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.
3,512
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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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2
answers
30
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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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33
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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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0
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8
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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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0
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37
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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 ...
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1
answer
25
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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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43
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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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38
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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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1
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33
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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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52
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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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1
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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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0
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23
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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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32
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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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1
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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 ...
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64
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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 ...
2
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0
answers
41
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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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0
answers
12
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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 ...
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1
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1
answer
33
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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
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1
answer
28
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Show that the spheres, which cut two given spheres along a great circle, all pass through two fixed points.
Show that the spheres, which cut two given spheres along a great circle, all pass through two fixed points.
let the given spheres be
x²+y²+z²+2$g_1$x+2$f_1$y+2$h_1$z+$d_1$=0 ...... (1)
and
x²+y²+z²+2$...
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40
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Show plane sections of the conicoid $ax²+by²+cz²= 1$ which are rectangular hyperbola and which pass through point $(\alpha,\beta,\gamma)$ touch a cone
Show that all plane sections of the conicoid $ax²+by²+cz²= 1$ which are rectangular hyperbolas and which pass through the point $(\alpha,\beta,\gamma)$ touch the cone
$\frac{(x-\alpha)^2}{b+c}$+$\frac{...
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0
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31
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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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1
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49
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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 ...
2
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0
answers
75
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Prove $\tan(\alpha/2)\tan(\beta/2)\tan(\gamma/2)\tan(\delta/2)+[\frac{(1+e)}{(1-e)}]^2 =0$
If the normals at the points of vectorial angles $\alpha,\beta,\gamma,\delta$ on
the conic $l/r=1+e\cos\theta$
meet at a point, then prove that
$\tan(\alpha/2)\tan(\beta/2)\tan(\gamma/2)\tan(\delta/2)+...
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2
votes
1
answer
29
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Change of angle between two skew lines during individual rotation
Given two skew lines, $\mathbf{x}_1 + \mathbf{u}_1 t$ and $\mathbf{x}_2 + \mathbf{u}_2 t$ where $\mathbf{u}_1$ and $\mathbf{u}_2$ are unit vectors along the lines, we might be able to rotate each line ...
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1
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76
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Generating lines
Show that the generators of the hyperboloid $\frac{x^2}{25}+\frac{y^2}{16}–\frac{z^2}{4}=1$ which are parallel to the plane $4x-5y-10z + 7 = 0$ are $x+5=0, y+2z=0$ and $y+4=0, 2x=5z$
My attempt
...
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1
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How could the ellipsoid formula form a 2-sheeted hyperboloid?
According to the sources, the ellipsoid formula is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
And the formula for a 2-sheeted hyperboloid is:
$$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{...
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How do you calculate an objects Y-Rotation relative to another plane in 3D space?
I am trying to program a robot for an FRC competition. One of the tasks this year is to balance on something similar to a see-saw/ramp. The root of our problem comes down to the way our robot knows ...
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On works regarding 3D surface or volume models represented by Dirichlet tessellations for various norms.
Background and context :
Dirichlet tesselations or Voronoi diagrams are used in 2D geometry to subdivide the plane into different sets.
As for Euclidean distances, I am quite convinced (although ...
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1
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Efficiently Determining Surface Intersections Along a Line Segment
Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
0
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1
answer
216
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Curved edges for a smooth three-dimensional object
Q) Looking at the surface of a smooth $3$-dimensional object from the outside, which one of the following options is TRUE?
$A)$ The surface of the object must be concave everywhere.
$B)$ The surface ...
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Rotating a 3D vector onto the XY-Plane around the x-axis
I'm currently working with an accelerometer data vector (x,y,z), where I want to "eliminate" the y component of the data while not scaling the vector.
I want to do this by rotating my vector ...
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1
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40
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Finding the angle between a prism's edge and one of its faces.
I’m having difficulty on how to start off the following question:
I know that forming a right-angled triangle AMJ with M as the midpoint of GH could be a good starting point, but I'm not sure how to ...
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How to convert the line $x+y=0$, $x+y+z=a$ into symmetric form?
I need to convert the line $x+y=0$, $x+y+z=a$ into its symmetric form. I've learned four different methods to convert a line from its general to symmetric form, but three of those methods didn't work ...
1
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0
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Visualizing Orthogonal Vectors in 3D
I am in a multivariable calculus course. Let a be the vector <-1 ,1, c> (with c any number) and b the vector <1, 1, 0>. The two vectors are orthogonal, as the dot product of a and b is 0.
(...
2
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Wolfram Mathematica: why is the plot not visible?
simple question regarding Wolfram Mathematica:
Why is the plot not showing?
(See screenshot)
(I am new to Wolfram Mathematica)
Thankful for any input.
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How can I check if a Point Intersect A Moving Sphere?
My problem is that I can't find all the points that intersect the sphere during its linear motion. What I did was to check starting with the starting point, from point to point until the final point, ...
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2
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0
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Volume of half an ellipsoid
I want to calculate the volume of the solid defined as follows.
$$ K := \left\{ (x,y,z) \in \Bbb R^3 : x^2 + y^2 + 3z^2 \le 4, z > 0 \right\} $$
I know that this is a half solid ellipsoid and that ...
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0
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Create a parametric representation of a 2d iterative function?
I have 2 functions X(x,y) and Y(x,y) which when given a coordinate pair (x,y) give a new coordinate pair's x, and y components respectively. For instance, given coordinate "(1,2)", "(X(...
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0
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Get the 3d coordinates for a given 2 points, angle and normal
Given 2 coordinates, 1 included angle and normal, find another point.
Given
point $A(1,2,3) $
$O(0,0,0) $
$n(0.0348, -0.832, 0.554)$ is the normal direction of the circle
$∠AOB = 100° $
Find the ...
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2
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How can I work out the angle between a face and base of a triangular prism
I'm struggling with a particular 3d problem - https://imgur.com/a/19EClHN (question 4)
To workout the length of $EM$, forming a right angled triangle from face $EAB$ I get: $$EM =\sqrt{4^2 - 2^2} = 3....
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4
answers
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Metric in terms of the connection
The Levi-Civita connection can be written in terms of the metric as:
$$
\Gamma^l_{jk}=\frac{1}{2}g^{lr}(\partial_kg_{rj}+\partial_jg_{rk}-\partial_rg_{jk}).
$$
Can this relation be inverted for the ...
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0
answers
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Determinant for of Area of triangle in 3D
As we all know that area of a triangle is given by absolute value of the determinant of this matrix$
A = \dfrac 1 2 {\left| \begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & ...
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0
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72
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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.
3
votes
1
answer
133
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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 ...
0
votes
1
answer
22
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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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0
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25
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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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0
votes
1
answer
48
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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 ...
- 3,052
1
vote
1
answer
33
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Maintain angle between two 3D vectors after rotation(s)
In a 3D cartesian space, I have two vectors. One vector is a static unmoving vector—let’s call it S. The other vector is the vector I need to rotate—let’s call it R. Both vectors have their tail as ...
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0
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0
answers
27
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Coordinate transform with known camera pose pairs
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 $...
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