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

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2
votes
1answer
49 views

The diameter of a convex hull.

I want to prove the following statement: Given $A\subset \mathbb{R}^n$ let $C(A)$ be its convex hull. Prove that $\text{diam }(A)=\text{diam }(C(A))$. I can suppose that $A$ is a bounded closed ...
2
votes
0answers
49 views

Rotation Matrix which maps a point to an specific point

How can I compute the rotation matrix which rotates an $n$-dimensional vector $\vec{A}$ around an $n-D$ vector $\vec{O}$, and maps it to a vector $\vec{B}$ (while $\vec{A}, \vec{B}, \vec{O}$ are known)...
2
votes
1answer
42 views

Largest enclosed (inscribed) circle in cloud of points

I have a set of points that approximately lie on a circle. I would like to compute the largest circle that does not contain any of the points. Of course, one could draw the circle far away from the ...
2
votes
2answers
46 views

How to sketch the region on the complex plane? [duplicate]

I am going through a basic course on complex analysis. I have a problem in understanding the following. E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...
0
votes
1answer
28 views

equation of tangent plane to sphere, given 2 points lying on plane?

i have sphere $x^2+y^2+z^2=r^2$ and a vector with points $P=(x_1,y_1,z_1)$ and $Q=(x_2,y_2,z_2)$, i need equation of tangent plane where above two points lyes and touches the sphere in one point only?
1
vote
1answer
21 views

Finding axis of a cylinder

I have to find axis of a cylinder that has the top in the origin and the points $A(-5,6,-4),B(-4,-1,2),C(-1,2,4)$ lie on its lateral area. Now I know that points A,B,C have the same distance to the ...
2
votes
1answer
32 views

Prove a specific property for tetrahedron

I have the following question. If the heights from vertices $A$ and $D$ in tetrahedron $ABCD$ intersect then $AD$ and $BC$ are perpendicular. I draw a sketch of the tetrahedron but I don't have any ...
1
vote
0answers
33 views

Is one diagonal of a convex quadrilateral always longer than at least 3 sides?

I am solving a problem which requires a proof of the statement in the title. So far, I was considering the following cases: 4 angles are at least 90°: rectangle, the statement holds. 3 angles are at ...
39
votes
0answers
4k views

What is Ptolemy holding in his picture on Wikipedia? [migrated]

I would like to know the name of the device Ptolemy is holding in his picture
0
votes
3answers
61 views

If $a^2 + b^2 = c^2$, then $a^3 + b^3 < c^3$, for $a$, $b$, $c$ the sides of a triangle

If $a$, $b$, $c$ are the sides of a triangle where $a^2 + b^2 = c^2$, prove that $a^3 + b^3 < c^3$. I've tried triangle inequality, but I am stuck.
0
votes
1answer
22 views

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum?

Given points P (2,3),Q (4,-2),R (a,0) what should be the value of a if |PR-RQ| Is maximum ? I tried that maybe the points are collinear but I'm getting wrong answer applying collinearity condition i....
1
vote
1answer
35 views

graphing a circle in the complex plane? [on hold]

The ellipse seemed rather simple: Defining the equation of an ellipse in the complex plane But Wolfram won't graph it with equal axes. http://www.wolframalpha.com/input/?i=abs{%28x%2Biy%29}%2Babs{%...
2
votes
2answers
94 views

Surface described by the equation $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$

Given the equation : $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$. Check if the surface described by that equation has a center of symmetry and then by making the correct coordinate system change, find ...
2
votes
1answer
19 views

Non-trivial 3D curve that projects as a line or a segment onto the faces of the quadrant

I want to illustrate how high dimensional objects may have misleading projections. Examples are for instance given with HiSee software, with nD bouquets of circles. Are there non-trivial (not a 3D ...
1
vote
1answer
48 views

Is there an easier way to solve a “Find the locus” problem?

Note: I am not concerned with the accuracy of my solution so you don't need to redo any of my calculations. Original question: Suppose $ABC$ is an equilateral triangular lamina of side length unity, ...
1
vote
1answer
109 views

Largest rotated ellipse inscribed in a rectangle

Let's say I have a parametrized ellipse $$x (t) = a \cos(t) \cos(r) - b \sin(t) \sin(r)$$ $$y (t) = a \cos(t) \sin(r) + b \sin(t) \cos(r)$$ Where $r$ is the rotation around the axis and $t \in [0,2\...
2
votes
1answer
65 views

Is the angle between a and b is equal to the angle between b and a?

This was a question in an exam: Calcualte tan of the angle between a and b if:a = (4,3) and b = (5, -12) There are two answers to this question: Some students devided the dot product of a and b by ...
0
votes
0answers
27 views

$N$-dimensional volume (of revolution)

Consider the system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ and an n-dimensional shape such that, in $\{x_{1},x_{n}\}$ (and $x_{2}=x_{3}=...=x_{n-1}=0$) it is inside the lines $x_{n}=ax_{1}+b$ and $...
0
votes
0answers
20 views

What are symmetric, spacefilling volumes in arbitrary dimensions?

Do you know how to find many/all spacefilling volumes in arbitrary dimensions? The requirement is that they should be convex and invariant under coordinate permutation. Are there infinitely many of ...
0
votes
3answers
46 views

Determine if the following vectors are coplanar.

I have no idea to start with this question, I know how to find if vectors are coplanar when the values of the vectors are given to me, but I do not know how to manipulate coplanarity properties well ...
0
votes
2answers
86 views

Why weren't “degrees” replaced with a more intuitive angle measure?

$\bf History$ It is speculated that the seemingly arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$ and $60 \times 6 = 360$. ...
0
votes
2answers
51 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
0
votes
1answer
77 views

Collinearity of triangle vertices on circular paths

Suppose as a function of time, the vertices of a triangle move with constant speed along circular trajectories: $$ \vec p_i(t) = a_i \left( \begin{array}{c} \cos(b_i t+ c_i)\\ \sin(b_i t+ c_i) \end{...
0
votes
0answers
23 views

Family of (closed) parametric 2D-curves with bounded curvature

Let's consider the set of (closed) parametric 2D-curves $(x(s),y(s))$ such that the curvature and its derivative are bounded at any point, i.e., $|\kappa(s)|\leq b_1$, $|\kappa'(s)|\leq b_2$. Do you ...
1
vote
0answers
23 views

Small circles on sphere: finding angles for constant “cosine” onto a parallel.

My problem can be best explained starting from a 2D example: Imagine having a circle and wanting to discretize N points on the circumference of the circle so that the difference of the cosine of each ...
0
votes
2answers
33 views

extend a line in both way

I have a line segment, passing through the points $A = (100,100), B = (200, 200)$ and I would like to extend it by a certain length in both way. I can get the length of the current line segment by $$...
0
votes
0answers
10 views

Probability Distribution over successive circular arcs.

So I'm looking at a problem sketched out below: so here what happens is you have a particle which moves at a constant speed and has some current orientation. At each timestep it can change it's ...
0
votes
2answers
47 views

Finding Perimeter of Shape

"Two circles of radii 5cm and 12cm overlap so that the distance between their centers is 13cm. Find the perimeter of the shape." This question was from a chapter about circle measure under the length ...
8
votes
1answer
112 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
3
votes
2answers
124 views

If the sides of a triangle satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$, and if one angle is $48^\circ$, then find the other angles.

In triangle $ABC$ one angle of which is $48^{\circ}$, length of the sides satisfy the equality: $$(a-c)(a+c)^2+bc(a+c)=ab^2$$ Find the value in degrees the other two angles of the triangle. I ...
0
votes
0answers
18 views

Find the new axis length of an ellipsoid that has changed its volume by delta [on hold]

I have an ellipsoid with 3 semi-axis $a$ $b$ $c$. The volume of the ellipsoid will be $V^o=4/3\pi abc$. For some reason I have to change its volume by $\Delta V$ so the new volume will be $V^n=V^o+\...
0
votes
1answer
14 views

relation between tangent plane and rectifying plane

Tangent plane is perpendicular to normal. Rectifying plane is plane containing tangent and binormal and is perpendicular to normal. I want to know if both are same. I have not read anywhere the ...
0
votes
0answers
35 views

General equation of a cone

What is the general equation of a cone in $\mathbb{R}^3$ space? There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.
15
votes
5answers
459 views

Visualization of surface area of a sphere

I help mentor some really young, bright kids in mathematics. We were looking at geometric properties of various shapes, and one of the kids noted that the surface area of a sphere $S = 4\pi r^2$ ...
0
votes
0answers
24 views

How does inversion affect the angle subtended by a circular arc?

Say that I describe a circular arc $A\subseteq\mathbb{R}^2$ using an ordered triple $(p_1,p_2,c)$, where $p_1,p_2$ are the endpoints of the arc and $c$ is its center. (Technically this also describes ...
0
votes
0answers
25 views

Poncelet's closure theorem

Need some help understanding the proof made by Kneebone and Semple in "Algebraic Projective Geometry". I loose it in the sentence about the (2,2) correspondance. As I understand it, they setup an ...
0
votes
2answers
36 views

Find length of a side from given mesurements

Source: gradestack.com This is a problem I am trying to solve for a long time. But still not able to proceed. After spending some time, I got a doubt whether this question is correct. Because, in a ...
0
votes
1answer
23 views

Method to find out how distributed are a certain set of data?

Assume I have array of $A_{3\times120}$ Each row of matrix A corresponds to a shape which is generated by its three row elements as below: $r=1+a_1\cos(\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)$ ...
0
votes
0answers
23 views

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$ Let the plane touches the cone at $(\alpha,\beta,\gamma)$. We know that ...
0
votes
2answers
27 views

Distance between incenter and centroid

i have problem about solving this homework In any given triangle find the distance between the centroid and the incenter . I have no idea which properties to use to find it can you help me please :( ...
1
vote
0answers
19 views

Plane containing two points $a,b \in \mathbb{S}^3$ but avoiding two other antipodal points

Let $\mathbb{S}^3$ be the three-dimensional subset of $\mathbb{R}^4$ given by $$ \mathbb{S}^3 = \{ (x_1,x_2,x_3,x_4) | x_1^2 + x_2^2 +x_3^2 +x_4^2 = 1 \}.$$ I want to construct a plane through the ...
-1
votes
0answers
28 views

Area of cross section in the intersection of two cylinders

A Steinmetz solid "B" is formed on the intersection of two cylinders, given as: x^2 + z^2 = r^2 and y^2 + z^2 = r^2 Although, referring to Gardner's work, I have successfully calculated the total ...
-1
votes
0answers
11 views

Change in azimuth of target after I execute a roll [closed]

Given the azimuth of an object from an aircraft, x°, what would be the new azimuth of the object if the aircraft executes a roll of y°?
3
votes
1answer
35 views

Counter-clockwise angle between edges

I have a couple of connected lines (or rather edges) in the form of coordinates, that is for each edge a starting point $(x_s,y_s)$ and end point $(x_e,y_e)$. and I want to know a specific angle ...
2
votes
2answers
53 views

How to find the restrictions of side length on an obtuse triangle

Question: In Triangle ABC, the angle ∠ABC is an obtuse angle. The Side AB is 1cm, and the side BC is 3cm. Side AC is (3x+10)/(x+3) cm Find the restriction(s) on x. I have tried a few different ...
0
votes
0answers
22 views

Global holomorphic vector field on a two-sphere

I'm sure this question has been asked before... Till today, I thought that one cannot define a global holomorphic vector field on a two-sphere due to the hairy ball theorem. However, here's an ...
1
vote
1answer
24 views

Collinear Points in 3-Dimensions

The points A(3, -1, z), B(1, 2, 6), and C(x, 8, 14) are collinear. Find the values of x and z. I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling ...
4
votes
2answers
44 views

Area of a circle on sphere

On a (flat) Euclidean plane, the area of a circle with a radius $r$ can be described by the function $A(r) = \pi r^2.$ But how can one describe the area of the same circle on a spherical manifold? ...