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

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4
votes
3answers
37 views

Check if a point is inside a rectangle (not knowing the coordinates, but knowing distances to vertices)

I have to solve the following problem: I have 4 points (A, B, C, D) which form a rectangle, but I do not know their coordinates. I have another point (X), I do not know its coordinates either, but I ...
6
votes
4answers
2k views

Prove concurrency of triangle altitudes with vector algebra?

I know how to do it in normal Euclid geometry, but is it possible to do it with vector algebra?
10
votes
6answers
258 views

Difficult Coordinate Geometry and Calculus Question

I was given this question by a friend and after working tirelessly on it I have not come up with anything substantial. I was hoping someone in the community could provide a pointer or possibly a ...
1
vote
1answer
52 views

Find the equation of a cylinder

Find the equation of the cylinder that has directrix the curve: $x(t)=t, y(t)=t^2/2, z(t)=0$ and the generatrix is parallel to the line $${x-1\over 1}={y+2\over 1}={z\over 3}$$ I would really ...
4
votes
1answer
244 views

What is the simplest 3D shape? [on hold]

What is the simplest 3D shape? I am asking with respect, particularly, to computers, which represent all shapes as triangles. It seems a single triangle apparently cannot be considered 3 dimensional ...
0
votes
0answers
23 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
2
votes
4answers
97 views

I call them squares. They called them arrays. What do they mean?

So I was in C++, and we had third graders come today to play our programs. Whilst the others just drilled them with problems, my game was subtract a square. It was fun watching them discover that ...
2
votes
1answer
19 views

Interior Angle Embedded in a Triangle Embedded in a Circle

With only knowing the angles of $B$, $C$, and $D$ (shown above), is it possible to find the interior angle $A$? And if so, how?
-2
votes
0answers
23 views

Almost independent vectors- Where do they live on a manifold [on hold]

I am new to this thing. I am having the next question : Almost independent vectors- Where do they live on a manifold? In a manifold with larger dimmension? Tnks!So don't be tuff with me cause I am ...
2
votes
1answer
31 views

n points can be equidistant from each other only in dimensions $\ge n-1$?

2 points are from equal distance to each other in dimensions 1,2,3,... 3 points can be equidistant from each other in 2,3,... dimensions 4 points can be equidistant from each other only in ...
1
vote
0answers
12 views

Hypersphere central angle

For a sphere, the relationship between steradian of a patch on the surface, and the central angle of the cone subtending that patch, is given by ...
3
votes
2answers
294 views

Calculate arc length of a logarithmic spiral between two points.

Its hard for me to put into words so please bear with me. Given a line of a certain length, how could I calculate the the arc length of a logarithmic spiral given that it intersects the line at two ...
0
votes
1answer
29 views

Where can I find a proof of this result?

Does anyone know where I can find a proof of the underlined statement? Newman states it without a proof, and I could see how he gets $\dfrac {\sigma}{n} + O\left(\dfrac{1}{n^{3/2}}\right)$. Any ...
1
vote
0answers
8 views

lattice : hypervolume and its associated hypersurface

Imagine I have a lattice with a point every one unit length in each direction. I'm interested in knowing how many points I have on the border (hypersurface) in any $D$ dimension. $1D$ for a length ...
0
votes
0answers
6 views

Mapping theorems in higher dimensions

The Poincaré mapping theorem states that given any two simply connected open domains of $\mathbb C$ that are neither empty nor equal to $\mathbb C$, there exists a unique conformal mapping from one to ...
-2
votes
2answers
32 views

looking for Radius /distance equation [on hold]

If I have a circle with the circumference of 40,000 km and I travel 100 km around, how far have I traveled along the radius? What is the equation for this? Many many thanks Phil
1
vote
0answers
48 views

Abc is a triangle

Abc is a triangle (drawing of the triangle with measurements up the side of each side) Make a full size drawing of triangle abc in the space below The line AB has been drawn for you. Leave in all ...
16
votes
1answer
86 views

Are there spaces that 'look the same' at every point, but are not homogeneous?

A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space ...
28
votes
5answers
2k views

I think I see mysterious lines inside triangles—how to prove their existence?

Lately I've been fooling around with points inside a triangle and the sum of their distances from all sides. This was when I noticed a weird behaviour: For each point I chose there always seemed to ...
0
votes
1answer
22 views

Reducible cubic surface are always singular.

I want to prove that Any reducible cubic surface are always singular. A possible way may be to take a look at the intersection of the irreducible components. But I don't know how. Thanks for any ...
1
vote
1answer
33 views

Prove that $x^2-y^2+xy-1=0$ is a ruled surface

I am studying for an analytic geometry, final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on ...
2
votes
0answers
33 views

Is there an easier way to show that a kite has perpendicular diagonals using scalar products?

I want to use scalar products to prove that a kite has perpendicular diagonals. My attempt : Let $a,b,c,d$ vectors with $a+b+c+d=0$ and $a^2=d^2$ and $b^2=c^2$ Then, we get ...
4
votes
2answers
168 views

Making cuts on a spiral so that all segments are of the same length

The issue we have is this: We have rolls of magnetic strip (about 2 cm in width) and they are rolled in a roll with about 30 windings and about 10m length. The rolls are about 2cm in width as i said. ...
0
votes
1answer
18 views

finding volume of an n-dimensional pyramid numerically

In my experiment I need to compute hypervolume/area from a set of points, let's start with a base case -- Triangle: In this case, I have 3 points in a 2D space and they make a triangle, $p_1 = ...
2
votes
1answer
45 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
-1
votes
1answer
23 views

Two circular tangents

The total area of both circles is $230$ $m^2$, i need to find the radius of each circle. The circles are externally tangential and the distance from their centers are $11$$m$. Unable to upload ...
1
vote
0answers
15 views

How to visualize orientation of 3d objects

The way I visualize orientations of $1$- and $2$-dimensional objects is by an ant walking along a path. For a $1$d object (like a line/ line segment/ etc), just place the ant on the line and confine ...
1
vote
1answer
750 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
1
vote
2answers
86 views

Edge length of a Dodecahedron

Good morning, If I have a $12$ sided regular pentagonal structure - that is, a Dodecahedron - and the widest point is $3.5m$ in diameter, what is the length of an edge (if they are all the same). ...
3
votes
4answers
75 views

Why do disks on planes grow more quickly with radius than disks on spheres?

In the book, Mr. Tompkins in Wonderland, there is written something like this: On a sphere the area within a given radius grows more slowly with the radius than on a plane. Could you explain ...
4
votes
1answer
1k views

Can you make an equilateral triangle from 3identical trapezoids?

Is it possible to make an equilateral triangle from 3 identical trapezoids? If so, what angles would be needed in the trapezoids?
0
votes
1answer
17 views

Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
0
votes
0answers
28 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
1
vote
1answer
27 views

Lines on a quadric surface

I want to show that: Let $Q$ be a quadric surface of rank 3 in $\mathbf{P}^3$(over $\mathbf{C}$), then any line in $Q$ must pass through the only singular point of $Q$. I know that up to a transform ...
2
votes
2answers
59 views
+50

What will be the equation of side $BC$.

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively . What will be the equation of the side $BC$ if the area of the triangle ...
2
votes
1answer
17 views

Decomposition of hyper-rectangles into congruent simplices

Let $(a_1, \ldots, a_d) \in \mathbb{N}_+^d$ be positive integers and the semi-axes of the $d$-dimensional $\ell_1$-ellipse $$ E_{\bf a} := \{{\bf x} \in \mathbb{R}_{\geq 0}^d: \sum_{j=1}^d ...
3
votes
2answers
24 views

Find the number of possible points $R$.

$P(3,1),Q(6,5)$ and $R(x,y)$ are three points such that the angle $\angle PRQ=90^{\circ}$ and the area of the triangle $\triangle PRQ=7$.The number of such points $R$ that are possible is . $a.)\ ...
1
vote
3answers
52 views

A point D in a triangle ABC such that $\angle DAB= \angle DBC= \angle DCA$

I got this question from a student of mine, who is participating in a math olympiad competition: How can we construct a point D in a triangle ABC such that $\angle DAB= \angle DBC= \angle DCA$? I've ...
-1
votes
2answers
29 views

Construct a regular hexagon of specific height?

Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are ...
2
votes
4answers
41 views

Geometric progression, how to find $x$

I tried many ways to solve this problem, but I can't! Please, someone explain how to solve this problem: If the sequence: $8, x, 50$ is a geometric progression, then $x = ?$
1
vote
1answer
9 views

How to find point on sphere from pitch and heading

I have a sphere of radius R and I would like to draw some vector positions on it given pitch and heading. I have a heading between 0 and 360 (0 being +x direction), and a pitch between -90 and 90 (90 ...
0
votes
1answer
661 views

Geometry Find the Radius of a circumcircle given the area of the triangle

Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle. I also know that to find the radius I ...
2
votes
2answers
45 views

Find h in terms of r

A sphere and a cylinder have equal volumes. The sphere has a radius 3r. The cylinder has radius 2r and height h. Find h in terms of r. I'm only 15, someone walk me through this as simple as ...
1
vote
3answers
30 views

How to find the radius if two circles intersect in two distinct points?

Question- if two circles $(x-1)^2+(y-3)^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points , then find the range in which r exists I have these two circles $(x-1)^2+(y-3)^2=r^2$ and ...
6
votes
4answers
289 views

Coordinates of the center of the circle

I am stuck on this problem: If the lines $y=x+\sqrt{2}$ and $y=x-2\sqrt{2}$ are two tangents of a circle and $(0,\sqrt{2})$ lies on this circle then what is the equation of the circle? I ...
2
votes
1answer
35 views

Is there an equidissection of a unit square involving irrational coordinates?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky's theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. ...
0
votes
0answers
5 views

Question regarding Calibration while using Phase Measuring Profilometry (PMP)

We are using PMP to create the 3d model of a real world object in a summer project. However, to actually use PMP we need to relate the camera and the projector parameters and coordinates. To ...
-1
votes
1answer
20 views

Standard integral Kähler form on $\mathbf{CP}^1 \times \mathbf{CP}^2$

$\newcommand{\Proj}{\mathbf{CP}}$What is the "standard integral Kähler form" on $\Proj^1 \times \Proj^2$? Does that mean Fubini-Study form on $\Proj^1$ and $\Proj^2$?
1
vote
2answers
17 views

How to find the area of the triangle formed by the lines $y=ax$ , $x+y-a=0$ and the $y$ axis?

I found out the intersection points $A( \frac{a}{1+a}$ , $\frac{a^2}{1+a} )$ and $B$ as $(0,a)$. Now, I don't know what to do next. Please explain in easy steps. Thank you!
14
votes
2answers
436 views

cutting a cake without destroying the toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake. I want to ...