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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1answer
6 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
25 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
11 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 ...
0
votes
0answers
16 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
12 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
26 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
0answers
14 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 ...
-5
votes
0answers
22 views

tessellation of an arbitrary shape [on hold]

Are there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate it by some shapes?
3
votes
0answers
34 views

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. ...
2
votes
2answers
23 views

Find the equation of base of Isosceles Traingle

Given the two Legs $AB$ and $AC$ of an Isosceles Traingle as $7x-y=3$ and $x-y+3=0$ Respectively. if area of $\Delta ABC$ is $5$ Square units, Find the Equation of the base $BC$ My Try: The ...
0
votes
2answers
30 views

Find perimeter and angle of triangle using three 3d vectors .

Given the following, three vectors: $$\vec{a} = 3\mathrm{i} - 2\mathrm{j} + 5\mathrm{k}\\\vec{b} = \mathrm{i} - 6\mathrm{j} + 6\mathrm{k}\\\vec{c} = 2\mathrm{i} + 3\mathrm{j} - \mathrm{k},\\$$ find ...
1
vote
1answer
16 views

Finding point where angular bisector meets circumcircle in complex plane

If $A(z_1)$,$b(z_2)$ and $C(z_3)$ are vertices of a triangle. It is inscribed in circle |z|=2. If internal angular bisector of A meets the circumcircle at $D(z_4)$. Find $z_4$ interms of $z_1$,$z_2$ ...
0
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2answers
28 views

Co-ordinate geometry and area of triangle

When a straight line $ax+by+c=0$ forms a triangle with the axes $x$ and $y$, what is the general formula for the area of the triangle?
8
votes
3answers
1k views

N gunmen in a field

Let n be an odd integer. In some field, n gunmen are placed such that all pairwise distances between them are different. At a signal, every gunman takes out his gun and shoots the closest gunman. ...
-1
votes
1answer
22 views

The limit incentral triangle is equilateral

I found a nice problem of geometry but I don't know how to prove it. Given a triangle $T_0$, we build $T_1$ by considering the projections of the incenter of $T_0$ on the sides of $T_0$. In the ...
2
votes
0answers
20 views

What shapes fit evenly inside a Hexagon

So I'm designing a board game that uses a number of adjacent hexagonal boards. These boards need to be divided up into spaces (tiles) that players move through. I've been playing with using Hexagons ...
1
vote
1answer
51 views

What geometric object is given by this equation?

What geometric object is given by this equation? $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$ Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the ...
0
votes
1answer
12 views

Delaunay Triangulation on Convex Polytopes — Uniform Sampling

My goal is to uniformly sample from a convex polytope. I know that for the simpler case, where I have to uniformly sample from a simplex, I can use Bayesian Bootstrap, discussed in these posts: ...
0
votes
1answer
50 views

Which slope is greater: one going very slightly downward, or one going quite steeply downward? [on hold]

I've been arguing back and forth with someone about this for a while now. It seems to me like a number closer to positive infinity is considered greater, no matter if you are talking about slopes or ...
3
votes
1answer
26 views

spirals around cone

I have multiple spirals running around a cone. The spirals are $$r_\Delta = r_b - r_t$$ $$x(z) = r_b \cos(z) - r_\Delta z \cos(z)$$ $$y(z) = r_b \sin(z) - r_\Delta z \sin(z)$$ $$d(z) = ...
0
votes
0answers
16 views

To circumscribe a square about a given circle.

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a ...
-1
votes
0answers
25 views

Area of a circle using Area =(D x 0.8862268332656149)Squared

I started off for fun to work out the area of a circle by using a percentage of its diameter and converting this into a decimal. Area =(D x 0.8862268332656149)Squared. then I found that if I squared ...
0
votes
3answers
67 views

Prove sum of $\sin$ of angles is greater than $\sin$ of sum of angles

It seems that $\displaystyle \sum_{x_i \in X} \sin\left(x_i\right) \geq \sin\left(\sum_{x_i \in X} x_i\right)$ where $X$ is a set of angles where $\displaystyle \sum_{x_i \in X} x_i \leq \pi$ radians ...
-3
votes
0answers
34 views

what is the distance [on hold]

I have a hill, I want to build a three piece retaining wall. One wall at the bottom, one wall half way up and one wall cresting the top. The hill is 10 foot high and 10 foot wide. What is the distance ...
1
vote
5answers
49 views

How to determine if 2 line segments cross?

Give two line segments, each defined by $2$ points in $x,y$ space, such as $L_1 = (x_1,y_1)-(x_2,y_2)$ and $L_2 = (x_3-y_3)-(x_4,y_4)$, and that these points are the result of sampled data (they are ...
1
vote
3answers
180 views

Is a ball noncompact?

A compact manifold usually refers to "a manifold without a boundary", for example the usual 2-sphere $S^2$. What about a manifold with a boundary? Intuitively, I think such an example, e.g. a ball ...
0
votes
1answer
17 views

Intersection of a cone and a plane

In $\mathbb{R}^3$, given the cone $K$ and the plane $E_c$ with the equations $4x^2=y^2+z^2$ and $z=c(1-x)$. How do I find out which different geometric objects I get for all $c\geq 0$ if I intersect ...
0
votes
1answer
47 views

Minimal area of triangle

We have the points $A(2, 3-m), B(m+2, -1)$ and $C(m, 2-m)$. Where $m$ is a real number. Find $m$ for which the area of triangle $ABC$ is minimal. So I've tried to find the equation of line $BC$(the ...
-1
votes
0answers
32 views

Point in the circle [on hold]

I have a circle with center $(x, y, z)$. Circle lies in plane which normal also known $(n_1, n_2, n_3)$. How calculate points on circle for given angle?
0
votes
2answers
22 views

Intersection of perpendicular and plane

I have a point $(x,y,z)$ and normal to plane with normal vector $(n_1, n_2, n_3)$. How can I calculate the intersection point of perpendicular from point to that plane? Thank You!
0
votes
1answer
29 views

Determine the lines of curvature of $z=xy$

I have to find the lines of curvature of $z=xy$ I calculate Weingarten Matrix as described below $p_u = (1, 0, v),p_v=(0,1,u),\nu =\frac{1}{\sqrt{1+u^2+v^2}}(-v,-u,1)$ so, $E=1+v^2,F=uv,G=1+u^2$ ...
0
votes
2answers
40 views

How do I solve using the line equation through (2,3) and (4,1)?

My teacher just taught me Coordinate Geometry and she is very fast :3 I cant really cope with the phase and i got loads of homework now. One of the question is "Find the equation of the straight ...
2
votes
0answers
28 views

how to find the edges emanating from a given vertex in a polyhedron

Suppose my polyhedron $P$ is defined as $P={ x\in \mathbb{R}^n \mid Ax=b, x\geq0 }$ I have $x_0$, which is a vertex of $P$. How to find the edges emanating from $x_0$? In other words, I want to find ...
0
votes
1answer
22 views

Prove that the median of this isosceles trapezoid is tangent to both circles.

What the problem really asks me to prove is that a circle can be inscribed in $ABCD$, or that $AB + CD = AC + BD$ or $2AB = AC + BD$ or $AB = \frac{AC+BD}{2} = PK$. I can do that assuming that $PK$ ...
0
votes
0answers
23 views

initial height = 60“. There is 5 degree decline over 163.5”. What is ending height?

I'm building a roof for a structure and need to get the ending height correct. The initial height is 60". The adjacent length (the ground) is 163.5". The decline is 5 degrees. I have gotten the ...
0
votes
1answer
102 views

Angles in triangle proof

Let $ABC$ be an acute triangle. The lines $\iota_{1}$ and $\iota_{2}$ are perpendicular to $AB$ at the points $A$ and $B$, respectively. The perpendicular lines from the midpoint $M$ of $AB$ to the ...
1
vote
1answer
25 views

Prove that a circle can be inscribed in the quadrilateral $ABCD$.

Prove that a circle can be inscribed in the quadrilateral $ABCD$. The problem is reduced to prove that $AC + BD = AB + CD$ My attempt: $1)$ $\triangle OBD \sim \triangle OBD$, by SAS congruency ...
1
vote
0answers
22 views

Domain enclosed by a simple closed curve with infinity many symmetry axes must be disk?

Let $D\subset\mathbb{R}^2$ be a domain. Suppose that (1) the boundary $\partial D$ is a simple closed curve; (2) the domain $D$ has infinity many symmetry axes. The domain $D$ must be a disk? If ...
2
votes
0answers
15 views

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
13
votes
6answers
2k views

How is the area of a circle calculated using basic mathematics?

Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is $$ A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n $$ which $$ ...
1
vote
1answer
24 views

List all sets of points in a plane that are enclosed by circles with given radius

My problem is: Given N points in a plane and a number R, list/enumerate all subsets of points, where points in each subset are enclosed by a circle with radius of R. Two subsets $S_i$ and $S_j$ should ...
0
votes
1answer
53 views

About the vertices of a regular polygon in the plane having rational coordinates

I have to prove that, except in the case $n=4$, the vertices of a regular $n$-agon in the Euclidean plane cannot have all rational coordinates $(x,y)$. Some idea?
8
votes
2answers
133 views

What is happening in the picture

I came across the picture below through random means. What is being demonstrated? All I could think of is maybe the center of the triangle is moving back and forth between the focii of the ellipse, ...
0
votes
1answer
21 views

Mobius map problem [on hold]

In computer science, a neural network (NN) is a digital representation of a brain. It can have any number of numeric inputs, any number of numeric outputs, and can be trained to do pretty much ...
3
votes
1answer
44 views

How to find intersection of two hypotenuses

I am a web developer who is bad with mathematics. I have never needed some math/geometry formulas before. But now I realize it is needed for more advanced design tecniques. I decided to learn math but ...
2
votes
1answer
37 views

Slopes of curves from complex derivative [on hold]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
-5
votes
0answers
46 views

what's the volume e? [on hold]

I have below picture. There are two cones with an intersection in between. I want to calculate the common area "e". What other information is needed? are these enough to calculate the common area "e"? ...
1
vote
1answer
57 views

How to make an icosahedron from 20 tetrahedra?

To make an icosahedron out of Sierpinsky tetrahedrons is difficult because regular tetrahedra can't tile in space. The dihedral angle of a tetrahedron is ~70.53. So the first step would be to make ...
3
votes
2answers
60 views

Definite solution for the mean distance from an external point to the surface of a sphere.

Sphere, radius $E$, is centred at point $O$ $[0,0,0]$. External point $Q$ is at $[D,0,0]$. I can slice the sphere by making multiple planar cuts parallel to the $YZ$ plane to produce circular ...
0
votes
0answers
44 views

Packing problem for circles [on hold]

Can the packing problem for big circle in which it has circles with equal radius be true if: the number of circles is even and every 2 circles are symmetric via the center of the big circle,there is ...