Questions tagged [3d]

For things related to 3 dimensions. For geometry of 3-dimensional solids, please use instead (solid-geometry). For non-planar geometry, but otherwise agnostic of dimensions, perhaps (euclidean-geometry) or (analytic-geometry) should also be considered.

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How to calculate the 3-D area of a n-ogon?

Let n = # vertices represented by [x,y,z] coordinates. Apply, the general formula to the coordinates below letting n = 5 Where the points are: [13.28, -14.6, 90.35] [0.16, 8.12, 81.12] [-1.75, 8.35, ...
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Finding the inverse of an $xyz$ Euler transform. [closed]

So I am trying to find the inverse rotation for the Euler $xyz$ rotation $(-0.4,-0.5,-1.0)$. I actually need this for Blender ($3D$ software) (In the geometry nodes for those who know!) Blender ...
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Tangent Space View Direction based factor value remap

I'm trying to setup a mask similar to what Fresnel produces. Unfortunately Fresnel gives pretty bad results at grazing angles so I ended up using this : ...
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Need help in quantitatively modeling solubility of carbon dioxide in beer.

I'm trying to model a 3D surface which reflects the solubility of $CO_2$ in beer. An empirically derived chart is available at this link. Solubility is empirically related to the pressure of $CO_2$ in ...
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3-D Heat Transfer equation with internal loss generation (cylindrical coordinates)

I want to further ask about the general solution to the 3-D heat transfer equation with constant internal loss generation in the cylindrical coordinates, as follows. $\frac{1}{r}\frac{\partial \...
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How to find length of vertical line perpendicular with the surface made of plane diagonal?

I have a cube. Let the length of the side be 6, cube of ABCD.EFGH. Then i have diagonals, BE and CH. Then i have a line drawn from the midpoint AB to F, in front and back . The intersection between ...
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$3D$ Heat transfer equation with internal loss generation (zero boundary temperatures )

I am sincerely asking the analytical solution to the $3D$ Heat transfer equation with constant internal loss generation. I don't know how to find the particular solution for the Poisson equation. The ...
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Setting home orientation of quaternion/euler angles

Background I have a physical device with an IMU that emits a quaternion of it's current orientation. I refer to euler angles / quaternions interchangeably in my question. Question Regardless of the ...
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How to calculate curvature of a 3 dimensional Bezier Curve?

The formula for curvature of a 2D bezier curve is as follows: $κ(t)=\frac{|B′(t),B′′(t)|}{||B′(t)||^{3}}$ The dividend is the determinant of two joined vectors (2x2 where each vector is a column), but ...
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Is there any connection between waves, fractals and 4D objects?

I know that images can be analyzed as combinations of waves with Fourier transform. In such analysis "waves" mean spatial frequencies. There're a lot of videos about it, for example this one ...
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What is the general name for shapes like sphere and cone?

What is the general name of the shapes you get by spinning a 2d shape around 360 degrees? e.g spinning right triangle around either short sides 360 degrees to get a cone, spinning a circle around the ...
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Finding slant length of any given point for a cone with an angled base

I'm trying to write a program that prints out the flat pattern for a cone with an angled bottom when given its height, base diameter, and base pitch like in the diagram above. My approach has been to ...
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1 answer
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I am given a $3$D vector $v=[x,y,z]$ and an amount I want to move it, by some constant say $c=5.$

Say I am given a 3D vector $v = [x,y,z]$ and an amount I want to move it, some constant say $c = 5$. I would like to move the vector in the direction it is pointing (or oriented). Is there a simple ...
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How to draw a 3D circle that is tangent to two lines?

I have two 3D space lines on the same plane (M-R and N-R), and I have two known point on the individual line (M and N). the angle between two lines is unknown. And now I want to draw a circle that is ...
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Slicing a tesseract

A friend of mine was recently struggling with visualising a problem involving a cube that had been sliced into 2 equal parts, diagonally in all 3 dimensions. The result leaves a cut surface which is ...
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Finding locus of a point in a cylindrical body that travels tangent to a conical body

I have a cone of height h, small radius a and big radius b. the axis of the cone is along y axis, origin is at the center of small radius a, it expands along negative y. A plane parallel to z axis at ...
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A tough $3$ dimensional geometry problem

I just started practicing $3$ dimensional geometry problems but was stuck due to this probably indestructible question. I am a beginner in this field so please guide me and help me out with this rigid ...
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Can quaternions be converted into positional data

I have begun a basic incursion into understanding quaternions. I have read the following: https://en.wikipedia.org/wiki/Quaternion and https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation. I ...
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Fit rigid piecewise linear body to point cloud

I have a set of 3D points, each of which can be considered a point lying inside a rigid body of known dimensions. The actual object is straight cylindrical rods arranged as in the image below. The ...
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1 vote
1 answer
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All possible interpretations of a rotation matrix in euler angles

I have searched as much as i could here, but I couldn't find the answer. Don't hesitate to let me know if i have overlooked some post covering this.. Given a rotation matrix of an object in 3d space, ...
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Finding an oblique pyramid with a rectangular base which admits an inscribed sphere

I want to find the apex (or locus of the apex) of a pyramid with a given (known) rectangular base, that will have an inscribed sphere. To that end, I've written a computer code, to implement the ...
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1 vote
1 answer
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Is it possible to square vector like this?

I'm wondering if squaring using a dot product is correct. For example, let $a,b,c$ be vectors if $(a+b) = c$ then does squaring it mean that $(a+b) \cdot (a+b) = c \cdot c$ i.e does $(a \cdot a+2a \...
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How to rotate an object in-place along an arbitrary axis in three dimenstions

Consider the following cube floating in 3D space. How do I rotate it in-place along the axis specified by the vector $\vec{V}$? I know how to rotate a point about an arbitrary axis, but it doesn't ...
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2 votes
1 answer
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Calculating the volume of an arbitrary mesh

Given a mesh (vertices, edges, faces), I need to calculate the volume of the formed mesh. By volume, I mean the amount of space that a substance or object occupies. The resulting volume does not ...
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1 answer
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Determine equation of plane that intersects an ellipsoid which results in a specified eccentricity of the resulting intersection ellipse

Suppose you're given the ellipse $$ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $$ I would like to find the direction of the normal to the cutting plane to this ellipsoid that will ...
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Calculate remaining filament based on rotation

This might not be possible at all but I can't get my head around it, not really mathematically minded. A 3D printer spool obviously depletes as it is used. Knowing the diameter of the filament, the ...
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1 answer
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Why is the sum of cosines of angles defining a point [duplicate]

Consider a point P in 3D space which makes an angle $\alpha$ to the $x$-axis an angle $\beta$ to the $y$-axis and an angle $\gamma$ to the $z$-axis. The sum of the squares of the cosines of these ...
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A variable plane at distance $3p$ from the origin cuts the coordinate axes at $A$, $B$, $C$. Show that the locus of $\triangle ABC$'s centroid is ...

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of $\triangle ABC$ is $$\frac {1}{x^2}+ \frac {1}{y^2} ...
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1 answer
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What is the equation describing earth's orbit around the sun in 3 dimensional space?

I'm trying to draw a 2d ellipse in 3d space, which describes earth's orbit around the sun. such as image of a 2d ellipse in 3d space or the same image but different perspective. I'd like to be able to ...
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1 answer
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Converse of Parametric equation of plane

Suppose a plane passes through a point A ( whose position vector is a ) and parallel to two vectors b and c . Then , to any general point on plane with position vector r , I can find the equation of ...
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2 answers
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How to find the points that are in-between 4 planes

I have a set of points, I want to select only a few of those points. For that, I have 4 planes equations in the general form and I want to be able to check in a look if a given point would exist in ...
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1 vote
1 answer
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Sum of "angles" of a 3D tetrahedron

We know that the sum of angles of a triangle equals the straight angle (180 degrees). Can we convert a 2D theorem to 3D? e. g. We can generalize the triangle to a tetrahedron, angles of the triangle ...
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1 vote
3 answers
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Shadow of a rod

AB is a rod which is held such that $A=(1,-2,3)$ and $B=(2,3,-4)$ . A source of light is at the origin. Find the length of the shadow of the rod on a plane screen whose equation is $x+y+2z=1$ I ...
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1 vote
0 answers
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Jacobian of converting Euler angles to rotation vector or rotation matrix

please consider this paper: A Primer on the Differential Calculus of 3D Orientations - Bloesch 2016. Equations 27, 29 and 30, for example, give nice results about differentiating the rotation of a ...
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3 votes
1 answer
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Calculating the angle between 2 one-sided surfaces.

For a piece of software that I am writing, I need to determine the angle between two 3-dimensional one-sided surfaces that intersect at a line. The surfaces are defined as triangles with a normal ...
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Recovering matrix from rotated version

I'm dealing with matrices that came from a software, 3ds Max. It uses 4x3 matrices to represent transformations https://documentation.help/3DS-Max/idx_AT_matrix_representations_of_3d.htm $$\begin{...
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1 vote
1 answer
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Restoring third coordinate for triangle by its orthogonal projection and similar triangle

Suppose we have triangle $\Delta OAB$ lying on plane $z=0$ with coordinates $O(0,0,0), A(x_a,y_a,0), B(x_b,y_b,0)$ Also there is triangle $\Delta EFG$, but we know only coordinates of its orthogonal ...
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1 vote
1 answer
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Locus of middle points of the chords of conicoid which are parallel to $xy$ plane and touch the given sphere

I have the following conicoid before me: $ax^2+by^2+cz^2=1$ I have to find the locus of the middle points of the chords which are parallel to the plane $z=0$ and touch the sphere $x^2+y^2+z^2=a^2$. ...
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1 answer
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how to offset a rotation contained in a unit quaternion rotation from the origin of a rigid object.

I'm using Unity3D for a project. The way it handles sorting transformations is with a 3vector-unit quaternion-3vector "sandwich" (the 1st vector for position, the quaternion for rotation, ...
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2 votes
0 answers
33 views

Identify the intersection between a Quadric and a plane

I am considering a given quadric $ r^T Q r + b^T r + c = 0 $ and its intersection with a given plane $ n^T (r - r_0) = 0 $ and I want to identify the intersection curve between the two. My attempt: ...
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1 vote
1 answer
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Hyperbola dimensions - intersections cone and plane

I’d like some help to understand how to derive the relationship of the intersection of a cone with a plane parallel to the cone axis. See below dimensions of the shapes involved. What I want to ...
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0 answers
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Calculate normal vectors for each element of a grid in Python

How can I quickly calculate the normal vectors of each mesh of my grid? (grid is defined by the three matrices Mx, My & <...
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3 votes
1 answer
68 views

Volume of a triangular prism with 2 different bases

How do I arrive at a formula to calculate the volume of the following 3D shape? Does this shape have a proper name? It kind of looks like an irregular triangular prism with 2 similar triangles as ...
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0 votes
1 answer
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How to add shadow effect to 2D fractals?

My question is simple: how do you add a shadow effect (like the one in Kalles Fraktaler 2)? I have tried distance estimation, but have failed at creating it reliably. I would like a relatively fast ...
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3 votes
1 answer
74 views

How many distinct ways to flatten a cube?

Think of cutting open a cubical box with the smallest possible cuts to lay it flat. A cube has 12 edges and it seems in all the possible meshes, you have to cut along 7 edges. So, the most possible ...
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11 votes
2 answers
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What does a flattened Teserract look like?

This question is best asked with a picture: In words, we can flatten a cube into 2-d space and get a set of flattened squares like in the top right of the picture where five of the edges have stayed ...
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1 vote
0 answers
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Practical geometry problem: Maximum dimensions of a box that can be moved down a particular flight of stairs that make a U-turn

I really hope you can help me out with a (hopefully basic) very practical problem. I've bought a two floor house. The first floor is connected to the second with a double flies of stairs that make a U-...
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3 votes
2 answers
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Finding the vertex of the parabola parameterized by $p(t)=P_0+P_1t+P_2t^2$ for vectors $P_0, P_1, P_2$

A parabola can always be described in parametric form by position vector $p(t)$, $p(t) = P_0 + P_1 t + P_2 t^2 $ where $P_0, P_1, P_2$ are vectors in $2D$ or $3D$. I would like to prove that the ...
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0 votes
2 answers
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Given two points$ P_{1}(0, 0, 0)$ and $P_{2}(2, 2, 0)$ what is the plane equation equidistant from $P_1$ and $P_2$? [closed]

I have given two points $P_{1}(0, 0, 0)$ and $P_{2}(2, 2, 0)$ what is the plane equation equidistant from $P_1$ and $P_2$? How can I find this?
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1 vote
1 answer
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3D forms of a two variables separated function

I am looking for all the possible forms in 3D space of the function defined as $$ \Psi(x,t) = \psi(x) e^{-it}$$ There is this funny constraint: $$|\Psi (x,t)|^2 = \psi^{\ast}(x)\psi(x) e^{it} e^{-it}$$...
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