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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28
votes
3answers
868 views

will 3 lights illuminate convex solid

Can 3 lights be placed on the outside of any convex N dimensional solid so that all points on its surface are illuminated?
14
votes
8answers
22k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
13
votes
3answers
958 views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that ...
13
votes
1answer
951 views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
12
votes
7answers
25k views

Recommended (free) software to plot points in 3d

I am looking for (preferably free) software to: 1) plot 3d points read from a file. A scatter plot would be fine. 2) Optionally color the points by a property - also read from the file It would be ...
12
votes
2answers
1k views

Overlap volume of three spheres

Given three spheres and their coordinates with equal radii that are known to have a triple overlap (a volume contained within all three spheres), is there a known closed form for the calculation of ...
12
votes
2answers
194 views

Mathematical description of a corncob

I'm trying to figure out how I can make a paper model of the corncob water tower in Rochester, Minnesota for my N-scale train layout. The best I can find is this: ...
11
votes
5answers
15k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
10
votes
4answers
2k views

Find whether two triangles intersect or not in 3D

Given 2 set of points ((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) and ((p1,q1,r1),(p2,q2,r2),(p3,q3,r3)) each forming a triangle in 3D space. How will you find out whether these triangles intersect or not? ...
10
votes
1answer
257 views

What is $f(x, y) = |x| - |y|$ called?

$f(x,y)=x^2-y^2$ is your friendly neighbourhood hyperbolic paraboloid. $f(x, y) = |x| - |y|$ naturally has similar appearance. Do shapes of the latter form have a name?
10
votes
0answers
307 views

Visualizing a Calabi Yau

I would like to understand how I can visualize the quintic threefold $$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$ For a similar problem, Hanson proposes the following: ...
9
votes
5answers
326 views

Does this interesting property characterize a sphere?

Consider 2-d surfaces in 3-d (at the suggestion of a comment, let's say closed connected 2-dim smooth manifolds, embedded in dimension 3) with finite area. A sphere has the interesting property that ...
8
votes
3answers
227 views

Eating a cake from the inside.

Imagine that you are in the centre of a cube of cake with a known size. In order to move you must eat the surrounding cake but you can only move within the restraints of the six obvious directions ...
8
votes
2answers
6k views

How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?

I googled around a bit, but usually I found overly-technical explanations, or other, more specific Stackoverflow questions on how 3D computer graphics work. I'm sure I can find enough resources for ...
8
votes
2answers
471 views

Coloring a sphere with minimum colors (with constraints)

This is a problem we've been considering in our undergraduate math club, and I thought it would be nice to get further thoughts on the subject. I will start with a two dimensional case and then ...
7
votes
1answer
710 views

To find the volume of tetrahedron by using all surfaces areas?

I am looking for a formula: $V=f(S_1,S_2,S_3,S_4)$, where $S_1$, $S_2$, $S_3$, and $S_4$ are the areas of the four faces. We know ...
7
votes
2answers
1k views

Combining Two 3D Rotations

Every rotation in 3D space can be defined by a rotation axis and an angle. Now let's say we have two rotations R1(axis1, angle1), R2(axis2, angle2). I remember that Rotation operator is closed under ...
7
votes
3answers
442 views

Rigid-body matching algorithm and clustering algorithm with groups of lines in 3D

I've been struggling with this problem for weeks, and couldn't find an appropriate algorithm to solve it. Could you guys please give me some advices or suggestions in addressing this question. Or if ...
7
votes
2answers
595 views

Cross section is a regular hexagon.Is it a cube?

One of the cross sections in a rectangular box is a regular hexagon.Prove that the box is a cube I tried to prove that certain lengths were equal by showing that certain triangles are congruent but ...
6
votes
4answers
3k views

Rotating one 3-vector to another

I have written an algorithm for solving the following problem: Given two 3-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
6
votes
1answer
844 views

3D Rotation Matrix Uniqueness

Given a 3D rotation matrix R in a basis B. Can we consider R as being unique in B? Is there any other 3d rotation matrix R' representing the same 3D rotation in B? How could I prove that? Note: I do ...
6
votes
1answer
403 views

Torus: Circle cut

Given a Torus $T$ with major and minor radius $R$ and $r$, respectively, I can obtain a circle lying in $T$ by cutting $T$ with a bi-tangential plane. Now I don't want circles, but Tori with major ...
6
votes
1answer
77 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
6
votes
2answers
1k views

Averaging quaternions

Given multiple quaternions representing orientations, and I want to average them. Each one has a different weight, and they all sum up to one. How can I get the average of them? Simple multiplication ...
6
votes
2answers
404 views

How to divide a $3$ D-sphere in “equivalent” parts?

My goal is to put $n$ points on a sphere in $\mathbb{R}^3$ to divide it in $n$ parts, so that their disposition would be as "equivalent" as possible. I don't exactly know what "equivalent" ...
6
votes
0answers
104 views

Analytic caustics for 3D objects

Is it possible to efficiently calculate caustics for a given 3D object, like a torus, or a cube? To be more precise: let's assume that we have a 3d torus, resting on a 2d plane and a single light ...
5
votes
4answers
3k views

Can the Surface Area of a Sphere be found without using Integration?

When we were in school they told us that the Surface Area of a sphere = $4\pi r^2$ Now, when I try to derive it using only high school level mathematics, I am unable to do so. Please help.
5
votes
2answers
774 views

Tranforming 2D outline into 3D plane

I am writing a program where I would like to allow the user to draw 4 connecting lines, such as: And convert this shape into a 3D plane. Is this possible? Is there an existing algorithm to do so? ...
5
votes
4answers
2k views

What's the best 3D angular co-ordinate system for working with smartphone apps

This is very much an applied maths question. I'm having trouble with Euler angles in the context of smartphone apps. I've been working with Android, but I would guess that the same problem arises ...
5
votes
1answer
80 views

Find all such functions defined on the space

$f:\mathbb{R}^3\to \mathbb{R}^{\ast}$ is such that for any non-degenerate tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : $$f(O)=f(A)f(B)f(C)f(D) $$ Prove that $f(X)=1$ for ...
5
votes
3answers
792 views

Largest Triangle with Vertices in the Unit Cube

How would one find a triangle, with vertices in or on the unit cube, such that the length of the smallest side is maximized? And what is that length? A lower bound for the length is $\sqrt{2}$, by ...
5
votes
1answer
358 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
5
votes
3answers
318 views

move a point up and down along a sphere

I have a problem where i have a sphere and 1 point that can be anywhere on that sphere's surface. The Sphere is at the center point (0,0,0). I now need to get 2 new points, 1 just a little below the ...
5
votes
3answers
137 views

Which solids are characterized by their orthographic projections?

If I know the orthographic projections of a given solid in Euclidean 3-space onto the $xy$, $xz$ and $yz$ planes, under which circumstances can I reconstruct the solid based on that information alone? ...
5
votes
2answers
563 views

“Normalizing” Points on a Sphere

I have a set of points on a unit sphere representing different orientations: Now I need to apply rotation(s) such that the points will lay on the horizon as tightly as possible: The ideal ...
5
votes
1answer
505 views

Find minimum in a constrained three-variable equation

After my last question I have worked through the math quite a bit and now I'm stuck again. This time my question is less wordy. I have two equations for $t$, one with respect to each $a_{x}$ and ...
5
votes
0answers
30 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
5
votes
0answers
138 views

Random 3D points uniformly distributed on an ellipse shaped window of a sphere

How can I generate random points uniformly distributed on the surface of a sphere such that a line that originates at the center of the sphere, and passes through one of the points, will intersect a ...
5
votes
0answers
54 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
5
votes
0answers
307 views

How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?

I am trying to numerically solve the Possion PDE in cylindrical coordinate system. $$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
4
votes
3answers
20k views

how to calculate area of 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
4
votes
5answers
274 views

Moving on the surface of a cube

A $3 \times 3$ cube is composed of $27$, $1 \times 1$ cubes. Moving along the surface of the larger cube, how many ways are there to get from the closer top-left vertex, to the further bottom-right ...
4
votes
3answers
150 views

How do you detect if a point is in a plane?

Let's say we have 3 points: (-2,7,4), (-4,5,2), (3,8,5) and we want to see if a fourth point, (2,6,3), is in the plane that the previous 3 points made. How would I go about doing this?
4
votes
3answers
536 views

What is the tangent plane equation on the 3 spheres?

3 spheres are on $z=0$ plane and touch each other as shown in the picture. Coordinates of their centers are $O_1=(0,0,5),O_2=(0,y_2,3),O_3=(x_3,y_3,2)$. What is the tangent plane equation on 3 ...
4
votes
3answers
839 views

Analytically compute signed distance of ellipsoid

I'm trying to generate a 3d signed distance field for a origin centered ellipsoid. For a sphere this is pretty easy: $$\sqrt{x^2 + y^2 + z^2}-r$$ where $r$ is the radius. I'm not sure what the best ...
4
votes
1answer
564 views

Hyperbolic geometry. 3 dimensions. What is not well understood?

According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions. http://mathworld.wolfram.com/HyperbolicGeometry.html What isn't well understood about ...
4
votes
2answers
691 views

3d transformation two triangles

I have two triangles in 3d. I need to calculate transformation matrix(3X3) between two triangles in 3D. 1)How can I calculate the transformation matrix(rigid) while fixing one of the points to the ...
4
votes
1answer
101 views

Any interesting properties of Fermat's Last Theorem Surfaces?

I wonder if there are any interesting geometric (as opposed to number-theoretic) properties of what might be called Fermat's Last Theorem surfaces, i.e., $x^d + y^d = z^d$. Below are the surfaces for ...
4
votes
2answers
176 views

Rotation of a point in 3d space

I'm trying to rotate a point around a single axis of a 3D system. Given $P=\begin{pmatrix} 101 \\ 102 \\ 103 \end{pmatrix} $, And the rotation matrix formula for rotation around the X axis only, I ...
4
votes
1answer
353 views

Uniform distributions on the space of rotations in 3D

I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions. The Haar measure on $SO(3)$. The uniform ...