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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How to prove that the shortest path between two points in sphere is a part of the great circle?

It is a known fact that the shortest path between tow points in sphere is a part of the great circle. but I don't know the proof of this claim so I tried to rigorously prove it myself. My attempt: ...
pie's user avatar
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How to get a 3D vector in the same direction as another vector limited by an angle?

If I have two 3D vectors as shown by the blue and red vector in the image below, where the red vector is at an angle of more than 20 degrees from the blue vector, how could I calculate a new vector (...
Mori's user avatar
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Planes tangent to a space hyperbola

Given the hyperbola specified in $3D$ as $ H(t) = V_0 + V_1 \sec t + V_2 \tan t $ And the family of parallel planes $n \cdot r = d $ where $ r = [x,y,z]^T $, and $n$ is the normal vector which is ...
of course's user avatar
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Planes tangent to a space ellipse

Given an ellipse in $3D$ space specified parametrically by $ E(t) = V_0 + V_1 \cos t + V_2 \sin t $ And given a vector $n$, the normal vector of the family of planes $n \cdot r = d $ where $ r = [x, y,...
of course's user avatar
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is there a set of great circles on a hypersphere analogous to Buckminster Fuller's 31 in 3 dimensions?

I am addressing points on a sphere using great circles, and am investigating using Buckminster Fuller's "31 great circles." I am considering the viability of doing the same on a 3-sphere (in ...
Travis Well's user avatar
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A detailed explanation of the finite element method [closed]

the Finite Element Methods for the solution partial differential equation
أسامة أبو يحيى's user avatar
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Regarding solving for the coordinates of reflection points in 3D space

In space, there exists a smooth plane, and additionally, there are two points, denoted as $A (x_a, y_a, z_a)$ and $B (x_b, y_b, z_b)$. A beam of light is emitted from point A, reflects off the mirror ...
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dihedral angles of non-regulare icosahedron given its coordinates

I have the coordinates (3D) of the vertices of an icosahedron (not a regular one) and I want to find all dihedral angles. I am fine with just equations to which they need to fulfill as I would think ...
koen ruymbeek's user avatar
-5 votes
0 answers
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3d geometry hvvtvgvvkvuv [closed]

Consider the tetrahedron formed by the planes y+z=0,z+x=0,x+y=0, x+y+z=a . The direction cosines of the shortest distance lie between the planes y+z=0 and z + x = 0 is:
Dev Patel's user avatar
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Determine the equation of the right circular cone whose intersection with the $xy$ plane is a known ellipse

There is a known ellipse $ (r - r_0)^T Q_e (r - r_0) = 1 $ that is the intersection of an unknown right circular cone with the $xy$ plane. Determine the equation of the cone, i.e. determine the ...
of course's user avatar
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Determine position of sound source from arrival times of a blip ($3d$ version)

As an extension to this problem, I am now considering the problem in $3d$. The setup is a follows: You have four sound receivers (microphones) at known locations $P_1(x_1,y_1, z_1), P_2(x_2,y_2,z_2), ...
of course's user avatar
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3 votes
4 answers
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How to calculate the area of a projected path onto a plane? (Bee traveling problem) [closed]

How do you "orthogonally project" the shape? After I drew multiple graphs, I still had no idea. The correct answer is $12$. I guess it is $3 \sqrt 2 \cdot 2 \sqrt 2$ according to the gist of ...
Mintylemon66's user avatar
2 votes
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Volume of water in a tilted hemispherical bowl

Suppose you have a hemispherical bowl that is open at the top. You tilt the bowl by an angle $\theta$ in any direction. Then you fill it with water till the water surface reaches the lowest point on ...
of course's user avatar
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Hyperbolic equations with three terms [closed]

The equation of a hyperbola is $ x^2 - y^2 = r $ Assuming $ a = b = 0 $ Suppose now the equation $ x^2 - y^2 + z^2 = r $ What kind of curve would this equation parametrize ? What is its geometry? ...
Tomás's user avatar
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Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$

Problem: Find the minimum value of the expression $2a^2 + b^2 + c^2$ if the point $(a,b,c)$ lies on the plane $4a + 3b + c = 7$. I couldn't get very far, but I do know how would I solve a simpler ...
Shivansh Jaiswal's user avatar
1 vote
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Volume of a sphere enclosing six tetrahedron edges [closed]

Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given. What relation should be there in order that a tetrahedron can be enclosed among the edge side lengths? What is the volume $V$ of the ...
Narasimham's user avatar
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Inverse square law-type integral over two line segments

This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in ...
Mel's user avatar
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Unique affine transformation, sending $P_1$ to $P'_1$, $P_2$ to $P'_2$, $P_3$ to $P'_3$ where $\| P_i P_j \| = \| P'_i P'_j \| $

Given three points in $3d$ space $\{ P_i, i = 1, 2, 3 \} $, and corresponding three points $\{ P'_i, i = 1, 2, 3 \} $ that are their images, respectively, such that $\| P_i P_j \| = \| P'_i P'_j \| , ...
of course's user avatar
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1 vote
1 answer
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Finding an affine transformation that will map one set of three given points $P_1, P_2, P_3$ to another given set of points $P'_1, P'_2, P'_3$

Suppose you given three points with known coordinates $P_1, P_2, P_3$, and also another set of three points $P'_1, P'_2, P'_3$. Now does there always exist an affine transformation $ f( r ) = A r + b ...
of course's user avatar
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2 votes
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Equation of the plane that is at given distances from three given points $A,B,C$

Suppose you have the coordinates of three points $A,B,C$. You want to construct a plane $\pi$ such that $ d(\pi, A) = r_1 $ $ d(\pi, B) = r_2 $ $ d(\pi , C ) = r_3 $ where $d(\pi,v) $ is the ...
of course's user avatar
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0 votes
1 answer
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Generate random point on the ellipsoid in 3d.

I want to generate the trajectory of point M in space, such that the sum of its distances to two other points, P and Q in space, remains constant, that is, |PM| + |MQ| = constant. Therefore, I believe ...
ZHIHA's user avatar
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Parameterizing (potentially non-planar) equilateral polygons in 3d

This is similar to Parameterizing equilateral polygons but for 3 dimensions. I have $n$ points in $\mathbb{R}^3$ ($n \geq 3$), labeled $1$, $2$, ..., $n$. Point $1$ is unit distance from point $2$, ...
Mel's user avatar
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2 votes
2 answers
73 views

3d fractal helix modeling

I'm trying to build a 3d visual to illustrate a concept. Imagine a circular helix. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We ...
nnuuurrrrcc's user avatar
3 votes
1 answer
69 views

Shortest distance between vertex of a circular cone and a quarter of its conical helix

I was given with the question below: ...
kingking's user avatar
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-1 votes
0 answers
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Confusion with a Cylinder's revolution [closed]

I have come up with this problem, yet I might be quite stuck finding the appropriate answer. Let there be a cylinder of length 'L' and radius 'R'. Let the cylinder lie on it's curved surface and be at ...
Nischal_Kolli's user avatar
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Calculate Up Vector of Object on Surface Given Points and Normals

Context I want to find the up vector an object would have if it were leaning on a surface, given a large amount of points on that surface and their associated surface normals (i.e., an equation that ...
natSegOS's user avatar
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1 answer
52 views

Spatial curve with certain condition

In a certain exercise I have been asked to find all spatial curves such that, given the curve $\gamma$, it is satisfied that $\gamma'' = \gamma' \times a$, $a$ is a certain constant vector. I am aware ...
Daniel C.'s user avatar
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Are two parallel projections sufficient to prove that four points are the vertices of a parallelogram?

This is an analytical proof of Varignon's theorem (of course it is a well-known proof) I came up with an idea, which is to take advantage of the two-dimensional situation to prove the three-...
زكريا حسناوي's user avatar
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24 views

higher dimensional analogue of volume difference of 2 three-dimensional cubes

In the illustration a way of expressing the difference in volume of 2 different sized 3-dimensional cubes is given: The volume of a black cube of size b is diminished by the volume of a red cube with ...
plopper's user avatar
1 vote
0 answers
21 views

Minimum and Maximum distance to the intersection of an ellipsoid with a plane

Given the ellipsoid $ (r - C)^T Q (r - C) = 1$ and the plane $ a^T r = b $ where $ r = [ x,y,z]^T $, $C \in \mathbb{R}^3$ is the center of the ellipsoid, and $Q \in \mathbb{R}^{3 \times 3} $ is a ...
of course's user avatar
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3 votes
1 answer
87 views

Interpretation of change in direction cosines of a variable line: Pythagorean theorem for small angles?

Consider a variable rotating line passing through a fixed point. The angle between two successive/adjacent positions of the line is a small angle $\delta \theta$. If the change in the direction ...
Cognoscenti's user avatar
0 votes
1 answer
23 views

Closest and Farthest point on curves of intersection between two ellipsoids to the origin

A curve is the intersection of two ellipsoids that are given by $ (r - C_1)^T Q_1 (r - C_1) = 1 $ $ (r - C_2)^T Q_2 (r - C_2) = 1 $ I'd like to find the points on this curve of intersection that are ...
of course's user avatar
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0 votes
0 answers
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Equations with nodes 72 degrees apart

I am interested the nodal ($z=0$) behavior of the following functions for a small research project: $$z=x(x^2-3y^2)$$ $$z=y(3x^2-y^2)$$ When graphed, there are nodes 60° ($n=6$) apart from each other. ...
Kevin Freddo's user avatar
2 votes
2 answers
210 views

Rotation of $3$-dimensional space vectors

Assuming there are two vectors $P(x_p,y_p,z_p)$ and $Q(x_q,y_q,z_q)$ originating from the origin. Now, if I rotate vector $P$ to another position in space and obtain the coordinates $P'(x_p',y_p',z_p')...
ZHIHA's user avatar
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0 votes
1 answer
46 views

Minimum and maximum distance of a $3D$ circle from the origin

A $3d$ circle is the intersection of two spheres given by $ (r - C_1)^T (r - C_1) = R_1^2 $ $ (r - C_2)^T (r - C_2) = R_2^2 $ I'd like to find the points on this circle of intersection that are at a ...
of course's user avatar
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6 votes
1 answer
79 views

Volume of the two parts of a pyramid cut by a plane [closed]

I have to translate the task from german so forgive my spelling mistakes if they occur. We have a pyramid with the points $A(-3,-3,0)$, $B(3,-3,0)$, $C(3,3,0)$, $D(-3,3,0)$ and $S(0,0,9)$ and a plane ...
Math Jesus's user avatar
1 vote
2 answers
64 views

In three-dimensional space, when the sum of the distances from an unknown point to two known points is constant, what is the trajectory of the point?

The unknown point P (x, y, z) in three-dimensional space is at a constant sum of distances dPA and dPB from two known fixed points A and B, i.e., dPA + dPB = constant. How can we express the ...
ZHIHA's user avatar
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0 votes
1 answer
56 views

Finding the minimum and maximum distance between the origin and the intersection curve between a cone and a sphere

Suppose you have the cone with its vertex at the origin given by $ r^T Q r = 0 \tag{1}$ where $r=[x,y,z]^T $, and $Q$ is a $3 \times 3$ symmetric indefinite matrix. And you have the sphere centered at ...
of course's user avatar
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0 votes
0 answers
60 views

Finding subset in $\mathbb{R}^3$ with under some condition.

Let $S$ be the subset in $\mathbb{R}^3$ of all vectors forming an angle $ \leq 45^{\circ}$ with the vector $(1 , 0 , 1)^T.$ I have no idea how to approach this problem? Is this the right approach, $S=...
Student of Mathematics's user avatar
-1 votes
1 answer
60 views

Volume of tetrahedron detected by four planes in $\mathbb R^3$ [closed]

Given the equation of $4$ planes in $\mathbb R^3$, find the volume of the tetrahedron. The planes are: $$\begin{align}&P1: x+3y+z=2\\&P2: x-2y+z=2\\&P3: -x+z=4\\&P4: x=3\end{align}$$ I ...
Julian Navarro's user avatar
1 vote
1 answer
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cant find the orthogonal proyection of the line on a plane. plane: 10x-6y-12z=7, line: (8-15t,9t,5+18t). [closed]

i have the following: when replacing x,y,z values of the line on the plane equation: 10(8-15t)-6(9t)-12(5+18t)=7, t=13/420.Then if we replace "t" in the equation of the line, we obtain the ...
Julian Navarro's user avatar
1 vote
0 answers
15 views

Computing the Upward Direction(Z-Axis) Vector from a 3D Rotation

I was tasked on creating a C Program that needs to process this case: Convert the Initial Rotation Matrix(InitialRotationMatrix) into 3D Rotation(Rotation3DResult1) Convert the 3D Rotation(...
Drin John's user avatar
2 votes
0 answers
56 views

get dihedral angles of octahedron given all triangles

An octahedron (not necessarily regular) consists of 8 triangles. You can see it as two pyramids glued together (for now on I only consider this case). Call the triangles in the upper pyramid $T_1, T_2,...
koen ruymbeek's user avatar
1 vote
1 answer
86 views

Solving a Lagrange multiplier optimization problem

I have the Lagrange multiplier problem where the objective function is $ f(r) = r^T r $ where $ r \in \mathbb{R}^3 $ subject to $r^T Q r = 0 $ where $Q$ is a $3 \times 3$ symmetric indefinite matrix, ...
of course's user avatar
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0 votes
3 answers
84 views

Find the equation of the sphere with points P such that the distance from P to A(−2, 4, 4) is twice the distance from P to B(6, 3, −1).

As you can tell from the title, I need to find the equation of the sphere with points $P$ such that $2|PB| = |PA|$. The coordinates for $A$ are $(-2, 4, 4)$ and the coordinates for $B$ are $(6, 3, -1)$...
baron's user avatar
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0 votes
0 answers
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Viewing a paraboloid from a point outside it

Suppose you're given the paraboloid $ z = a x^2 + b y^2 + c $ which you're viewing from the point $A$ that lies outside. What will be the equation of the cone of view of the paraboloid from $A$? My ...
of course's user avatar
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1 vote
0 answers
51 views

Determine equation of cone of view of an ellipsoid

Given the ellipsoid $ (p - C)^T Q (p - C) = 1 \tag{1}$ where $ C $ is the center of the sphere, $p $ is a point on the ellipsoid surface, and $Q$ is a $3\times3$ symmetric and positive definite matrix....
of course's user avatar
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0 votes
1 answer
57 views

Elliptic cone dimensions such that mutually perpendicular axes can be drawn on its surface

Given the elliptic cone $r^T Q r = 0 $ , where $ r= [x, y, z]^T $ and $$ Q = \begin{bmatrix} \dfrac{1}{a^2} && 0 && 0 \\ 0 && \dfrac{1}{b^2} && 0 \\ 0 && 0 &...
of course's user avatar
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2 votes
2 answers
84 views

Locus of point from which three mutually perpendicular lines tangent to an ellipsoid can be drawn

Inspired by this problem, I would like to find the locus of all points from which three mutually perpendicular tangent lines can be drawn to a given ellipsoid. The ellipsoid is given by $$ \dfrac{x^2}{...
of course's user avatar
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4 votes
2 answers
245 views

Intersection of a circle in 3D with horizontal planes?

Let's assume that we have two vectors in $\mathbb R ^3$, $v_1$ and $v_2$, such that $v_1 \perp v_2$ and $|v_1|=|v_2|=1$. Since the two vectors emerge from the origin, they can define a circle in $\...
Physician's user avatar
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