What is Cartesian coordinates: Definition and 90 Discussions
A Cartesian coordinate system (UK: , US: ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.
Given a cartesian coordinate system with a fixed point of origin and three axes, it is a fact, that the coordinates of a point P change, when the coordinate system is rotated around its point of origin. The distance between the origin and point P is of course unaffected by such a rotation. What...
So given the Helmholtz equation $$\nabla^2 u(x,y,z) + k^2u(x,y,z)=0$$ we do the separation of variables $$u=u_x(x)u_y(y)u_z(z)= u_xu_yu_z$$ and ##k^2 = k_x^2 + k_y^2 +k_z^2## giving three separate equations; $$\nabla^2_x u_x+ k_x^2 u_x=0$$ $$\nabla^2_y u_y+ k_y^2 u_y=0$$ $$\nabla^2_z u_z+ k_z^2...
It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
Problem : We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##.
Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using (plane) polar coordinates.
(1) In (plane)...
I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?
As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics.
To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below.
$$x=\rho \sin \phi \cos \theta$$
$$y= \rho \sin \phi...
i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right?
The surface occupied by the cylinder can be described by all vectors ##\vec x =(x,y,z)## so that...
I was trying to construct locally Euclidean metrics. Consider the sphere with the usual coordinate system induced from spherical coordinates in ##\mathbb R^3##. Consider a point ##p## in the Equator having coordinates ##(\theta_0, \phi_0) = (\pi/2, 0)##. If you make the coordinate change ##\xi^1...
Summary: I can't figure out how the solver carries out the conversions from cartesian to cylindrical coordinates and vice-versa.
I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of...
Problem Statement: How do you calculate the rotational inertia of a hollow sphere in cartesian (x,y) coordinates?
Relevant Equations: I=Mr^2
My physics teacher said its his goal to figure this out before he dies. He has personally solved all objects inertias in cartesian coordinates but can't...
In section 1-5 of the third edition of Foundations of Electromagnetic Theory by Reitz, Milford and Christy, the authors give a coordinate-system-independent definition of the divergence of a vector field:
$$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...
I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system).
I need to integrate this function over a part of spherical shell of radius R. This part is...
I have 2 points expressed in (latitude,longitude) and I want to calculate the angle with respect to the north pole.
Since the two points are very near (like hundred of meters), is it possible to consider the two points in the carthesian system simply as:
x=longitude
y=latitude
Then...
I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates.
A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
Hello,
I'm having a visualisation problem. I have a point in R3 that I want to rotate about the ##y##-axis anticlockwise (assuming a right-handed cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows:
$$z =...
Homework Statement
Homework EquationsThe Attempt at a Solution
I am having a bit of trouble visualizing the vectors for these type of problems. The angles they give are very ambiguous and so I am not sure why they are there. For the 45 degree angle, how do i know that this is used for finding...
Hello, I have 6 equations in Cartesian coordinates a) change to cylindrical coordinates b) change to spherical coordinate
This book show me the answers but i don't find it
If anyone can help me i will appreciate so much!
Thanks for your time1) z = 2...
Relations between vectors in cylindrical and
Cartesian
coordinate systems are given by
\vec{e}_{\rho}=\cos \varphi \vec{e}_x+\sin \varphi \vec{e}_y
\vec{e}_{\varphi}=-\sin \varphi \vec{e}_x+\cos \varphi \vec{e}_y
\vec{e}_z=\vec{e}_z
We can write this in form
\begin{bmatrix}...
Homework Statement
It's just an example in the textbook. A vector in cylindrical coordinates.
A=arAr+aΦAΦ+azAz
to be expressed in cartesian coordinates.
Start with the Ax component:
Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax
ar⋅ax=cosΦ
aΦ⋅ax=-sinΦ
Ax=ArcosΦ - AΦsinΦ
Looking at a figure of the unit vectors I...
Homework Statement
Homework EquationsThe Attempt at a Solution
the answer given is the same but without the negative sign, I don't understand because the crossproduct of unit vectors
when using a Cartesian coordinates of the directions given by the right-hand rule? Is the positive z...
Suppose we wish to use Cartesian coordinates for points on the surface of a sphere. Then all derivatives of the metric would vanish and so the Riemann curvature tensor would vanish. But it would give us a wrong result, namely that the space is not curved. So it means that if we want to get...
I am trying to convert this polar equation to Cartesian coordinates.
r = 8 cos theta
I type the equation into wolfram alpha and it gives me a graph, but no Cartesian points.
If somebody could help me find the cartesian points, I would appreciate it.
Thank you.
First, I'd like to say I apologize if my formatting is off! I am trying to figure out how to do all of this on here, so please bear with me!
So I was watching this video on spherical coordinates via a rotation matrix:
and in the end, he gets:
x = \rho * sin(\theta) * sin(\phi)
y = \rho*...
Homework Statement
Hi everybody! I would like to discuss with you a problem that I am wondering if I understand it correctly:
Find expressions for the cartesian components and for the magnitude of the angular momentum of a particle in cylindrical coordinates ##(r,\varphi,z)##.
Homework...
Hi all,
i am having a problem with question 3, as its not clear if i should use the Z value for the camera as 15 or 25 m or... Could you suggest me. Cheers
The goal of this project is to obtain some understanding of the camera’s motion in space.
Also, based on the camera’s motion, we will...
Ok, so randomizing three random variables, X, Y and Z, each from a standard normal distribution, then plotting these in an ordinary cartesian coordinate system gets me a spherically symmetric cloud of points.
Now I want to create this cloud having the same probability distribution but by using...
Hello! Good morning to all forum members!
I am studying general relativity through the wonderful book: "General Relativity: An Introduction for Physicists" by M.P. Hobson (Cambridge University Press) (2006). My question is about Riemannian manifolds and local cartesian coordinates (Chapter 02 -...
Homework Statement
Provide an expression in Cartesian coordinates for a plane wave of amplitude 1 [V/m] and wavelength 700 nm propagating in u = cosθx + sinθy direction, where x and y are unit vectors along the x and y-axis and θ is the measured angle from the x axis.
Homework Equations...
What is the physical interpretation of Cartesian coordinates in GR? Say, e.g., a system centered at the center of a spherical mass. What are x,y, and z physically, i.e., how are they measured?
Homework Statement
Consider a unit sphere centered at the origin. In terms of the Cartesian unit vectors i, j and k, find the unit normal vector on the surface
Homework Equations
A dot B = AB cos(theta)
A cross B = AB (normal vector) sin(theta)
Unit sphere radius = 1 The Attempt at a...
Homework Statement
Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field.
Homework Equations
From wolfram alpha a Beltrami field is defined as v x (curl v)=0
The Attempt...
The usual change of variables in this case (mentioned in the title of this topic) is this:
##x = rcos(t)##
##y = rsin(t)##
When I rewrite (say my integral) in polar coordinates I have to change ##dxdy## to ##rdrdt##
My question is why can't I just compute dx and dy the usual way (the already...
Hello
I have this problem - From a generator, I get a compton scattering with the electrons theta and phi angles.
where I having the following equations for a particle
px = E_particle * sin (theta) * cos (phi);
py = E_particle * sin (theta) * sin (phi);
pz = E_particle * cos (theta)...
Homework Statement
The problem and its solution are attached as TheProblemAndTheSolution.jpg.
If you don't want to view the attached image, the cartesian-coordinate version that the problem wants me to convert to a polar-coordinate version is the following (let "int" = "integral").:
int int (1...
Homework Statement
A cannon shoots a ball at an angle θ above the horizontal ground. (a) Neglecting air resistance, use Newton's second law to find the ball's position as a function of time. (Use axes with x measured horizontally and y vertically.) (b) Let r(t) denote the ball's distance...
I have a function to plot the orbits of planets based on their orbital elements (Semi-major Axis, Eccentricity, Argument of periapsis, Inclination, and longitude of ascending node). I have the x and y coordinates working great using only the semi-major axis, eccentricity, and argument of...
How to translate r = 2 /(2 - cos(theta)) to cartesian coordinates:
so far:
r = 2 /(2 - cos(theta))
r = 2 /(2 - cos(theta)) |* (2 - cos(theta)) both sides
r (2 - cos(theta))= 2
2*r - rcos(theta) = 2 | know x = rcos(theta)
2*r - x...
Hello
how can Convert Cartesian coordinates to spherical with shape?
for clear my question i explain a way to convert my coordinates in different spherical.
for example i use this diagram to convert Cartesian coordinates to Cylindrical(with image to axises)
for example:
now how can i do...
Homework Statement
It's given that absolute permitivity is a coordinate function: ε (x, y, z) = Asin(x)cos(y), where A=const
Homework Equations
We need to find an electrostatic field potential function \varphi in Cartesian coordinate system.
The Attempt at a Solution
I tired to solve, but...
Homework Statement
Hi
The expression for the magnetic field from an infinite wire is
\boldsymbol B(r) = \frac{\mu_0I}{2\pi}\frac{1}{r} \hat\phi
which points along \phi. I am trying to convert this into cartesian coordinates, and what I get is
\boldsymbol B(x, y) =...
I was given the problem r=2sin(2(θ)). I'm supposed to write the equation in the Cartesian Coordinates. I understand the basics to this but I'm not really sure how I'm supposed to write the equation when I have x=2sin(2(θ))cos(θ) and y=2sin(2θ)sin(θ).
Homework Statement
An airplane flies at an air speed of 300 miles per hour, in the direction toward southwest. There is a head wind of 75 mi/hr in the direction toward due east.
(A) Determine the ground speed.
(B) Determine the direction of motion of the plane, expressed as an angle...
Ok I have a quick question. I have this problem that is doable with polar coordinates and triple integrals but I was wondering if it would be possible to do this problem in the cartesian coordinate system (odd question I know...).
Homework Statement
A sprinkler distributes water in a circular...
Homework Statement
Homework Equations
The Attempt at a Solution
Do you see that 2 between A and the integral? There's no 2 in the above equation. I don't see where that 2 came from. Everything else is fine.
Homework Statement
The magnetic field around a long, straight wire carrying a steady current I is given in spherical coordinates by the expression
\vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi} ,
where \mu_{o} is a constant and R is the perpendicular distance from the wire to...
we all know that triple integral can be solved by either cartesian coordinates , spherical ,or cylindrical coordinates
i just need like some advice in knowing when the variable used is constant and when it is not
for example : r in cylindrical coordinates can it be constant or not?? because i...
1. The position vector of a particle at time t ≥ 0 is given by r = sin(t)*i + cos(2t)*j. Find the cartesian equation for the path of the particle.
2. I was told that the answer is:
y = 1 - 2x^2
But I don't know how to obtain that solution.
3. r = sin(t)*i + cos(2t)*j
At first I...
Homework Statement
The following equation describes a surface in spherical coordinates. θ =pi/4
Write the equation in the cartesian coordinates?
that is, (r,θ,Ø) to (x,y,z)
Homework Equations
x=rsinθcosØ
y=rsinθsinØ
z=rcosθ
r=sqrt(x^2+y^2+z^2)
θ=cos^-1(z/r)
Ø=tan^-1(y/x)
The...