Articles of coordinate systems

In my textbook,the coordinate (x,y) by sine and cosine addition formula seems to form a circle,is that a coincidence?

In my textbook,the coordinate (x,y) by sine and cosine addition formula seems to form a circle,is that a coincidence ?Since the addition formula once was defined by acute case and to prove by geometry with acute case,why does the obtuse and reflex case also work to form a circle? @J.M-my high school trig book @joriki-it […]

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: 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.

Find the circle circumscribing a triangle related to a parabola

This question already has an answer here: A question on the parabola.. 1 answer

Is it possible to find function that contains every given point?

Let say we have a arbitrary number of given points and there is at least one function, for which every point lies on its graph. Is it possible to find that function using only X and Y coordinates of every given point? Example: We are given points $A(0,1)$; $B(0.27;0)$ and $C (3.73;0)$. All of these […]

How to determine the direction of one point from another, given their coordinates?

If I have the coordinates of two points, how would I determine what direction the second point lies in, relative to the first point? Specifically, I’m writing an application that involves basically drawing an arrow starting at a certain area on the screen, pointing in the direction of the mouse.

conversion of laplacian from cartesian to spherical coordinates

In cartesian coordinates, the Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\qquad(1)$$ If it’s converted to spherical coordinates, we get $$\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial}{\partial \theta}\left(sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^2 sin^2 \theta}\frac{\partial^2}{\partial \phi^2}\qquad(2)$$ I am following the derivation (i.e. the method of conversion from cartesian to spherical) in “Quantum physics of atoms, molecules, solids, nuclei […]

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point $A$ the Point $B$ moves in an arc upward (i.e. along positive $y$-axis) consistently maintaining the distance $L$ from point […]

What is the explicit equation that converts Cartesian coordinates to elliptical coordinates?

All of the solutions that I have seen so far require solving an implicit equation after substituting in whatever $x$, $y$, and $z$ are. What are the explicit equations for each elliptical coordinate in a two dimensions and three dimensions

Difference between the Jacobian matrix and the metric tensor

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial e_j}+\dfrac{\partial y}{\partial e_i}\dfrac{\partial y}{\partial e_j}+\dfrac{\partial z}{\partial e_i}\dfrac{\partial z}{\partial e_j}$). As far as I understand it is used when transforming the arc element ds from one coordinate system e.g Cartesian to another one e.g. cylindrical […]