Questions tagged [geometry]

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

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23 views

Finding the shortest path through $N$ points [closed]

Say you have an $xy$ grid, and you start at a randomly generated initial position $x_i, y_i$. There are $n$ points on the grid that are also randomly generated. You need to construct the shortest path ...
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Distances between Countably Many Points in the Plane

If one has a set of countably many points in the plane $\mathbb{R}^2$, is it necessarily true that the distance between any pair of these points needs to be finite? I see a similar question someone ...
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1answer
27 views

Prove this point is on the circumcircle

I have a triangle $DCE$ with $DC>DE$. The exterior angle bisector of $\angle CDE$ meets the perpendicular bisector of $CE$ at $H$. How do I prove that $H$ always lies on the circumcircle of $CDE$?
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Given a circle, is there a relationship between the cycloid it generates and its circumscribed square?

This question is motivated by an answer I provided to a question here on the arc length of a cycloid. I noticed that the ratio of the circumference of the generating circle (which is also the ...
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Show that the kernel of a bundle morphism with constant rank is a sub-bundle.

I'm trying to solve the following problem: Let $M$ be a smooth manifold and $\pi: E \to X$ and $\pi ':F \to X$ be smooth vector bundles of $X$. Suppose we have a (smooth) bundle morphism $f:E\to F$ of ...
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Traingle inequality with a point Lying inside the triangle [closed]

I used AM and GM inequality here but how to move further, I'm not getting any idea about it
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11 views

Test if Line “Outside” of Concave Polygon

I'm trying to find an efficient test/algorithm that can determine if any line segment between two vertices in a simple, concave polygon contains points that lie outside the domain of the polygon. I ...
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1answer
28 views

Vector analysis on stokes theorem [closed]

How can I approach this kinda problem .Verify stokes theorem for $\vec{F} = z i + x j + y k$ . Curve is the one quadrant of the hemisphere $x^2 + y^2 + z^2 = 1$ .
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Scalar functions - Gram-Schmidt Orthogonalisation

I'm reading Chapter 11 (Normal Modes) of Classical Mechanics (5th ed.) by Berkshire and Kibble and came across this on pg. 253: The kinetic energy in terms in terms of the generalised coordinates is ...
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1answer
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Do there exist uniform triangular prisms with all vertices in $\mathbb Z^3$?

It's quite easy to find a regular square prism (cube) or a regular triangular antiprism (octahedron) with vertices in $\mathbb Z^3$. Take for instance, take the convex hulls $$ \begin{align*} &\...
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1answer
40 views

Given a point on a circle, find it on a given square

Math is not my forte so I apologize. I have a seemingly simple problem and I'm unable to figure out the formula for this. I'd describe the problem as given the center point of a circle and a point on ...
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Find how many dm3 water are in well. Cylinder challenge and logic problem surface and volume. [closed]

In the well with a depth of 15m and a diameter of 1.8m, the water reaches up to 10m in height. a) how many cubic meters of soil have been extracted from the well. b) How many liters of water does ...
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1answer
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Drawing vectors in 3dimension

The vectors $\vec{a}, \vec{b}, \vec{c}$ are as follows: with $a=2\ cm, \ b=2.5\ cm, \ c=3\ cm$. Construct the vector (a) $\vec{u}=2\vec{a}+\vec{b}-3\vec{c}$ (b) $\vec{v}$ such that $3\vec{a}-2\vec{b}+...
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4answers
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Prove that the area of a right triangle is no more than the square of the hypotenuse divided by 4

I'm struggling with this proof. Let $a,b,c$ denote the base, height, and hypotenuse of a right triangle. Let $A$ denote area of this triangle We have the following relations: $$ a^2 + b^2 = c^2 \\ A =...
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1answer
22 views

Contruction of vectors

Draw the vectors $\vec{a}$ and $\vec{b}$ with $|\vec{a}|=a=2\ cm$ and $|\vec{b}|=b=3\ cm$, these have angle $40^{\circ}$ between them. Contruct the vectors: (a) $\vec{c}=2\vec{a}-\vec{b}$ (b) $\vec{d}=...
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1answer
28 views

How to prove $C'$ is reflection of $C$ when we know $\triangle ABC\cong \triangle ABC'$?

We know $\triangle ABC\cong \triangle ABC'$ (the triangles are congruent). We may assume $C\neq C'$. How do I prove $C'$ is the reflection of $C$ in the line $AB$? I know I first need to prove the ...
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Mathematics behind simultaneous 3d rotations

I'm trying to simulate the effect of two simultaneous rotations acting on a sphere, more specifically: rotation and precession of the earth. These two rotations act on different rates and different ...
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1answer
24 views

Quaternion and Euler angle conversions: $f(q + \delta q) = f(q) +$ what? [closed]

Say we have function $f$ that converts quaternion to Euler angle ($f(q)= \theta)$. I want to know function p where: $f(q + \Delta q) = f(q) + p(\Delta q)$ where $\Delta q$ is a quaternion difference. ...
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1answer
33 views

Show that the curve is contained in a plane.

If all osculating planes of a regular curve pass through one point, show that the curve is contained in a plane. The osculating plane equation is as follows: $$\begin{vmatrix} X-x & Y-y & Z-z ...
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Shortest and longest paths between polygons

first of all I have to say, I am not a mathematician, I am software developer. Right now I am working on the app for pilots. Part of the app is creation of the navigation task. The task is defined by ...
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Analytical description of rotation operator

Let $\delta$ be the rotation with center $(1,2)$, that maps the point $(-1,3)$ to the point $(3,3)$. (a) Describe $\delta$ analytically and determine the rotation angle. (b) Let $g$ be the line with ...
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1answer
53 views

Possible length of median and height

In a triangle $ABC$ let $D$ be the intersection point of the side $AB$ with the angle bisector of the inner angle at $C$. It holds that $|AC|=4$, $|CD|=3$. Let the inner angle at $A$ be equal to $\...
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The use of modern geometries in classical Euclidean geometry

My question is whether any problem inherent in classical Euclidean geometry (plane geometry) can be solved by some modern geometry, such as differential geometry or if there are problems that only ...
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1answer
23 views

I need a formula for the volume of an “irregular antiprism” [closed]

I need help with finding a suitable formula that involves the VERTICAL HEIGHT (I mean the height from the centre of the pentagonal base straight up until it reaches the centre of the top pentagonal ...
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The Question with full description is given below and find the size of angle alpha [closed]

Find the size of the angle alpha. I tried many ways but cant find a proper method
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5answers
52 views

How to find range of values for where line cuts circle? [closed]

I am doing a question on circles. The question is: The circle has equation $x^2 + y^2 - 6x + 10y + 9 = 0 $ The line with equation $y= kx$ , where is a constant, cuts at two distinct points. Find ...
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13 views

Addition formulas for direction cosines?

I will quickly explain what I am expecting from this question. Given a point $p=(x,y)$ in the plane, we can look at the ratios $$A_1(p)=\dfrac{x}{r},\qquad A_2(p)=\dfrac{y}{r},$$ where $r$ is the ...
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1answer
55 views

How to prove that the line outside a convex polygon, having the minimum sum of distances is one of the edges?

As part of a computational geometry question, I need a result for an intermediate step. Suppose there is a convex polygon S, with finitely many points inside the polygon. Now, we want to find the line ...
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1answer
16 views

Given a plane, is there a vector that points towards the highest dz when moving by dx and dy? What is that vector called?

The question came to my mind when trying to explain snowboarding up a ramp. When you don't use an edge and the snowboard is flat, it is only stable if your momentum is going straight up the ramp. You ...
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1answer
22 views

Is the number of enclosed areas always equal to number of intersections plus one?

For this shape, the number of intersections is 4, and the number of enclosed areas/regions is five. For this shape, the number of intersections is 6, and the number of enclosed areas is 7. It seems ...
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1answer
13 views

Can segment sphere intersection be view as point cylinder intersection

If we have a sphere with origin at $p$ and radius $r$, and a segment with endpoints $a$ and $b$. And we want to check if they intersect. Would it be correct to assume that we could also view this as ...
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34 views

Explanation on derivation of angle from isosceles triangle

Problem and diagram: static system Problematic approach: "To write the balances, we will also need to determine the angle which the stick makes to the horizontal line, which is $\phi = \frac{β−α}{...
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1answer
18 views

Computing the parametric equation of the line of intersection of two planes

I'm given the following problem: Find the parametric equation of the line of intersection of the two given planes: $y-6z = 7$ and $9x - 8y = 5$ My attempted solution follows: I take the cross ...
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2answers
27 views

How to calculate this angle from 2 points in 3d space?

How do I find the following angle $a$ given $2$ points $(x, y, z)$ in $3$-dimensional space? I've drawn $2$ points, one in green, one in red. The curved black line being the earth, and the normal ...
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1answer
29 views

How to find set of points to satisfy a vector paralel to a known vector?

I am given two points: $C(4,-2,3)$ and $U(1,0,-2)$. How do I find the set of points $P(x,y,z)$, knowing that $\vec{UC} \cdot \vec UP=0$? I know that $\vec{UC}$ and $\vec{UP}$ should be parallel, but ...
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1answer
37 views

Circle of Apollonius; proving a tangent line

Quick recap, since there are several circles of Apollonius: Given a triangle with fixed base and the other two sides in known ratio, the circle of Apollonius (of the first type) gives the possible ...
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28 views

Intersection of a sphere and cylinder (Viviani´s Curve)

Let $-1<a<1$ and $C_a$ be the curve given by the intersection of the sphere of center $\textbf{0}$ and radius 2 and cylindre at a distance $(1-\cos(\pi a))$ from $\textbf{0}$ and of radius $(1+\...
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Finding values to make this figure work

I want to construct a quadrilateral with integer side lengths and with transcendental angles. BUT three of the angles should be integer multiples of an arc cosine transcendental number. So it would ...
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1answer
24 views

Is the image of a curve under the projection morphism of a fiber product again a curve?

If $X$ is a variety defined over the rationals and $f:X_\mathbb{C}\to X$ is the projection map of the fibered product, then is $f(C)$ is a curve in $X$? (Where $C$ is a curve in $X_\mathbb{C}$.)
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1answer
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Parametrizing straight lines intersecting a circle

I've got a point $ P (-1,1,2) $ and the circle $ C: ~ (x-1)^2+(y-1)^2 = \dfrac {25} {4} $ (which is contained in the plane: $ z=0 $). I need to write the parametric equations of all straight lines ...
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1answer
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Find a locus of a centroid point of a triangle.

When doing another geometry proof, I have this problem: Let $A$ is the point outside the circle $(O;R)$ such that $OA=2R$. The line $d$ goes through $A$ and cut the circle at $2$ distinct points $E$ ...
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2answers
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Intersection of two vectors with tails?

I'm working on a 2 degree of freedom simulation of gas dynamics. Suppose you have two vectors in the x-y plane. Each vector has two tails, or coordinates where they start. Because I'm looking for a ...
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1answer
28 views

How should I find the four vertices of a rectangle if I have its center of gravity and it length and width?

basically all I want to know is in the question. I know that in a square if I have the length of it and its center of gravity I can find the vertices by this formula: If the coordinates of the center ...
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1answer
41 views

How to transform a line to circle curve.

I am facing a problem of the transformation of a line to a circle curve as shown in the attached figure. The line segment $A1Z1$ which has many points$(A1(x_A,y_A), B1(x_B,y_B),...P1(x_P,y_P),...)$ ...
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0answers
11 views

Orientation of Cylinder

Consider the a cylinder $M = S^1 \times [0,1]$ with counterclockwise orientation when viewed from the exterior. Let $C_0 = S^1 \times \{ 0 \}$ and $C_1 = S^1 \times \{ 1 \}$ be its boundary curves. I ...
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4answers
37 views

Conics Section: Equation of a circle

I have a question that states Find the equation of a circle with center $(2,-3)$ and passing through $(3, -5)$. I arrived at $x^2 + y^2 -4x + 6y - 5 = 0 $. It was marked wrong. The answer is said to ...
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1answer
29 views

How do I find the value of k to make the equations of a line and a plane be parallel?

Given the equation of the line $r$: $r:\frac{x-2}{k}=\frac{y-1}{2}=z$ And the equation of the plane $\alpha$: $\alpha:3x-ky-z-2=0$ How do I determine $k$ such that the line $r$ and the plane $\alpha$ ...
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0answers
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Is it true that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity is just a line represented by $\mathbf{v}$?

Is it true in $\mathbb{P}^3$, that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity $\mathbf{\pi_\infty}$ is just a line represented by $\mathbf{v}$? In the ...
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2answers
41 views

Is the Cardinality of all Lines in the Extended Euclidean Plane $\mathfrak{c}$?

Consider the surjection $[0,2\pi)*\left(\mathbb{R}^2\cup\ell_\infty\right)\rightarrow L$ (such that $L$ is the set of all lines determined by the binary operation $*$ between a point and an angle). ...
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2answers
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Path difference on a sphere due to geodesic deviation

Between two points of same latitude but different longitudes [spherical coordinates $(\theta,\phi)$] we measure two lengths of arcs $(s_1,s_2)$ along (1) the geodesic or red great circle (a graphic ...

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