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

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Jumps in a flow to the outermost area enclosure of a surface

While studying some geometrical properties of some flows of surfaces, I encountered this problem: I consider some surfaces $E_t$ flowing to infinity. I also define $E'_t$ to be the outermost minimal ...
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0answers
9 views

Elementary 3D geometry

This is surely trivial, but my old brain can't remember how to do it. Assume a plane. A second plane intersects, forming line $AB$. The angle of intersection is $\theta$. A third plane intersects, ...
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0answers
12 views

Construction of the Area Function

I am following calculus by Tom M Apostol in which he has given the Axiomatic definition of the Area Function We assume there exists a class M of measurable sets in the plane and a set function a, ...
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0answers
7 views

Variant Lemoine's problem

You can see Lemoine's problem: Kiepert triangle: Let $ABC$ be a triangle, $BCA_0$, $CAB_0$, $ABC_0$ be three isosceles triangles constructed on the sides of $ABC$ with base angle $\alpha$. We called ...
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2answers
44 views

Preplexity of Pi [on hold]

Why should we use the area of circle as $\pi$ multiplied by square of radius ? Can't we use another formula to get the specific answer? We know that $\pi$ does not equals to $\frac{22}{7}$ or ...
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2answers
42 views

Show that among all quadrilaterals of a given perimeter the square has the largest area

Show that among all quadrilaterals of a given perimeter the square has the largest area. By Ptolemy's theorem we have that if $a,b,c,d$ are the side lengths of the quadrilateral then $ac+bd \geq ...
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1answer
31 views

Rotating a sphere

I'm trying to rotate a sphere, and I'm having a bit of a problem calculating the angle to rotate it by. I wonder if anyone can help me? On my sphere I've marked three points. If the centre of the ...
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0answers
11 views

Calculate the Angle between two vectors in 3d Spherical Coordinates

I have two vectors in spherical coordinates, both originating at the origin and both with the same magnitude equal to one. One is vertical: {1,0,0} and the other undefined: {Ms,Mt,Mp}. The other one ...
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0answers
32 views

what is relation between topology and geometry? [on hold]

what is relation between general topology and geometry ?( and example in this relation .) is there simple book in relation between general topology and geometry ?
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1answer
14 views

Length of one internal tangent with two external tangents

Two circles with radii 25 and 16 are tangent externally. The two common external tangents intersect the larger circle at A and B and the smaller circle at C and D. The common internal tangent ...
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0answers
32 views

Find the volume of the following room [1] [on hold]

I was working on a project which required me to calculate the volume of the room. The picture of the room is given below: I tried splitting the shape across the diagonals but each time end up ...
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0answers
9 views

Mathematical theory for equally distributed dipole structures with inner equilibration

I'm looking for a mathematical theory for equally distributed dipole structures with inner equilibration. I know, that there exist two magnetic clusters, where the north and the south poles equally ...
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0answers
37 views

is the followng function $f$ surjective?

$f$ is a function mapping $x$ axis to plane $V$ defined by if $P(x,0)$ then $f(P) = (x,x^2)$ ? I am not so sure for my following answer : to show that $f$ is surjective, for every $A(x,x^2)$ there ...
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0answers
17 views

Torus Cylinder intersection [on hold]

Two surfaces in 3D .. a Torus and a Cylinder with parametrizations respectively as: $$ \{ b + a \cos(u) ) \cos(v), (b + a \cos(u) \sin(v), a \sin(u )\}, $$ $$ \{a + b \cos(p), b \sin(p) ...
2
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0answers
33 views

what exactly is arc length element $ds$ or area element $dA$

I am reading a book on complex analysis and it has something like: The spherical arc length element on the Riemann sphere ($S^2$) works out to be $ds=\frac{|dz|}{1+|z|^2},$ and the spherical area ...
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1answer
12 views

3D Shape Name Recognition

I have 1: a right angled triangle, 2: an isosceles triangle, 3: a rectangle, 4: a parallelogram, and 5 a trapezium in the 2D picture which I am rotating about the red axis next to them. They produce ...
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1answer
46 views

How do I prove that sine is dependent only on the angle? [on hold]

I did the following: Taking one triangle and writing the pythagorean formula to it: $a^2+b^2=c^2$ and hence: $$\frac{o}{h}=\frac{\pm \sqrt{c^2-a^2}}{\pm \sqrt{a^2+b^2}}$$ I took another ...
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0answers
30 views

Given triangle w/ two congruent perpendicular bisectors, is it isosceles?

Given any triangle with two congruent perpendicular bisectors (the segment that is inside the triangle), must it be isosceles? Why? Trying to construct a non-isosceles one doesn't seem to work. I ...
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3answers
64 views

Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
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1answer
17 views

prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
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1answer
18 views

Conditions for convex hulls

What are the conditions for there to exist a convex hull on a set $X$ of points? I know that there exists a unique convex hull for a set of $X$ points, but must no $3$ points be collinear in ...
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1answer
34 views

How could I use the centroid and size of a triangle to find the coordinates of its vertices?

I'm making a 3D graphics program in OpenGL and I'm making a function to automatically place the centroid of a triangle on the specified xy coordinates. The triangle will be of a set size. Essentially ...
2
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1answer
26 views

Local isometry between simply connected manifolds

Suppose $D:\tilde{M}\rightarrow N$ is a local diffeomorphism between two simply connected smooth manifolds $\tilde{M}$ and $\tilde{N}$. $D$ is onto. In the case of $D$ being a covering map, it follows ...
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1answer
25 views

A proof in Desargues' geometry

The question is: Prove in Desargues' geometry that if a, b, c are three lines where a is parallel to both b and c then b and c intersect at the pole of a. Desargues' Geometry has the following ...
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2answers
51 views

minimising sum of distances

I have three points $A(-3.5, 0), B(2,0), C(0,3)$. I am looking for the fourth point $D(0,d)$ such that $AD + BD + CD$ is minimal. Fermat does not work here due to $D$ lying on the y-axis. I thought ...
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1answer
30 views

Centroid of heart shape

A heart shape is constructed using two identical circles with radius $r$. A line is drawn from point $T$ to $C$. There, two tangents are constructed, one to each circle, through $AC$ and $CB$. (The ...
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1answer
22 views

Size of a 3D object with relation to reference point [on hold]

I have a simple image of a table. I placed a reference ($10 \times 10$ rectangle) on top of it. I know the size of the rectangle and I want to calculate the size of the table. If I try simple ...
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3answers
40 views

The line $x+\sqrt{3} y-10=0$ makes an angle of $150$° with the positive sense of the $x$-axis. How can this be proven?

I cant figure out how this is correct. I know that $\tan(a)=m$ of a line but I cant figure this out. Could someone show how to prove the line makes an angle of $150$° with the positive $x$-axis? I ...
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0answers
22 views

tensor product of a line bundle with $\bigwedge^k T^*M$

I am reading a source that says: Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$. Can someone explain thoroughly what this means? What does it mean ...
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1answer
31 views

Equation of a curved line from a graph

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation. The second graph shows how it is used. The ...
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4answers
30 views

Problem based on circle geometry - related to circumcircles and angles finding the angles within a circle

Let the vertex of an angle $ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that the angle $ABC$ is equal to the half the ...
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1answer
20 views

Convert quaternions to xyz degrees

I knew quaternion for the first time a few days ago and I still don't get the way it works even when reading explanations. All I want to do is to make a subtraction between two quaternions and convert ...
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1answer
21 views

Maximum hyperrectangle

Is there a way to determine the coordinates of the maximum hyper-rectangle in n-D space subject to linear constraints and $x_i\ge0$ ? Example: Argument Maximum of $x_1 x_2 x_3$ Given ...
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0answers
8 views

Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
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1answer
44 views

Proving the volume of sphere by using tiny volumes

How can I prove the volume of sphere, by using many cones starting at the center of the sphere? It doesn't have to be cones, pyramids also work.
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0answers
28 views

Describe curves by words

I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth ...
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1answer
24 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$ [on hold]

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
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1answer
37 views

Two tangents BC and BD are drawn. Prove that Ob=2BC

Two tangent segments BC & BD are drawn to a circle with centre O such that $\angle$CBD=120$^{\circ}$. Prove that OB=2BC. What I've tried, BC=BD[two tangents drawn from a single point to the ...
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3answers
57 views

Prove a geometry related equation.

$\Delta ABC$ is an isosceles triangle. $\angle BAC$ is an right angle. $BC$ is its hypotenuse and $P$ is any point on $BC$. Prove that, $PB^2+PC^2=2\times PA^2$. I have tried it in many ways and ...
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1answer
22 views

ABCD is a square of side 4cm. E is a point in the interior of the square such that CED is equilateral. Then find the area of ACE in sq.centimeters.

The given answer is $4*(\sqrt{3} - 1)$ I tried all the methods but could not match the answer. Please tell if the question is wrong. Thanks in advance.
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4answers
49 views

Move a dot along a path

I have a multi-point, straight line path - to keep it simple it has three points, A B & C. A = 60,410 B = 127.5,410 C = 195,240 This is the 'template' path, I need to animate a dot moving ...
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2answers
18 views

Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
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2answers
56 views

Third Ailles Rectangle

The Ailles rectangle (named after an Ontario high school teacher, D. S. Ailles) is a rectangle of size $\sqrt{3}\times\sqrt{3}+1$ with three kind of triangles, like below. We have triangle ...
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1answer
57 views

Show that the line $KD$ bisects $\angle{EKF}$

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each ...
2
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1answer
24 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
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1answer
25 views

Is the interior of a simple polygon, simply-connected?

This may be trivial, but I want to be sure I understand correctly: Is it true that the interior of a simple polygon is always a simply-connected subset of the plane? I.e, is it eligible for the ...
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2answers
22 views

Algorithm to cover maximal number of points with one circle of given radius

we have a plane with some points on it. We know coordinate of each point apriori. We also have a circle of unit radius. I need an algorithm that determines optimal/sub-optimal position of a circle ...
0
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0answers
15 views

find all straight-lines that goes through a point $p$ in projective and affine space [on hold]

I have an exam tomorrow in projective Geometry and I am lacking of many basic skills. For instance this one: How can I find all straight lines that go through $p$ in the projective space $\Bbb ...
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1answer
21 views

How can I prove triangle ABC is congruent to triangle MaMbMc? [on hold]

Ma is a point set halfway on AB, Mb halfway on BC, Mc halfway on CA.
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0answers
53 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...