# 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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### what is the best method for modeling 3D point?

The helicopter blade deforms during rotation (Figure.1)enter image description here and I want to model the structure of the deformed blade. I have the 3D coordinates of the markers mounted on the ...
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### Probability of the surface area of a pentagonal prism being greater than the volume

Consider a unit cube. Points $WXYZ$ are on the sides of the bottom face of the unit cube. Point $I$ is inside of the bottom face of the unit cube such that $WXYZI$ forms a pentagon. Point $M$ is ...
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### Trying to graph $y = (2-z)^{1/2}$

I tried to graph $y = (2-z)^{1/2}$ and I got half of a paraboloid that appears upside down from the point of $z = 2$ for $y$ values greater than or equal to $0$. However, when I tried this on Geogobra,...
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### Locus of pyramid vertex with solid angle

What is locus surface for apex P when constant solid angle $\Omega$ of central position $$\Omega = 4 \sin^{-1}\left( \sin a/2 \cdot\sin b/2 \right)$$ is subtended by a rectangular base pyramid with ...
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### Maximal volume "surrounded" by a constant surface area

I am sorry for the bad notation, see comment in the end. For any closed surface $S \subset \Bbb R^3$, let $V(S)$ denote its volume. I guess $V$ is a function that gets "a manifold" and ...
1 vote
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### For a continuous rotation representation in 3D, do you need at least 4 real variables?

I'm pretty sure I have read this somewhere, but I just can't get to find this theorem anywhere. Is there a theorem that states that for a continuous rotation representation you need at least 4 real ...
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### Find the volume of the given solid

Find the volume of the solid $$\{(x,y,z) : x ≥0,y ≥0,z ≥0,z ≤2 −e^y, x + y + z ≤10\}.$$ Not sure if my workings for this is correct? Any help would be appreciated, thanks! workings
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### Tangent plane for sphere containing a line [closed]

Find the equation of the tangent planes to the sphere $$x^2+y^2+z^2-2x+4y-6z+10=0$$ which pass through the line $$\frac{x+3}{14}=\frac{y+1}{-3}=\frac{z-5}{4}.$$ Find also angle between these planes. ...
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### find quaternion to rotate a point towards another point in 3d space

I have a point in a 3D space rotated by a quaternion. The point has a fixed length from the rotation itself (shown in image). What I'm trying to achieve is to rotate my point towards a goal position ...
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### When is a 3D polyhedron stacking aperiodic. [closed]

Suppose you find a 3D "einstein". For me, its stacking has no dominant continuous planes*. How do you determine whether you can or can't make a periodic stacking (3D tiling) with it? (*) In ...
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### How can I find the normal vector of a plane passing through the origin with its roll, pitch, and yaw? [closed]

I want to find the normal vector of a plane that is centered at the origin. I know the roll, pitch, and yaw, but I don’t know how to convert those to the angles used to find a plane. It would also be ...
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### How many 2D "slices" fit into a 3D object?

I assume that 3D objects are comprised of 2D cross-sections, but how many of these cross-sections fit into a finite 3D object? I know that infinite 2D cross sections would not be enough to have depth ...
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### Given 2 vectors in 3d find a condition that satisfies an inequality.

given $u=[1, 0, 1]$ and $v=[1, 2, -1]$. Let $x = sv + t\sqrt{2} \ u \times v$, where $s ∈ R$ and $t ∈ R$. Find the condition on $s$ and $t$ that renders $||x|| \leq \sqrt6$. Answer should be a ...
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### Jones polynomial of a knot in terms of its Seifert matrix

It is well known that the Alexander polynomial of a knot can be written in terms of the Seifert matrix of the knot by a simple relationship $$\Delta(t)=\det(V^T-tV),$$ where $t$ is a formal variable ...
1 vote
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### Converting a sphere equation to a homogenous one to obtain the equation of a cone

A variable plane parallel to the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0$ meets the coordinate axes at $A, B, C$. Find the equation of the cone whose vertex is at the origin and the guiding curve ...
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### How many points are needed to uniquely determine a cone?

The general equation of a cone that passes through the origin is $$ax^{2}+by^{2}+cz^{2}+2fyz+2gzx+2hxy=0$$If I'm given $5$ points on the cone, I should be able to get $5$ equations and be able to ...
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### Need help on Spherical sweep calculation based on animated 3D camera.

I need some maths help to crack this problem, so I can implement something in Maya (3D animation program. If we have an animated 3D camera ( -z is aiming direction, y is up, x is right) and focusing ...
1 vote
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### 3D rotation matrix between coordinate systems when knowing new x and y axis?

Given $x,y,z$ axis of an original coordinate system as $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, assume that we have new $x$ and $y$ axis defined as $(a,b,c)$ and $(d,e,f)$ where $x,y$ are orthonormal, of ...
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### Independent axis rotation in 3D space for a point.

I'm working on a sensor fusion application. In our vehicle, 4 radars (fixed, non-rotating radars) are placed at 4 corners of the vehicles at 45 degrees, such that the axis of the Radar are inclined at ...
1 vote
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### Geometrical area relationship using vector cross product

In order to show the result, the textbook quotes that: Area of parallelogram ABEF = Area of parallelogram ABCD + Area of CDEF + Area of BCF - Area of ADE I cannot see why this relationship is true - ...
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### Rotation Matrix with local control? [closed]

Is there a rotation matrix that provides local control at each axis? I'm not entirely sure how to phrase what I'm asking - the best analogy I can give is imagining taking a physical version of the ...
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### 3D logarithmic spiral using Fibonacci sequence

I would like to get an equation of a 3D logarithmic spiral using the Fibonacci sequence so I can use it in GeoGebra. I'm an artist, not a mathematician. This is beyond my understanding. Thank you.
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### Projection of a 3D circle onto a 2D camera image

Assume that I have a 3D circle with a center at $(c_1, c_2, c_3)$ in the circle coordinate frame $C$. The radius of the circle is $r$, and there is a unit vector $(v_1, v_2, v_3)$ (also in coordinate ...
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### Connected components of complement of double napped cone

How may connected components does the complement of the conic surface (extending on both sides) have in the three dimensional space? I think the connected components is two, but am not convinced about ...
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### Minimum number of times needed to cut a cube [duplicate]

How many times would I have to cut a large cube to form 64 individual, equally sized smaller cubes? To get $4$ equally sized cubes, it would take $3$ cuts. $4^3$ is $64$ so my guess is $3^3$ which is ...
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### Find the parameterization of shadow cast on plane

Suppose there is a light source at $(0, 0, 8)$. Consider the line segment $r(t) = \langle 3-3t, 1+t, 2+2t\rangle, 0\le t\le1$. Find the shadow cast by the line segment on the plane $x+y-2z=8$ I have ...
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1 vote
Let's say that we have two $3D$ coordinate systems, $A$ and $B$. Now I want to get the coordinate transform matrix $T_{AB}$ such that for any points $p_A$ in $A$, we can get its coordinate in $B$ by \$...