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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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 ...
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2answers
29 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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1answer
17 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 ...
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1answer
38 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
25 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 ...
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2answers
89 views

Gradient of an angle in terms of the vertices

Let $\theta(\vec p, \vec q, \vec r)$ be the angle theta between 3D real vectors $(\vec{q}-\vec{p})$ and $(\vec{r} - \vec{p})$. What is a simple expression of $\nabla \theta$ in terms of $\vec{p}$, ...
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1answer
17 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 ...
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0answers
16 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 ...
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1answer
24 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
850 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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3answers
179 views

Volume of overlap between two convex polyhedra

I have two convex polyhedra represented by triangle meshes. I can easily determine if they are in contact or not, but when they are in contact then I would like to determine the volume of their ...
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3answers
115 views

Considering a convex polygon lying on a plane in 3D space, how can I know if a point on that plane lies inside or outside that polygon?

I have a plane in space and a polygon in it. I know the position of each vertices making the polygon. I also know the position of the point on the plane. How can I know whether the point is inside or ...
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1answer
552 views

Calculating normals for a polygon mesh (3D computer graphics)

I want to write a program to generate arches, a common architectural form, and export them to a wavefront object format for sharing with various three dimensional graphics editors. To do this, I need ...
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1answer
39 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. ...
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0answers
16 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 \\ ...
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1answer
29 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 ...
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1answer
33 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 ...
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0answers
8 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
28 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
35 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 ...
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1answer
5k views

How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i ...
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0answers
14 views

How can I find UV coordinate of a 3d plane? [closed]

I have four vertices, when the plane is a square ( example 2x2 ), the UVs will be: (0,0) (0,1) (1,1) (1,0) And it will look like this: enter image description here The plane could have it's vertices ...
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2answers
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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 ...
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1answer
29 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 ...
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1answer
69 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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7answers
11k views

How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
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4answers
52 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
30 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 ...
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0answers
975 views

Helix around helix parametric equation?

I know the parametric equation for a $3D$ helix is: $x = R \cos t$ $y = R \sin t$ $z = h t$ Can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix ...
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2answers
5k views

How to find line parallel to direction vector and passing through a specific point?

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the ...
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1answer
382 views

Counting the number of cubes in an isometric view

I had seen questions in a sample aptitude test, where an isometric view of an object made up of cubes was given, with some of the cubes removed. We were supposed to count the number of cubes present ...
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1answer
20 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 ...
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1answer
25 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
13 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
34 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 ...
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1answer
38 views

What is the initial velocity height of a projectile with destination vector D and gravity G?

I am doing a modification of Unreal Tournament 1999. Normally the game's jump pads' velocity applied to pawns that reach it's radius is defined by a velocity vector, which is a true pain to change and ...
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2answers
174 views

Catenary equation in 3D

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is lowest point of the catenary curve. I only know z-coordinate of this third point. I need to find ...
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1answer
379 views

Best fitting circle to points in 3D

I have a set of n ≥ 3 points in 3D that are measurements of a possible circle. The measured points are "noisy" so best-fitting algorithms are involved. I'm programming in C# and have put together some ...
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2answers
1k views

What is the line of greatest slope on a plane?

Let $P$ be a plane in $\mathbb{R}^3$ that is inclined (neither horizontal nor vertical). When considering lines lying on $P$, it is sometimes said "$L$ is a line of greatest slope of $P$". What is ...
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1answer
24 views

Locus of a point on a variable plane

A variable plane passes through a fixed point $(a,b,c)$ and meets the coordinate axes in A,B,C.The locus of the point common to the planes through $A,B,C$ parallel to coordinate planes is? Ok ...
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0answers
21 views

Given square pyramid, find all skew lines with line AB

Given square pyramid $ABCDE$, find all skew lines with line AB. Here is drawing: It's kind of obvious for me that those lines are $DE$ and $CE$, however I don't know how to prove it. Note: I can't ...
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2answers
21 views

What does the homogeneous system of equations represent under certain conditions?

Consider the following linear equations $ax+by+cz=0,bx+cy+az=0,cx+ay+bz=0$ 1) $a+b+c \neq o$ and $a^2+b^2+c^2=ab+bc+ca$ 2) $a+b+c \neq o$ and $a^2+b^2+c^2 \neq ab+bc+ca$ 3) ...
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5answers
422 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
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1answer
20 views

For the points A,B,C,D is given that C belongs to AB, M belongs to AD and D doesn't belong to AB, prove that the plane (ABD) is the same as (CDM)

For the points $A,B,C,D$ is given that $C$ belongs to $AB$, $M$ belongs to $AD$ and $D$ doesn't belong to $AB$, prove that the plane $(ABD)$ is the same as $(CDM)$. Here is drawing: I tried to prove ...
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2answers
41 views

Coordinates of circumcentre of an isosceles triangle in 3D

I have an isosceles triangle in 3D and I need to find the coordinates of the circumcentre of this triangle. I know the coordinates of the three vertices. One method I thought of is to solve equation ...
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0answers
43 views

Text book on solid geometry/stereometry, without involving analytic geometry

As the title says I'm searching for a textbook, about solid geometry, without involving analytic geometry. The material which the book should cover is the stereometry learned in the eastern bloc. An ...
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1answer
45 views

Obtain plane equation from the rotating angles that generated it

Consider an $(x, y, z)$ system where positive $x$ points to the right, positive $y$ points upwards, and positive $z$ points outside of the screen. I create a new system $(x', y', z')$ by applying two ...
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2answers
34 views

Prove that if through three given points two planes can be drawn, then infinitely many planes throught these points can be drawn.

Prove that if through three given points two planes can be drawn, then infinitely many planes through these points can be drawn. I don't get how this is possible, since there is unique plane passing ...
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1answer
34 views

How to extract an equation from transformation matrix multiplication?

I am trying to rotate a point in a 3D space in the 3 axis together around a specific origin point. Unfortunately I can't use matrices in my application,All I can do is just the basic math operations ...
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0answers
15 views

Check Vector3 points on one line using a Matrix

I know that for 3 Vector2 points (say points a, b, c) the determinant of the following ...