# Tagged Questions

The (3d) tag is for things related to 3-dimensions. For geometry of 3-dimensional solids, please use instead (solid-geometry). For geometry that is not on a plane, but otherwise agnostic of dimensions, perhaps (euclidean-geometry) or (analytic-geometry) should also be considered.

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### Quaternion Rotation

I am modelling rotations of a rectangular box (3 dimensions) in Matlab using Quaternion theory. Using the theory found on https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation I have ...
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### In 3dimension, How to move the vector to the point? [on hold]

In 3D, I want to move my point to target point side. And my point has direction. By the way, I must move in direction of my point. If direction of my point same position of target, I have to move my ...
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### What is the equation for the number of combinations of 4 cubes that can be rotated on all axes

I have been trying to work out the number of possible unique combinations of 4 cubes where they can be rotated on any axis. So for example if all the faces of all the cubes where unique across the ...
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### Proof that there exists a 3d representation of all graphs

Below is a question and proof that I've done. I was wondering if there is a more formal way of concluding a point must exist that is not in a set composed of a finite number planes. Currently I am ...
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### The formula for 3D rotation of the perspective of an image in 2D space

Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of ...
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### Formula For 3D Dilation?

If I have a sphere on a 3D grid with it's center being at the origin, and I want to double the size of the sphere, where would it's poles be?
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### Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
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### Non-trivial 3D curve that projects as a line or a segment onto the faces of the quadrant

I want to illustrate how high dimensional objects may have misleading projections. Examples are for instance given with HiSee software, with nD bouquets of circles. Are there non-trivial (not a 3D ...
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### Translation by tensors

According to this question, quaternions would not be the right choice to handle both rotation and translation. In the case of tensors, one might assert that the rotation would be possible by tensors, ...
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### 3-Space Vertices of a Parallelogram

The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three ...
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### Determining angle of view from an image with a square or checkerboard in the background.

I take a picture of a square of known dimension (let's say 1x1 units) with the camera at an unknown angle relative to the plane of the rectangle. (The distance to the rectangle is also unknown, but ...
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### Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, ...
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### Counting balls in face centred cubic close packing

Possibly too easy for stack exchange, but... Consider a cubic close packing, or face centred cubic, arrangement of balls or radius $1$ in dimension $3$. Suppose that the origin is the centre of one ...
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### Euler Angle + Distance to XYZ coordinate

Background I'm familiar with Euler Angles and 3d space systems but I'm having trouble with Rotation Matrices. Scenario I've converted my Euler Angle to degrees for ...
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### Given set of points in 3D, find group of points closest to each other

Given a set of any 8 points in 3D space. I want to find a subset of points that are closest to each other. Application: Assume in a 3D space, I have any 8 colors(represented in RGB). I know how to ...
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### Draw a line between an observer and the current direction of the sun

My goal is to draw a line between an observer and the current direction of the sun. Given the observers coordinates (Lat, Lon) of (51.50442, -0.08630) a North of (90, 0), an Azimuth of 270 degrees ...
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### How to determine the equation of shortest path on any 3d surface between two given points?

I am working on draping of woven composite and I have to determine the equation of shortest path on 3D surface (i.e. $z=x^2+y^2$) between two given points in order to get the yarn path between two ...
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### Can two lines lying in different plane be parallel?

In two lines lie in different plane, can they be parallel to each other? I am thinking if two lines are parallel to each other, then their direction cosines must be same so two lines lying in ...
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### Find radius of curvature of a $3D$ surface at a point if the tangent vector at that point is given.

I have a $3D$ surface with equation $z=x^2 +y^2$ and need to find the radius of curvature at a point $P(x_1,y_1,z_1)$ using the tangent vector at that point. Since, there are many possible radii of ...
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### Plotting a 3D graph from explicit equation

I´m a 2nd year engineering student and today we learned how to plot 3d graphs from a $XYZ$ equation on paper. For example, I know ($\frac{X^2}{9}+ \frac{Y^2}{16} + \frac{Z^2}{9} =1$) will produce an ...
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### A 3d integral over a ball

Given $a \in \mathbb{R}^3$ and $r>0$, is it possible to compute $$\int_{B_r(a)} \frac{a\cdot x}{|x|^3} dx$$ where $B_r(a)$ is the ball in $\mathbb{R}^3$ with radius $r$ and centered at $a$.
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### Quaternion interpolation in 3D

I'm a chemist lost in the captivating world of mathematics thus if you could keep your answers simple it would be awesome! Here is my problem: I have two mobiles (A,B) in 3D. Ideally, I would like to ...
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### How to compute the pivot point of a rectangular cuboid to achieve a certain rotation?

Summary: For a video game project, I have an object (craft) that hovers the ground using a soft constraint. Imagine that on the picture below there is an invisible point above the craft whose ...
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### Find the equation of the sphere which touches the sphere $x^2+y^2+z^2+2x-6y+1=0$ at $(1,2,-2)$ and pass through $(1,-1,0)$

Find the equation of the sphere which touches the sphere $x^2+y^2+z^2+2x-6y+1=0$ at $(1,2,-2)$ and pass through $(1,-1,0)$ My Attempt: Let the equation of the sphere be $x^2+y^2+z^2+2ux+2vy+2wz+d=0$...
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### Find 2D plane in the center of nonlinear 3D object

I'm building a segmentation algorithm. I'm segmenting pieces of paper in a book that have been slightly crumpled. Imagine taking a piece of paper, crumpling it into a ball, and then trying to ...
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### Fitting point to plane??

As explained here, given a plane: ax + by + cz + d = 0 and a point x0=( x0 , y0, z0 ), the normal vector to the plane is given by: v = [ a ; b ; c ] and a vector from the plane to the ...
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### Cylinder ray puzzle

A set of rays (imaginary beams, no direct relation to other concepts) are shot from various points in a cylinder forward and and backwards until they reach the edge of the cylinder. The rays' forward ...