Questions tagged [3d]
For things related to 3 dimensions. For geometry of 3-dimensional solids, please use instead (solid-geometry). For non-planar geometry, but otherwise agnostic of dimensions, perhaps (euclidean-geometry) or (analytic-geometry) should also be considered.
3,856 questions
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A single-sided infinite surface, does that exist? [closed]
I just read a question about the surface of a sphere, and it just hit me:
The surface of a sphere is infinite: in every direction you choose, you can go on forever. On the other hand, the surface of ...
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Visualizing the volume of tetrahedral portion of a cube
Suppose I have a unit cube with one vertex at $(0,0,0)$, and I slice it via the plane with equation $x+y+z=1$. I know, algebraically, that this tetrahedral slice has volume $\frac16$, but I'm ...
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Finding rotations transform
Two vectors vec1 and vec2 were randomly rotated the same way about origin. The result of rotations are vectors ...
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generalizing space diagonals to all (or most) geometric solids
While writing about diagonals of shapes, I defined a space diagonal as a diagonal of a 3D shape connecting vertices that are not on the same face of the shape. I then realized that this definition ...
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Is this a known theorem?
Reference image ^^^
Edit: a person has answered this and I have rediscovered Euclid's Elements, Book 13, Proposition 15.
Ok so I think I might have found a new theorem or maybe rediscovered an old one....
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Rotating a unit vector to another vector using two consecutive axes
Given a unit vector $u$ and another unit vector $v$, I want to rotate $u$ into $v$ in two stages. In the first stage, I rotate $u$ about a given axis $a_1$ (by an unknown angle) to produce a vector $...
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Cutting a Möbius strip in thirds. Why are the resulting strips interlinked?
It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.
If instead, one begins ...
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How to rotate a 3d object drawn along the z axis so that it lies along some other arbitrary axis
There are several very similar questions to this one, and I have read them all, but they are all a generalized version of the problem, and full of Math language. I don't speak Math, so the answers to ...
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Bearing angle of great circular arc between Ottawa Canada, and Sarajevo, Bosnia
I have this problem that I have been working on today. I want to calculate the local direction of the great circle connecting Ottawa, Canada, and Sarajevo, Bosnia. I assume Earth is perfectly ...
user1709783
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Finding the minimum distance path on the surface of an ellipsoid [duplicate]
Suppose I have the ellipsoid
$$ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $$
And I have two points on this surface, $P_1 = (x_1, y_1, z_1)$, and $P_2= (x_2, y_2, z_2) $.
I am ...
user1709783
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Minimal possible area of a given union of polygons.
Consider $P$ to be the union of polygons inside a $3\rm{D}$ space. Find the minimal possible area of $P$ provided that the projection of $P$ onto the axis planes is a unit square.
This is a question ...
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Rotating a line in space to align it with another line
In my previous problem, I asked about rotating a plane into another plane.
In this question, I am given two lines in 3D space: $P_1(t) = r_1 + t v_1$ , $P_2(s) = r_2 + s v_2$. I am interested in ...
user1480757
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Rotating a plane into another plane
I am given two planes $n_1 \cdot (r - r_1) = 0 $ and $n_2 \cdot ( r - r_2 ) = 0 $ where $ r = (x, y, z), r_1 = (x_1, y_1, z_1) $ is a point on the first plane, and $r_2 = (x_2, y_2, z_2) $ is a point ...
user1480757
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Volume spanned by surface peeled from cylinder
Problem
In three-dimensional $xyz$-space, consider the cylindrical surface given by $x^2+y^2=1$, and let $S$ be its portion with $0\le z\le 2$.
A sheet of paper of negligible thickness is wrapped ...
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A formula for the diagonal of a skew quadrilateral
Let $i,j,k,m\in\mathbb R^3$. Write $\ell_{ab}=\|a-b\|$ for edge lengths, $A_{ijk}$ for the area of $\triangle ijk$, and let $\theta$ be the dihedral angle along edge $ij$ between the oriented ...
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