Articles of 3d

Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? http://biochemistry.utoronto.ca/steipe/abc/images/2/2b/CubeBasic.jpg I think I can just compute $\mathbf{n}\cdot e_{\mathbf{x}},\mathbf{n}\cdot e_{\mathbf{y}}$ and $\mathbf{n}\cdot e_{\mathbf{z}}$ and the area would just be the some of the 3 […]

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 which is minimum? That distance is thus but what are the points where it is achieved?

Quaternion – Angle computation using accelerometer and gyroscope

I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope) and am trying to calculate the angle of rotation around all three axes. I have tried many methods but am not getting the results as expected. Methods tried: (i) Complementary filter approach – I am able to get the angles using the […]

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 have provided equations in vector form which is far from my ability to understand. Pre-requisite: http://paulbourke.net/geometry/circlesphere/spheresphere1.gif Equation for intersection of two spheres having centres […]

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 algorithms for conversions between various coordinate systems and representations of transformations. Is there a single book that covers these.

Number of variables and dimension of a function

Why is a function $f(x)$ called a single-variable function if it has coordinates represented by $x$ and $y$? Can it be called a 1D function if its plot is 2D? Subsequently, can two-variable functions $f(x,y)$ (that have coordinates $x,y,z$) be called a 2D function if its plot is 3D? This is confusing and potentially ambiguous. […]

Finding the parameters of an ellipsoid given its quadratic form

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the angles of rotation. For the 2D case I found an answer here. I was wondering if someone can help me do the same for 3D.

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

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 meant by the perimeter of a Sector Why is the value of $\pi$ not exactly $3$? why is it $3.14$………. or […]

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 profile has to be an isosceles triangle rather than equilateral, rectangular rather than square, or elliptical rather than […]

How to find line parallel to direction vector and passing through a specific point?

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the line and the vector? Please give me some direction as where to go for this question. I am so lost