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.

learn more… | top users | synonyms

41
votes
13answers
73k 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 ...
31
votes
3answers
943 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?
21
votes
7answers
55k 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 ...
16
votes
5answers
28k 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 ...
14
votes
3answers
1k 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 $S$...
13
votes
7answers
61k 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. ...
13
votes
1answer
2k 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
4answers
441 views

Tricky 3d geometry problem

We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get? Astoundingly the answer is D. ...
12
votes
1answer
286 views

Every three of $n$ points is the vertices of an isosceles triangle. What is the max of $n$?

Suppose that we have $n\ (\ge 3)$ points in the three dimensional space and that every three of the $n$ points is the vertices of an isosceles triangle. Here, suppose that the vertices of an isosceles ...
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
240 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: http://www.korthalsaltes.com/model....
11
votes
5answers
809 views

Ellipsoid but not quite

I have an ellipsoid centered at the origin. https://en.wikipedia.org/wiki/Ellipsoid Assume $a,b,c$ are expressed in $mm$. Say I want to cover it with a uniform coat/layer which is $d$ mm thick ...
11
votes
5answers
377 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 ...
11
votes
0answers
577 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: ...
10
votes
1answer
1k views

Is the volume of a tetrahedron determined by the surface areas of the faces?

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 $V=\dfrac{S_1.h_1}{3}=\dfrac{S_2.h_2}{3}=\dfrac{S_3.h_3}{3}=\dfrac{...
10
votes
2answers
4k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-...
10
votes
2answers
13k 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 ...
10
votes
4answers
3k 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
250 views

How many spheres can fit in this box?

HASELBAUER - DICKHEISER TEST #15: What is the maximum number of one inch-diameter spheres that can be packed into a box ten inches square and five inches deep? My attempt to solve this: If i ...
10
votes
1answer
277 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?
9
votes
4answers
8k 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.
8
votes
7answers
17k 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} ...
8
votes
2answers
2k 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 $R_1 (\text{(axis)}_1, \text{(angle)}_1)$, $R_2 (\text{(axis)}_2, \text{(angle)}_2)$. I ...
8
votes
3answers
243 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 $(x+...
8
votes
4answers
186 views

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
8
votes
2answers
791 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 ...
8
votes
1answer
1k 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 ...
8
votes
2answers
2k 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 ...
7
votes
2answers
15k views

Formula to project a vector onto a plane

I have a reference plane formed by 3 points in $\mathbb{R}^3$ – $A, B$ and $C$. I have a 4th point, $D$. I would like to project the vector $\vec{BD}$ onto the reference plane as well as project ...
7
votes
2answers
119 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
7
votes
1answer
561 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 ...
7
votes
3answers
631 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
3answers
974 views

Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...
7
votes
2answers
1k 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 ...
7
votes
1answer
137 views

Photo image to find the screen orientation

I am trying to find the angle of tilts of a screen using projection of a circle from a source $S$. The light beam falls on the photo screen to expose it and what we get is an ellipse with major axis $...
7
votes
0answers
813 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 ...
6
votes
4answers
22k views

Calculate distance in 3D space

Imagine I want to determine the distance between points 0,0,0 and 1,2,3. How is this calculated?
6
votes
3answers
1k views

Why is wolfram alpha plotting this differently?

I have an equation for a cylinder as $x^2+(y-b)^2=a^2$ for some $a$ and $b$. so I just plugged in $b=2$ and $a=1$ and tried to plot it using wolfram alpha, and the 3D plot looked like half a cylinder, ...
6
votes
4answers
439 views

Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
6
votes
2answers
14k views

slope of a line in 3D coordinate system

Suppose I have $2$ points in a 3D coordinate space. Say $p_1=(5,5,5)$, $p_2=(1,2,3)$. How do I find the slope of the line joining $p_1$ and $p_2$? After getting the slope (which I assume will be an ...
6
votes
2answers
124 views

Sphere packing question

I'm a secondary school maths teacher, currently on my holidays working through some maths problems for fun. Here is one I have done, but it felt too easy, so if you could check if there's any mistakes,...
6
votes
1answer
81 views

Perfectly Round Sphere on Perfectly Flat Floor

I know that in practice it may be practically impossible to create the following situation, but suppose I did place a perfectly round ball of radius r (not that I think radius r is relevant) on a ...
6
votes
1answer
432 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
99 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
452 views

How to divide a $3$ D-sphere into “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
110 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
3answers
8k views

What is the formula for a 3D line?

Just like we have the formula $y=mx+b$ for $\mathbb{R}^{2}$, what would be a formula for $\mathbb{R}^{3}$? Thanks.
5
votes
7answers
23k views

Find if three points in 3-dimensional space are collinear

Find if the points joining $A=(6,7,1), B=(2,-3,1)$ and $C=(4,-5,0)$ are collinear. How to determine collinearity in three dimensions? In two dimensions, one can compare the slopes of segments $AB$ ...
5
votes
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 ...
5
votes
2answers
1k 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? ...