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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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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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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3 dimensions and movement [closed]

There are three points in the three dimensional space. Those points must always touch each other. There is a hypothetical sphere with the coordinates where those three points meet as its center and ...
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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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Perpendicular vector in a plane in spherical coordinates system

Let's suppose we have a unit tangent vector $\mathbf{\hat{t}}$ along a curve at point $\mathbf{P}$. We can construct a plane perpendicular to this unit vector which passes through $\mathbf{P}$. I need ...
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Finding point on a 3-dimensional surface at a fixed given distance from a starting point [closed]

I'm in a situation in which I have a starting point of known coordinates (x, y, z) and a 3-dimensional vector starting from that point of known components (and of known length L). I would like to know ...
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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 votes
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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1 vote
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Identify the intersection between a Quadric and a plane

So, here 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 ...
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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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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
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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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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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2 votes
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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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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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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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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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4 votes
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If a polyhedron's faces and vertex figures are convex, is the polyhedron convex?

Suppose a polyhedron's faces are convex polygons, and its vertex figures are convex spherical polygons (or convex cones, depending on definitions). Must the polyhedron be convex? The polyhedron may be ...
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Find the best fitting plane given n points that goes through the origin

My Problem I need to find the plane which best fits a given number of points (at least 3) and that must contain the origin. I'm supposed to do this with the least squares method, but i can also use ...
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Is there widely accepted notion of tangent plane(or similar) for a vertex of a polyhedron?

For a vertex v of a polygon, I think it is reasonable to define its tangent line as ...
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polyhedrons congruent if faces are all congruent + same connection status?

In 2 dimension, a unit square and a unit rhombus(of certain angle) has the same list of edges and connection status. But the unit square and the unit rhombus are not generally congruent. Can we find ...
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Identify the hyperboloid of one sheet that contains three skew lines on its surface

Suppose you are given three lines in parametric form in $3D$, described as follows $Q_i(t) = P_i + t\ d_i ,\ t \in \mathbb{R},\ i = 1, 2, 3 $ where $P_i$ is a point on the $i$-th line and $d_i$ is the ...
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How to find the expected value of an exponential curve?

I have a function of the form $$ y = \left(\frac{a}{b}\right)^{1/T} - 1$$ where $T$ is a future point in time, such that $$0 < T \leq \infty.$$ Question 1: Given constant values for $a$ and $b$, ex:...
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Show that $E,F,G,H,I$ lies on a sphere. [closed]

Let $ABCD$ a tetrahedron and $E,F,G,H,I,J$ on $AB,BC,CA,CD,DA,DB$ s.t. $AE\cdot EB=BF\cdot FC=CH\cdot HD=DI\cdot IA=AG\cdot GC=DJ\cdot JB$. Show that $E,F,G,H,I$ lies on a sphere. I know that is ...
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Offset a point on a curve in 3D space

I have a curve AB in 3D space in which I know the start A(x,y,z) and end point B (x,y,z). Now, I have a point O (x,y,z) which should be moved along the curve for some distance (D). If it's a straight ...
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0 votes
1 answer
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The locus of a line in 3D passing intersecting two other lines

A line through any point on the curve $x^2-y^2=1 , z=0$ intersects two lines $y=x, z=1$ and $y=-x, z=-1$. Required is the locus of the line . Just to clarify(as I have seen people question), the above ...
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0 votes
1 answer
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Is there a better way to find the "outline" path for a set of points in $3D$ space?

I am trying to create a $3D$ model. In $3D$ model file formats, a face is defined by ordering the vertices of that face clockwise as viewed from the center of the object. I have a script that ...
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0 votes
1 answer
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Rotate a body to align z axis in a particular direction

I have the orientation of a body in world frame, $_wP_b$. Let us say, a bottle with the z axis representing the height, lying on its sides on the table. Now, I have a direction vector $\textbf{v}$, ...
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1 answer
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In how many different ways can I draw them?

I have a cube and I draw a vertex, a middle of an edge and a diagonal of a face. In how many different ways can I draw them? Two cubes can look similar after a rotation. I don't know how to start.
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Inverse of a roto-translation matrix in 3D space

I want to create two roto-translation matrices. The first transforms point $P$ into point $P'$ by performing a translation $T=(x_t, y_t, z_t)$ and two rotations (one around the $x$ axis of $\alpha$ ...
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1 vote
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In any 3d object, how do you calculate the rotation of that object so the visible part with most surface area is pointing to the camera/eye?

Example: Take a sphere and cut it in half. If you point the curved part of the half sphere to the camera its the visible part with most surface area
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-1 votes
2 answers
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Possible to find rotation around x, y, z axes by knowing the polar angles $\phi, \theta$?

I'm working with a 3D cartesian system $\vec{e'_x},\vec{e'_y},\vec{e'_z}$ that moves around in a global coordinate system. I know the origo position and $\vec{e'_z}$ in global coordinates. There is no ...
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1 vote
1 answer
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Equation of bisector between two straight lines given in symmetrical form [closed]

Please help in solving the attached question. I know in 2D, it is solved as $\dfrac{ax+by+c}{\sqrt{a^2+b^2}}=\pm \dfrac{px+qy+s}{\sqrt{p^2+q^2}}$ Not sure if we use the same formula for 3D? And if yes,...
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How to get 3D coordinates (XYZ) of the sun given Azimuth and altitude/elevation?

I want to create an Augmented Reality(AR) application to show the sun's position where I need the 3D(XYZ) coordinates to plot in Augmented Reality(AR). Main question: I need to calculate the 3D ...
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0 votes
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Rotating a vector to make it parallel to the z axis

I have to rotate the $P=(P_x,P_Y,P_z)$ point so that the $OP$ vector is parallel with respect to the $Z$ axis. To do this I perform 2 rotations, the first around the $Z$ axis and the second around the ...
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Approximate 3D function that has certain points as peak

Given set of 3D points, how can I find a continuous function that has each point as the peak? You can imagine making a terrain editor that the inputs are 3D points and the output is a smooth mountain ...
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3 votes
2 answers
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Conditions on tetrahedron side lengths that guarantee existence of a sphere tangent to all its edges

I recently worked on the problem of finding a sphere tangent to the edges of an irregular tetrahedron. I found that if one of the triangular faces of the tetrahedron is an isosceles triangle, then it ...
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0 votes
1 answer
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Rotation of a frame reference with axis z lying on normal n

I am creating a rotation matrix capable of converting the reference system $A=(x_a, y_a, z_a)$ into a second reference system $B=(x_b, y_b, z_b)$ where the $x_b$ and $y_b$ axes lie in a plane with ...
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1 vote
0 answers
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Rotate a Translation and a Quaternion around the z axis of an arbitrary pose by an angle theta

I need to implement a rotation in a program but it's 15 years I haven't used rigid body motion maths. I use poses that are described by a translation T and quaternion Q. Everything is expressed in the ...
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2 answers
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Angle between two planes in a cube

Find the cosine of the angle between the planes $(A,B,C,D)$ and $(M,N,K)$ in the cube $ABCDA_1B_1C_1D_1$ where $M,N$ and $K$ are the midpoints of $BB_1,A_1B_1$ and $B_1C_1$, respectively. As we can ...
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2 votes
2 answers
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Computing Euler angles between two 3D points from Cartesian coordinates

We are given the three-dimensional cartesian coordinates of a point $A$, a point $B$ and a point $C$. The distance from $A$ to $B$ is the same as the distance from $A$ to $C$ ($|\vec{AB}| = |\vec{AC}|)...
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Reidemeister Torsion Example with Lens Spaces

In Andrew Ranicki's notes on Reidemeister torsion (https://www.maths.ed.ac.uk/~v1ranick/papers/torsion.pdf), he gives the following example with lens spaces: where I copied the beginning of the ...
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0 votes
1 answer
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Find a point on triangle and interpolated triangle normal which points to specific point in 3D

Suppose i have triangle in 3d with vertices $v1, v2, v3 \in R^3$. Each triangle vertex has associated normal vector $ n1, n2, n3 \in R^3, ||n_i|| = 1$. In computer graphics such vectors sometimes ...
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What is the face to face height of a Acute Golden Rhombohedron?

I am doing the CAD for a Acute Golden Rhombohedron, and I want to form it by having 2 parallel Golden Rhombus, and offset one of them by the long axis (X), and the face to face axis(Z). I am following ...
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0 votes
1 answer
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Calibrating a pinhole camera (finding $z_0$)

A pinhole camera is a very simple theoretical device for generating perspective images on a plane that a distance $z_0$ from the pinhole (a point) and whose normal vector is the direction vector at ...
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0 votes
0 answers
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Locating vertices of a known triangle in $3D$ from a single image

Suppose you have a labelled triangle with known side lengths, and you take one image of this triangle using a known pinhole camera (i.e. the focal length is known), from a point with known coordinates,...
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0 votes
1 answer
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Locating a point in $3D$ using two perspective images

Suppose I am given a point $P(x, y, z)$ where $x, y, z$ are unknown. I take two images (perspective projections) of this point using a simple pinhole camera with an unknown focal length $f$, from two ...
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