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
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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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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 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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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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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 ...
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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})...
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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
$$
m_{ij}=\frac{|V_i'V_j'|}{|...
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Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$. If P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum, find $a$ and $b$
Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$.
If point P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum,
then find the value of $a$ and $b$.
My Attempt:
If $P$ lies ...
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Parametrization of the space curve formed by intersection of two cylinders
My answer includes a 3d visualization of :
Intersecting circular/parabolic cylinders
Please help find 3d curve parametrization w.r.t. a single parameter $t$.
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Three dimensional geometry
This question was in my Oxford Scholarsip paper 56 years ago which I recently revisited:
Two parallel planes, $p$ and $q$, are at a distance a apart and the line $PQ$, perpendicular to them, meets ...
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Why does a line in 3D space (let's call it a 3D line), have 4 degrees of Freedom? [closed]
I’ve seen a lot of posts and read a few blogs about the DOF for a 3D line, but I still don’t quite get it. I understand that a point in 3D space requires 3 DOF ...
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Stereometry question regarding the insphere of the tetrahedron
The insphere of a tetrahedron $ABCD$ touches the faces $ABC, BCD,
CDA, DAB$ at $D′, A′, B′, C′$ respectively. Denote by $SAB$ the area of the triangle
$AC′B$. Define similarly $SAC, SBC, SAD, SBD, ...
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How to get an infinitely differentiable curve given a set of points
When working in $2D$, I can have a polynomial function
$$P(t) = c_0 t^0 + c_1 t^1 +c_2 t^2 + ... + c_n t^n$$
Which is infinitely differentiable and I can use it to fit a set of points $(t_0,x_0), ...,...
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Where’s the mistake? Find the side of the base of a regular triangular pyramid
In a regular triangular pyramid, the side edge is $5$ and the tangent of the angle between the side face and the plane of the base is $\frac{\sqrt{11}}{4}$. Find the side of the base of the pyramid.
...
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3D geometry : Direction ratio of diagonal of rhombus
If we are given the coordinates of extremities of one of the diagonal of a rhombus , is it possible to find the DR's (Direction Ratio) of the other diagonal?
I know that the diagonals are ...
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Given any set of $3$D points, can we always tetrahedronize them?
Given any set of $3$D points, can we always make non-overlapping tetrahedrons from them where the union of tetrahedrons exactly fill the convex hull of the input points?
AFAIK, given any set of $2$D ...
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Solid Angle Subtended By a Rectangle [closed]
Consider a rectangle with edges $2a$ and $2b$ in the $xy$-plane with its centre (the point of intersection of diagonals) at the origin. The problem is to find a closed form for the solid angle ...
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A line does not separate 3-space
I know that in 2d space, a line $L$ separates the plane into two disjoint nonempty portions, called half-planes, such that two points lie $P$ and $Q$ lie on the same half-plane iff the segment $PQ$ ...
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Exact Triangle Sorting for Orthographic Rendering of a Triangulated Surface
I want a definitive front-to-back triangle drawing order under orthographic projection.
This is an X-post due to inactivity on the other: https://tex.stackexchange.com/q/735053/319072
Note: I have ...
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Relationship between vertices and edges in platonic solids
The Question:
Given a list of the vertices in a platonic solid, is there a way to calculate which vertices are connected by an edge? I know that one could find edges using edge length or rotations but ...
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Prove total edge length of section is constant? A cross-polytope cut by a hyperplane parallel to a face.
The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of $(±1, 0, 0,\dots, 0)$. The cross-polytope is the convex hull of ...
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Two perpendicular contact unit discs rolling without slipping over the $xy$-plane.
I'm trying to solve a question posted in 2013 about the trajectory of contact points with the ground of two perpendicular contact unit discs rolling without slipping over the ground.
My progress:
$$\...
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Distance of a point to bilinearly interpolated surface
I have a 3D surface that is defined by bilinear interpolation of 4 points $Q_i$ and coordinates $u,v \in [0..1]$:
$S(u,v) = (1-v)(1-u) Q_1 + (1-v)u Q_2 + v(1-u)Q_3 + v u Q_4$
I would like to find the ...
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UV map of a sphere with perspective given perfect distance and fov
This is my first post here, and I'm nowhere near the level of math knowledge everyone else is, though I am still fascinated by 3d (and color, though here that's irrelevant) related fields.
I've been ...
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What is the probability that two random planes intersecting a sphere intersect each other inside the sphere?
I'm trying to calculate the expected number of 'pieces' a sphere will be cut into after n cuts. I've solved part of the problem, but I need to work out the probabilities that planes that 'cut' a ...
