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Questions tagged [3d]

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.

2
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3answers
41 views

Intersection of 3 planes along a line

I have three planes: \begin{align*} \pi_1: x+y+z&=2\\ \pi_2: x+ay+2z&=3\\ \pi_3: x+a^2y+4z&=3+a \end{align*} I want to determine a such that the three planes intersect along a line. I do ...
1
vote
1answer
24 views

Writing a Cartesian Equation for a plane.

A plane $\pi_2$ intersects $\pi_1$: $4x-2y+7z-3=0$ at right angles. Two points lie on $\pi_2$: $A(3,2,0)$ and $B(2,-2,1)$. Write a cartesian equation for $\pi_2$. I know that the normals of these ...
-1
votes
0answers
21 views

how fast does the sun travel in one day [migrated]

I always wanted to know how fast the sun travels in the universe per day. But then to find this out, you need direction. There is no direction in space. So how are trying to determine speed and what ...
0
votes
0answers
30 views

Defining/formalizing right-handedness in $\mathbb{R}^3$

Let $a$ and $b$ be points in $\mathbb{R}^3$. Let $o$ be the origin. Let $\theta\in[0,\pi)$ be the angle between $\overrightarrow{oa}$ and $\overrightarrow{ob}$. Formally, let: $$\theta=\cos^{-1}\frac{...
0
votes
0answers
40 views

Angle between lines in 3D space

If direction cosines of two lines satisfy the equations $$l+m+n=0$$and $$amn+bnl+clm=0$$then show that the angle between them is $$\frac{\pi}{3}$$ if $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$$ My ...
0
votes
1answer
20 views

finding the locus of the foot of the perpendicular

For the curve $r=(a\cos(t), a\sin(t), at)$, show that the locus of the feet of the perpendicular from the origin to the tangent is a curve that completely lies on the hyperboloid $x^2+y^2-z^2 = a^2$. ...
1
vote
0answers
51 views

What planar concave polyforms bound 3D space?

A square can bound a cube or a heptacube if a cube is glued to each face. An equilateral triangle can bound a tetrahedron and other shapes. The diamond (2 equilateral triangles) can make a ...
1
vote
0answers
19 views

Projective Transformation between two matched 3D point sets

I have two point sets of about 800 3D-points that are matched. That means, I know the corresponding points in the two point clouds. Now I want to minimize the distance between the corresponding points ...
1
vote
2answers
55 views

Finding the “simplest” rotation transform

I am trying to undo some simple rotation errors of my sensor by tracking a reference plane. I am using the following accepted solution: Calculate Rotation Matrix to align Vector A to Vector B in 3d? I ...
1
vote
3answers
61 views

3D rotation about an arbitrary axis (3d Math Primer)

I am reading a 3d math primer book and I don't understand the following paragraph. Please Help me. I have been stuck on this for 2 days. Notice that $\mathbf w$ and ${\mathbf v}_\bot$ form a 2D ...
0
votes
0answers
18 views

3D Cube rotation possibilities [closed]

A cube of 40mm has its centre at the origin and its edges paralell to the axes x,y and z. If this cube has rotations of 90 degrees about the axes, the set of possible rotations also forms a group. ...
2
votes
3answers
53 views

Combining two rotations

I'm working on a project, where I have to perform rotations of a point which is on the surface of a sphere of radius 1, around the center of the sphere. In order to do so, I have a function, let's ...
0
votes
3answers
38 views

Find sphere equation

How to find sphere equation with center in plane $\pi =\{x-5y+z-2=0\}$ and tangent to plane $\alpha =\{ 2x+3y-4z-1=0 \}$ at $M(1,1,1)$ Where I've stopped: Knowing the sphere equation is: $$(X-a)^2 + ...
0
votes
0answers
23 views

Dragging an object on a plane with respect to the camera

this is a long description but I hope the solution is simple: I have a 3D pointCloud with a box (displayed at 0,0,0) and I want to drag the box on the XY plane with my mouse cursor (Zero movement on ...
0
votes
2answers
53 views

Finding the locus of centroid of ∆ABC [closed]

Please help me to solve question no. 12 I don't know what to do. Just guide me please Thank you! I am just known to the formula of a point from plane I m unable visualize and analyse this problem ...
0
votes
2answers
38 views

Conventions about z axis

I was doing an exercise to learn about parametrization and I stumbled upon one that I thought had no answer, it asked for the parametrization of the curve formed by the intersection of $z=\sqrt(x^2+y^...
0
votes
1answer
17 views

Showing the direction cosines of line perpendicular to two lines direction cosines

The question is :- if $l_1$, $m_1$, $n_1$ and $l_2$, $m_2$, $n_2$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of ...
1
vote
0answers
13 views

area of a concave spherical polygon with radius 1

In class today our professor showed us the general area formula for the area of a convex spherical polygon with radius 1, which is $\text{area}(\text{spherical polygon}(\theta_{1},...,\theta_{n}))=\...
1
vote
1answer
34 views

Parametric equation of a circle on a 3D plane [duplicate]

How can I compute points along a circle that lays on a plane given the following: center $P$ = $(a, b, c)$ radius $r$ plane equation $2x - 8y + 5z = 18$ I need to find points along this circle from $...
1
vote
0answers
20 views

Curve tracing : paraboloid [closed]

I know the equation $ x^2+y ^2=-z $ is paraboloid along negative z axis passing through origin...but what if there is absolute constant term in this equation... eg. $ x^2+y ^2=-z+2 $ what type of ...
4
votes
0answers
75 views

Minimal surfaces for planar octagons and nonagons

4, 6, 8 triangles can make a tetrahedron and up. 6, 8, 9, 10 quadrilaterals can make a cube and up. 12, 16, 18, 20 pentagons can make a tetartoid or dodecahedron and up. 7, 8, 9, 10 hexagons can make ...
1
vote
1answer
30 views

Closed Graph from a Collection of Vertices

Say I have two types of vertices: (D) vertices have two edges (T) vertices have three edges Given a collection of these vertices, is there any way to prove whether they can be joined edge-to-edge to ...
0
votes
5answers
48 views

Equation of the line passing through $(3,-2,-5)$ and $(3,-2,6)$

Find the Cartesian equation of the line passing through $(3,-2,-5)$ and $(3,-2,6)$ in $3$D. The equation of the line through the points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is given by $$ \vec{r}=\vec{...
2
votes
2answers
54 views

Intuition behind directional derivative (geometrically)?

The directional derivative in the direction <0, 1> is < ∂f/∂x, ∂f, ∂y> ⋅ <0, 1>. This makes perfect sense because this comes out to taking ∂f/∂x and multiplying it to 1, and adding it to (∂f/∂...
0
votes
0answers
14 views

Distance formula of skew lines for intersecting lines

The distance between two skew lines $\vec{r}=\vec{a}_1+\lambda\vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu\vec{b}_2$ is given by: $$ d=\bigg|\frac{(\vec{a}_2-\vec{a}_1).(\vec{b}_1\times\vec{b}_2)}{|\vec{b}_1\...
0
votes
1answer
53 views

AIME I 2000 Problem 8: Calculating the height of the liquid given Fraction of the Volume

As the title says, I'm looking at problem number 8 from AIME I 2000. https://artofproblemsolving.com/wiki/index.php?title=2000_AIME_I_Problems/Problem_8 I'm currently looking at solution 3 and I ...
0
votes
1answer
31 views

How to transform Parabola to 3D graph and lift the vertex upon a vertical line (z axis)?

If I have a quadratic in the form of $f(x) =ax^2+bx+c$, how would I be able to transfer this onto a 3D Graph where $x$=horizontal, $y$=the other 2D dimension and $z$= height , and raise the turning ...
0
votes
1answer
55 views

How to get bounding box around objects?

I have coordinates for different geometric shapes like rhombus, polygon, rectangle and etc. I need exact coordinates for bounding box around them. My coordinates for points 1,2,3,4 & 5 can start ...
0
votes
0answers
19 views

Rotate Point Between Coordinate Systems

I have a point(s) P0(x,y,z)...Pn(x,y,z) and a plane described by two vectors and a normal. V1(x,y,z) ...
0
votes
4answers
96 views

Derive Equation of Plane passing through the intersection of two planes

How to derive the equation of the plane passing through the intersection of two given planes. Lets say we have given two planes $$ \pi_1 : \vec{r}.\hat{n}_1=d_1\\ \pi_2 : \vec{r}.\hat{n}_2=d_2\\ $$ ...
1
vote
2answers
67 views

Additional solutions and generalizations of an 8th grade Olympiad problem about a cube

This year, I was the author of the following problem which came at the county level of the Math Olympiad at 8th grade: Consider cube $ABCDA^\prime B^\prime C^\prime D^\prime$ with faces $\square ...
1
vote
1answer
34 views

calculating the location of a point relative to another origin

There are two sources of 3-dimensional coordinates, the second source can be traversed from the first origin at any angle and direction, and we also have a point with its coordinates relative to the ...
2
votes
0answers
108 views

Oriented bounding box vs. enclosing cylinder

Say we have a 3D cloud of dots. An oriented bounding box (OBB) can be constructed around them. It usually has a longest edge. Let's call a line parallel to the longest edge and passing through the ...
1
vote
1answer
34 views

Unprojecting a 2D point to 3D space on a plane with perspective.

I'm not a good mathematician, but I'm trying to unproject 2D screen coordinates to a plane in a 3D space with perspective. I first do an uniform scaling on the 3D scene. Then I rotate around X axis, ...
1
vote
2answers
26 views

How to tell which faces of a convex solid are visible.

I have a series of faces that describe a 3-d solid. If I draw these faces, I've drawn the solid (my light source is at infinity). Except, for most of the solids I'm drawing (platonic solids and ...
-1
votes
2answers
33 views

Displace $3$D vertices along a $2$ dimension plane using normals

I have this one triangle with arbitrary vertices positioned in a 3D space. It's also easy to find the normal for it with the plane equation. There is a very simple method for moving the vertices ...
0
votes
1answer
59 views

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola My attempt: let $\frac {x}{l} = \frac {y}{m} = \frac {z}{n}$ ...
2
votes
1answer
35 views

Can a manifold locally isomorphic to 3-sphere have an unbounded geodesic?

Consider a manifold, which is locally isomorphic to 3-sphere (three-dimensional sphere). Does it have to have the topology of 3-sphere? In particular, can it have a geodesic, which doesn't loop on ...
6
votes
1answer
87 views

The Dihedral Constant Center of a Tetrahedron

For opposing edges in a tetrahedron, define $p\otimes q = p^2 + q^2 + 2 p q \cot(\angle p)\cot(\angle q)$, where $\angle p$ is the inner dihedral angle of edge $p$. In tetrahedron ABCD, $AB\otimes ...
0
votes
1answer
57 views

Plotting XYZ coordinates, given an array of distances to n 3D vertices.

I believe this is more of a mathematical topic than a gaming one, hence the post here. For the sake of context: In a game called Subnautica, the player has the ability to move about in an underwater ...
0
votes
0answers
30 views

Checking for Skew Lines

How do we check given two vectors, say $a_1\hat{i}+b_1\hat{j}+c_1\hat{k}$ and $a_2\hat{i}+b_2\hat{j}+c_2\hat{k}$ are skew lines ? I understand that skew lines are those which are neither intersecting ...
0
votes
1answer
34 views

3d Rotation Calculation

I have a known 3d rotation matrix (orthogonal) for rotating an object, created from right, up, forward, and position vectors: | $r_1$ $u_1$ $f_1$ $p_1$ | | $r_2$ $u_2$ $f_2$ $p_2$ | | $r_3$ $...
0
votes
1answer
28 views

Z Intersection of Cylinder and Plane Base On Angle Around Cylinder

I'm trying to find the (x, y, z) intersection of a cylinder and a plane given a cylinder aligned to the z axis and a plane rotated around the y axis like that shown below. The other parameter ...
5
votes
1answer
110 views

Fermat Point of a Tetrahedron

Here's a curious set of vertices for a tetrahedron: {{-22, -25, 4}, {-12, 15, -6}, {8, 5, -6}, {18, -15, 24}} The Fermat point of a tetrahedron minimizes the total distances from the point to the ...
0
votes
3answers
69 views

Finding Point on Plane Closest to Curve

I was given the problem: Find the point on a plane $4x+5y+z=1$ that is closest to $(23,0,0)$ I am very confused on how to approach this - any ideas?
-2
votes
1answer
22 views

How to find parameters of 3D plane in new frame of reference?

Say I have a 3D plane in frame $A$ parametrized by $ax + by + cz = d$. Now I also have a frame $B$ and I know the $4\times4$ transformation matrix between $A$ and $B$. How do I obtain the parameters ...
0
votes
0answers
18 views

Find the differential equation of family of cones with vertex at $(\alpha, \beta, \gamma)$

Find the differential equation of cones with vertex at $(\alpha, \beta, \gamma)$. My attempt Equation of cone with vertex at (0,0,0) is $ax^2+by^2+cz^2=0$ Now shifting origin to new origin/vertex of ...
0
votes
2answers
40 views

nonlinear 3d system

I am trying to find all the critical points of my 3D Nonlinear System described as follows: \begin{array}{ll} \\ \dot{x}_1=\mu-x_1^2+x_3+x_2-2x_1 \quad (1)\\\\ \dot{x}_2=\mu-x_2^2+x_3+x_1-2x_2 \quad (...
2
votes
2answers
52 views

Flatten 3D triangle while maintaining edge lengths

I currently try to flatten a triangle in 3d space while maintaining the edge lengths. The triangle consists of 3 vertices, all with x,y,z coordinates and is drawn clockwise. The second vertex yields 1 ...
0
votes
0answers
56 views

Projection of a plane onto a surface of sphere

What is the effect on the plane equation of projecting a plane onto the surface of a sphere. Assume we are starting with the standard point-normal plane equation: $$ax + by + cz + d = 0$$ or ...