# 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: ...
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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 (...
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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 ...
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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 ...
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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 ...
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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? ...
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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 ...
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 ...
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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 ...
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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 ...
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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 ...
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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 $2$, ...
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### 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 ...
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### Shortest distance between vertex of a circular cone and a quarter of its conical helix

I was given with the question below: ...
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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 ...
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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 ...
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### 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 ...
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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-...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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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. ...
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### 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 ...
1 vote
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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 ...
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