1140 meters 21.5 km away. ...
15
votes
4answers
1k views

The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
0
votes
0answers
15 views

Find angles between sides of triangle and coordinate planes (xy,yz,zx planes) using three 3d vectors .

Given the following, three vectors: a⃗ =3i−2j+5kb⃗ =i−6j+6kc⃗ =2i+3j−k, find the angles between sides of triangle and coordinate planes. I calculated the sides to be 4.58 , 11.45 and 7.87. I also ...
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votes
1answer
27 views

Two full-length [on hold]

A high school basketball player is $75$ inches tall. The basketball hoop is $10$ feet above the court. Find the distance in feet between the top of the players head and the basketball hoop.
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3answers
46 views

Is it possible to find the vertices of an equilateral triangle given its center point?

I was wondering how to find the vertices of an equilateral triangle given its center point? Such as: ...
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votes
0answers
23 views

Find base coordinates for oblique pyramid with rectangular base with known apex coordinates, unit direction vectors and base side lengths

Given an oblique pyramid with a rectangular base and the following known parameters: apex coordinates $X$ the unit direction vectors $\hat{XA},\hat{XB},\hat{XC},\hat{XD}$, which go from $X$ in the ...
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votes
1answer
26 views

Find the equation of the ellipse [on hold]

This question is hard I really appreciate if anyone can solve it. Find the equation of the ellipse that has $e=1/2$, center in (0,0) and passes through $P(9/2,3)$.
2
votes
1answer
19 views

Number of components needed for 3D rotation

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real ...
0
votes
2answers
54 views

Determine the locus of a equation Quickly[Mental Math]

if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my ...
1
vote
0answers
18 views

Camera calibration: how does checkerboard size/numbers/placement affect accuracy

I am trying to calibrate a camera using a checkerboard by the well known Zhang's method followed by bundle adjustment, which is available in both Matlab and OpenCV. There are a lot of empirical ...
3
votes
1answer
36 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
0
votes
1answer
13 views

Determine Orthogonal and non orthogonal using Coordinates

Can we identify using coordinates that if Polygon is orthogonal or non orthogonal. data = [(100, 100), (100, 200), (300, 200), (600, 400), (1150, 400), (1150,300), (600,300), (300,100)](These ...
1
vote
1answer
27 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
0
votes
1answer
21 views

Proper definition of concyclic?

Let $z_1,z_2...,z_n$ be points in the complex plane, then if there exists $Z$ such that $$\vert Z-z_k\vert=a\in\{\text{Real Numbers}\}$$for all $k\in \{1, 2, 3...,n\}$, then $z_1,z_2...,z_n$ are ...