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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41
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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 ...
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.
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? ...
16
votes
5answers
29k 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 ...
2
votes
1answer
2k views

Finding Rotation Axis and Angle to Align Two “Oriented Vectors”

In general, one can align a 3D vector $\vec A$ to another 3D vector $\vec B$ by rotating $\vec A$ around the axis $\| \vec A \times \vec B \|$ by the angle $\arccos{(\| \vec A \| \cdot \| \vec B \|)}$....
2
votes
1answer
175 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
5
votes
5answers
7k views

Rotating one 3d-vector to another

I have written an algorithm for solving the following problem: Given two 3d-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$. However, I am not sure if the following ...
2
votes
2answers
14k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 ...
1
vote
1answer
2k views

Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
2
votes
2answers
75 views

What makes the inside of a shape the inside?

A question occurred to me when browsing SE this evening, just curious. What specifies the inside of a 3D construct? If I have a hollow sphere, what's to say that the world isn't the inside, and the ...
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-...
5
votes
1answer
86 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 ...
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 ...
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.
4
votes
1answer
597 views

Automation of 3D Paper Modeling

I recently saw this creative paper contraption online this prior weekend and wanted to see if I could automate the process of creating all ~35 layers of an equation. Essentially what I would want the ...
2
votes
3answers
2k views

3D to 2D rotation matrix

I have been trawling through this forum but am struggling to understand the maths a bit. Currently I have a 2D plane within a 3D space and I have the coordinates for them. I want to work on this 2D ...
2
votes
1answer
580 views

Closed-form for eigenvectors of rotation matrix

For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?
1
vote
3answers
883 views

Identify and sketch the quadric surface?

I'm stuck trying to figure out which type of quadric surface this equation is: $$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$ I have narrowed it down to a hyperboloid, but cannot ...
1
vote
2answers
149 views

Calculating new vector positions

I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
0
votes
3answers
1k views

Formula for gallons in a trough

I have a trough which is a circular container. How do I determine how many gallons of water it takes to fill up the trough? I was thinking that we measure the height and the width but I think it's a ...
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 ...
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. ...
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 ...
5
votes
4answers
4k 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 ...
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 ...
4
votes
2answers
13k views

Line and plane intersection in 3D

How would one calculate the intersection of a line and a plane in 3D ? Given for example are 4 points which form a plane (x1,y1,z1)...(x4,y4,z4) and 2 different ...
2
votes
2answers
9k views

How to multiply vector 3 with 4by4 matrix, more precisely position * transformation matrix

All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one ...
4
votes
1answer
565 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 ...
4
votes
2answers
11k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
3
votes
2answers
436 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
3
votes
1answer
247 views

Finding ray direction with smallest angle from a ray to a rectangle in 3D

In 3D I have been given a ray and a rectangle: $$Ray=\mathbf{r_o}+t*\mathbf{r_d}$$ $$Rect=\mathbf{p_o}+p*\mathbf{p_x}+q*\mathbf{p_y}$$ where $\mathbf{r_o}$ is the ray origin $\mathbf{r_d}$ is ...
1
vote
1answer
72 views

Book on coordinate transformations

I am looking for a book that covers various coordinate systems in 3 dimensions, various methods of representing rotations and other transformations like rotation matrices and quarternions, including ...
0
votes
0answers
28 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
11
votes
5answers
811 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 ...
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? ...
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
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" ...
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
973 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 ...
4
votes
2answers
2k views

How to calculate volume of 3d convex hull?

Convex hull is defined by a set of planes (point on plane, plane normal). I also know the plane intersections points which form polygons on each face. How to calculate volume of convex hull?
4
votes
5answers
1k views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?
3
votes
1answer
1k views

Plane intersecting line segment

I have a plane which is represented as a 3d point $\vec{p}$ with a normal $\hat{n}$. I also have a line segment specified by two points $\vec{v_1},\vec{v_2}$ . I want to get the intersection point (if ...
3
votes
0answers
291 views

Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible - either one ...
3
votes
1answer
601 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
3
votes
1answer
1k 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 ...
2
votes
2answers
215 views

Is the value of $\pi$ in 2d the same in 3d? [closed]

I am starting with my question with the note "Assume no math skills". Given that, all down votes are welcomed. (At the expense of better understanding of course!) Given my first question: What is ...
2
votes
2answers
1k views

Calculate coordinate of any point on triangle in 3D plane

I am really stuck and can't find right way to write a formula(s) that will calculate Z coordinate of point on triangle plane in 3D plane. I know all coordinates of triangle points ( Ax, Ay, Az, Bx, ...
1
vote
1answer
487 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
1
vote
2answers
183 views

All Intersection points of two spheres having arbitary centres?

I have read much about intersection of two spheres from spheres-intersect , circlesphere and collision-points but all are based on the assumption of spheres located at origin or $x$-axis or some ...