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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1answer
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Equation of plane passing through intersection of line and plane

Find the equation of the plane passing through the intersection of line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{2}$$ and the plane $$x-y+z=5$$ and parallel to a vector with direction ratios $<2,3,...
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3answers
95 views

One-sided submanifolds in Hempel's 3-Manifolds

Early on in Hempel's book 3-Manifolds, he discusses two-sided submanifolds: if $N$ is a manifold of dimension $n$, and $M$ is a submanifold of dimension $(n-1)$, then $M$ is two-sided if there is an ...
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3answers
49 views

Equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$

Find the equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ and passes through the point $(1,1,1)$. First I ...
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1answer
49 views

Get 4 points of a rectangle inside a plane

There is a plane defined by 3 points in 3D space. I need to get the 4 edges points of a rectangle that: Lies on the plane. Has Width and high of w and h. Centered at the Z axis. One of it side is ...
1
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1answer
52 views

Find the equation of the plane through a given point, with given normal vector [duplicate]

I need to find the equation of the plane through the point $(−1,.5, 3)$ with normal vector $𝐢 + 4𝐣 + 𝐤$. I know that the equation will look something like this: $1 (x + 1) + 4 (y - .5) + 1 (z - 3) =...
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2answers
26 views

Solving vector equations of planes

Find the line of intersection of two planes denoted by: $r=\overrightarrow{b}+\lambda(\overrightarrow{b}-\overrightarrow{a})+\nu(\overrightarrow{a}+\overrightarrow{c})$ $r=\overrightarrow{c}+\alpha(\...
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3answers
22 views

Vector equation to line equation

I've not read vectors in math yet but I'm done with those in physics. I want to find out a line equation from a vector equation. Say I've 2 points in 3D space: Point A with coordinate $(a, b, c)$ and ...
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1answer
43 views

Why do two different quaternions appear to have the same rotation?

When using a Quaternions I've noticed something I don't quite understand. If I'm rotation $\frac{\pi}{2}$ radians on the Y axis it goes from $[0,0,0,1]$ to $[0,\sqrt{2},0,\sqrt{2}]$. A rotation of $\...
2
votes
1answer
45 views

difference between 2 quaternions

I'm trying to calculate quaternions relative to a given orientation. It is easiest for me to explain my intentions by means of an example: Suppose you have a vector $v1=[0,0,1]$ and I want to rotate ...
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0answers
12 views

Dot Product of Position and Directiom Vectors

Assume that we have one normalised 3D vector (D) representing direction and another 3D vector representing a position (P). How can we calculate the dot product of D and P? If it was the dot product ...
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1answer
13 views

Retrieve Cubes that Share a Vertex [closed]

Suppose that we have a world with an infinite number of cubes, each of which are 1x1x1 in size and have integral coordinates, e.g. (1, 2, 3) Given that C is a cube at (XC, YC, ZC) and that P is a ...
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2answers
45 views

Matlab 3D plots

I am currently in high school and is writing a maths research paper on a calculus problem. In the conclusion, I would like to include a 3D plot of a function I found. It goes something like this: a, ...
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1answer
22 views

Find a vector from the origin that is a known length and is orthogonal to the plane defined by its endpoint and two other known points?

I have a mechanism that pivot on one point, which I'll call the origin, and is moved by pushing on two other points. These other points are not fixed on the mechanism, but I can compute where they ...
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0answers
16 views

How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
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0answers
19 views

Scaling 3D-Points depending on Groundtruth

I generated points of a trajectory of a plane, like: $$ \begin{array}{lcr} \text{x} & \text{y} & \text{z} \\ \hline 0.396950& -0.199959& -0.000336\\ 0.122995& -0.199975& -0....
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1answer
54 views

How to adapt “System of Circles” method to 3D for finding a sphere given 4 points?

I want to analyze (computational complexity & running time) of different approaches to determining a sphere in 3D given 4 points on its surface. To start I have been searching for different ...
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1answer
31 views

direction cosines to axis confusion

I'm asking for clarification on following question : Find the direction Cosines of AB and hence calculate the angle in degrees ,between AB and each of the positive coordinate axes. AB = = -5i,13j,-...
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1answer
25 views

Cutting a cube into pieces

A cube has four diagonal planes. Let them be $P_1, P_2, P_3, P_4$. $P_1$ and $P_2$ intersect at exactly two corners. The cube is cut by $P_1$ and $P_2$ diagonal planes. What are the volumes of the ...
2
votes
2answers
104 views

What's the mathematics behind 3D modelling? [closed]

I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too. I would like to learn what specific topics are behind it mathematically. As long as I ...
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0answers
10 views

Mathematical theory for equally distributed dipole structures with inner equilibration

I'm looking for a mathematical theory for equally distributed dipole structures with inner equilibration. I know, that there exist two magnetic clusters, where the north and the south poles equally ...
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1answer
36 views

How to find a point which lies at distance d on 3D line, given a position vector and direction vector?

I have a position vector $(p_x, p_y, p_z)$ and direction vector $(v_x, v_y, v_z)$. I need to find a point on along the direction vector which is at distance $d$ from $(p_x, p_y, p_z)$.
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1answer
47 views

Motion in 3D Space: Finding Velocity from Distance, Launch Angle

The question asks: A bullet is fired from the ground at an angle of $45°$. What initial speed must the bullet have in order to hit the top of a $130 m$ tower located $190 m$ away? (Recall that $g=...
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2answers
22 views

Find the equations of $x-y$, $x-z$ and $z-y$ planes.

Do the $y-x$ and $x-y$ planes have the same equations? I think that the equation of the $x-y$ plane can be $x+y+z=0$ or $x+y+z=4$ or $ax+by+cz=$ any real number and $a,b,c$ are arbitrary real numbers. ...
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1answer
500 views

Quaternion - Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope). And I am trying to calculate the angle of rotation around all the three axes. I have tried may methods but not getting the ...
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0answers
40 views

How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin

I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting. (I ...
2
votes
2answers
45 views

Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly

Quaternion has components X, Y, Z, and W. If you created a Quaternion with input being a 3D Vector representing the axis (X,Y,Z) and a floating point number representing the amount to rotate around ...
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2answers
38 views

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$ The equation represents the line of intersection of two planes. Using augmented matrix $$ \begin{bmatrix} 1 & 1 & 1 ...
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3answers
55 views

How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
2
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1answer
39 views

Equation of plane perpendicular to given plane

Find the equation of the plane which contains the line of intersection of the planes $x+2y+3z-4=0$ and $2x+y-z+5=0$ and which is perpendicular to the plane $5x+3y-6z+8=0$ By setting $z=0$ I found a ...
1
vote
1answer
27 views

Equation of line passing through origin

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\frac{\pi}{3}$ Now our required line should be $\frac{x}{a}=\...
1
vote
1answer
27 views

Rotation matrix between two similar cuboids using their upper sides ( and the planes defined by these sides)

I have two different images and with them an estimation of two planes ( defined in the same system). I would like to get the rotation matrix, quaternion or euler angles of a surface within this planes....
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0answers
45 views

Translate Pitch and Roll Angles of Object to those at different Yaw

I have been trying to find a method to translate the pitch and roll angles of one object to those of another connected object at a different yaw - i.e I have an IMU mounted on a quadcopter frame and a ...
1
vote
1answer
48 views

Conjugating rotation by another rotation

If $g ∈ \mathrm{SO}(3)$ is the rotation about axis $p$ by angle $α$, and $h$ is a rotation mapping $p$ to another line $q$, then $g$ conjugated by $h$ is the rotation about $q$ by the same angle $α$. ...
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1answer
52 views

Plane rotation: range of angles to produce all posible x'y' planes

Given an $(x, y, z)$ system I create a new system $(x', y', z')$ by applying two rotations $\theta$ and $\phi$. In the new system the $(x',y')$ plane, i.e.: the $z'=0$ plane, can be written as: $$ (...
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0answers
18 views

finding pixel coordinates

I'm trying to calculate pixel coordinates of 3d points Xw = [150 200 350] where R is given as \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}...
1
vote
1answer
49 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
1
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1answer
36 views

Does $(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$ represent a line or plane?

Does the equation $$(x,y,z) = (2,1,1) +s(-1,-1,-1) + t(2,-2,-2)$$ represent a line or plane? I claimed it is a plane, as the two direction vectors are not multiples and thus for any values of $s$ and ...
2
votes
1answer
47 views

Calculate sphere radius using two vector points.

Using accelerometers I have acquired two $3D$ vectors $V_1$, $V_2$ which both have $(x, y, z)$. Assume that these vectors are points ($P_1$ and $P_2$) on the surface of a sphere ($S$), so that the ...
2
votes
0answers
9 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?
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1answer
111 views

Give a geometric description of the following set of points

Give a geometric description of the following set of points: $x^2 +y^2 + z^2-8x+14y-18z>/= 65 $ So I completed the square and got the set to read: $(x-4)^2+(y+7)^2+(z-9)^2>/= 211 $ However ...
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0answers
50 views

can every object be represented mathematically?

I was just wondering if all 2D/3D objects/images/shapes could be represented by equations. For example, SpongeBob 2D curve and many more. How should I approach, as in, some theories that already exist?...
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2answers
72 views

Pizza Delivery along the shortest path

You are the Captain of the USS Gauss and you have been flying in the direction $2 i + j + k$ for quite awhile. You are currently at the point $(6, 3, 3)$. Your helper monkey Mojo just discovered ...
2
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1answer
38 views

rotation to quaternion matrix handeness

I've understand that quaternions do not have handness but rotation matricies derived from unit quaternions does. The following formula is given by wikipedia for quaternion to rotation matrix ...
4
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1answer
85 views

Complement of a knot that *isn't* rationally null-homologous

Let $K$ be a knot in a closed, oriented 3-manifold $Y$. It is a standard fact that if $K$ is (at least rationally) null-homologous, then $H_1(Y-K;\mathbb{Z})$ is isomorphic to $H_1(Y;\mathbb{Z})\oplus ...
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votes
1answer
33 views

Trouble understanding solution to exercise

Given: Right tetrahedron, find $\angle \alpha$, between surrounding edge(not sure if this is the right term in English, but those edges is AD, BD and CD). and the plane of the base, and $\angle \beta$...
2
votes
4answers
133 views

Plane of intersection of two spheres

What is the plane of intersection of spheres $$x^2+y^2+z^2+2x+2y+2z+2=0$$ and $$x^2+y^2+z^2+x+y+z-\frac{1}{4}=0$$ I am not sure of how to do this, i just subtracted the two equations and i got a ...
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1answer
25 views

Calculating rotations required to

I have a situation similar to a question asked on StackOverflow with the following image: I have an object that rests at the tip of the green arrow (point XYZ), and I'm using the following formulas ...
0
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1answer
28 views

Geometry problem with rectangular parallelepiped

Given right angled parallelepiped $ABCDA1B1C1D1$, with bases $ABCD$ and $A1B1C1D1$, which are squares with side $1$. if $\angle (B1C;D1A) = 60^\circ$ find the length of the surrounding edge (I'm not ...
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0answers
29 views

Project 4 cones onto a sphere

I have four cones. The angle of each cones is 140 degree. I need to project it onto a sphere(place it ) such that, the cones cover the maximum area with minimum overlap. I initially thought that ...
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1answer
35 views

Trouble with understanding a solution to an exercise

Given right triangular prism $ABCA_1B_1C_1$, the surrounding edge(not sure if this is the right term in English, but the surrounding edge are $AA_1, BB_1, CC_1$) are equal to $\frac{\sqrt{5}}{5}$ and ...