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.
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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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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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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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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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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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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$...
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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?...
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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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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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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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Can a laser beam traverse all points of $\{0,1,2\}^3 \subset \mathbb{R}^3$ using $12$ mirrors only if it is emitted from outside the open cube?
Let $G := \{0,1,2\}^3 \subset \mathbb{R}^3$, the $3\times3\times3$ grid consisting of $27$ integer points, be given.
We emit a laser beam (infinitesimally thin, traveling in affine Euclidean $3$-...
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New Conjecture Found In Geometry. Related to tetrahedron and vector 3D. Please review the statement and provide views. [duplicate]
Hi guys currently working on vectors I studied incenter of tetrahedron and the inradius for a tetrahedron was (3*volume)/surface area in modulus.
I checked for some other closed figures and the ...
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Finding the solid angle for a viewer in a movie theatre, from viewer to screen but at different heights...
I am doing a research paper to find the best seat in the cinema. I want to find the solid angle of all possible seats, so I searched for a formula to do so.
I came across someone asking a similar ...
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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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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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A unit sphere moving from an initial state to a final state, minimize the maximum trajectory length across all points.
A unit sphere moves from an initial state to a final state. This motion is described by
a rotation $R$ followed by a translation vector $T$ (displacement of the center).
Each point on the sphere's ...
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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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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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How to decide bounds in spherical coordinates especially for $\Phi$
[Find the volume of] The solid bounded below by the sphere $\rho=2\cos\phi$ and above by the cone $z=\sqrt{x^2+y^2}$.
I'm a bit skeptical on how to decide bounds for the integral in spherical ...
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Classification of a stationary point for multi-variable function
I have this multivariable function $f(x,y) = (y-x^2)(y-2x^2)$. Obviously the first step to finding any stationary points would be to find where the derivatives $f_x$ and $fy$ are zero. And this occurs ...
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Rotate a vertical plane about two axis to achieve a set an angle in each rotation
I am running a 3D cad program. Z +up, Y +to left, X + into page.
If I want a new plane (A) that is 20 degrees from the YZ about Y, no problem. Simple rotation about Y axis and enter 20 degrees.
If I ...
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Given the 3D polar coordinates for 2 vertices of an origin-centered, regular tetrahedron, how do I find the coordinates of the remaining 2 vertices?
There is a trivial case of this which I can intuitively understand.
Assuming the tetrahedron is inscribed in a unit sphere and the given points are $$v_1 = (1, 0°, x°)\\v_2 \approx (1, 109.5°, x°)$$ ...
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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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