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.
3,853 questions
6 votes
2 answers
478 views
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 ...
- 3,735
2 votes
4 answers
154 views
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 ...
- 123
0 votes
1 answer
83 views
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
0 votes
0 answers
25 views
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
2 votes
1 answer
267 views
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 ...
- 355
0 votes
0 answers
61 views
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
2 votes
3 answers
124 views
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
1 vote
0 answers
49 views
Volume of the cylinder surface
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 ...
- 31
0 votes
1 answer
55 views
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 ...
- 695
0 votes
1 answer
99 views
$6$th Angle Between $4$ vectors [closed]
If I am given $4$ unit vectors in $3D$ space, there are $6$ angles between them. If I have $5$ angles out of them, I think I should be able to find the $6$th angle(intuitively it sounds like that)(...
1 vote
0 answers
51 views
Condition for two tetrahedra to be related by inversion
Given two labeled tetrahedra $$ V_0V_1V_2V_3 \quad\text{and}\quad V_0'V_1'V_2'V_3', $$ define their opposite-edge pairs $$ (01,23),\quad (02,13),\quad (03,12). $$ Let $$ m_{ij}=\frac{|V_i'V_j'|}{|...
- 695
2 votes
0 answers
61 views
Volume of polytope spanned by vertex pairs $(A_i,A_i')$ is invariant under translation of diagonals $A_iA_i'$
2D case Let a convex quadrilateral $Q= \operatorname{conv}\{A_1,A_1',A_2,A_2'\}$ have vertex pairs $$(A_1,A_1'),\quad (A_2,A_2'),$$ and define the “diagonal vectors” $$d_i = \overrightarrow{A_iA_i'} = ...
- 695
1 vote
2 answers
86 views
Finding the curvilinear length of quadratic Bézier curves in 3D
A quadratic Bézier curve is defined by three points in 3D, $P_0$, $P_1$, and $P_2$. The equation for the Bézier curve is defined through the parameterization of $t$, which has the range $0\leq t\leq 1$...
- 13
0 votes
2 answers
98 views
Direction cosines of a line
Suppose let there be this condition: Direction cosines of the line joining $A(0,7,10)$ and $B(−1,6,6)$ are $x,y,z$. When we find out the value we get $(- 1/(3\sqrt{2}), - 1/(3\sqrt{2}), - 4/(3\sqrt{2})...
0 votes
2 answers
67 views
Linear dependency of 4 vectors in 3D space
Suppose we have 3 vectors $(a, b, c)$ in the x-y plane and a fourth $(d)$ in the y-z plane. If 4 vectors in 3D space are always linearly dependent, how do we express the fourth in terms of the other 3?...
