This is a familiar problem recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Convert from rectangular to spherical coordinates.Convert from spherical to rectangular coordinates.Convert from rectangular to cylindrical coordinates.Convert from cylindrical to rectangular coordinates.$\rho=$ constant is a sphere of radius $\rho$ centered at the origin. Sphere of radius 3 centered at the origin. Most people don't have trouble understanding what $\rho=3$ means. Y &= \rho\sin\phi\sin\theta\label, we see it is only a half plane because $\rho\sin\phi$ cannot be negative. In summary, the formulas for Cartesian coordinates in terms of spherical coordinates are Since $r=\rho\sin\phi$, these components can be rewritten as $x=\rho\sin\phi\cos\theta$ and $y=\rho\sin\phi\sin\theta$. As $\theta$ is the angle this hypotenuse makes with the $x$-axis, the $x$- and $y$-components of the point $Q$ (which are the same as the $x$- and $y$-components of the point $P$) are given by $x=r\cos\theta$ and $y=r\sin\theta$. In the right plot, the distance from $Q$ to the origin, which is the length of hypotenuse of the right triangle, is labeled just as $r$. The cyan triangle, shown in both the original 3D coordinate system on the left and in the $xy$-plane on the right, is the right triangle whose vertices are the origin, the point $Q$, and its projection onto the $x$-axis. The distance of the point $Q$ from the origin is the same quantity. The length of the other leg of the right triangle is the distance from $P$ to the $z$-axis, which is $r=\rho\sin\phi$. As the length of the hypotenuse is $\rho$ and $\phi$ is the angle the hypotenuse makes with the $z$-axis leg of the right triangle, the $z$-coordinate of $P$ (i.e., the height of the triangle) is $z=\rho\cos\phi$. ![]() The pink triangle above is the right triangle whose vertices are the origin, the point $P$, and its projection onto the $z$-axis. We can calculate the relationship between the Cartesian coordinates $(x,y,z)$ of the point $P$ and its spherical coordinates $(\rho,\theta,\phi)$ using trigonometry. Lastly, $\phi$ is the angle between the positive $z$-axis and ![]() The angle between the positive $x$-axis and the line segment from the origin Point $Q$ is the projection of $P$ to the $xy$-plane, then $\theta$ is The coordinate $\rho$ is the distance from $P$ to the origin. Spherical coordinates are defined as indicated in theįollowing figure, which illustrates the spherical coordinates of the Relationship between spherical and Cartesian coordinates On this page, we derive the relationship between spherical and Cartesian coordinates, show an applet that allows you to explore the influence of each spherical coordinate, and illustrate simple spherical coordinate surfaces. The following graphics and interactive applets may help you understand sphericalĬoordinates better. But some people have trouble grasping what the ![]() If one is familiar with polar coordinates, then the angle $\theta$ isn't too difficult to understand as it is essentially the same as the angle $\theta$ from polar coordinates. Spherical coordinates determine the position of a point in three-dimensional space based on the distance $\rho$ from the origin and two angles $\theta$ and $\phi$. Spherical coordinates can be a little challenging to understand at first.
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