spherical coordinates graphing calculator

Note that if x = 0, x = 0, then the value of θ θ is either π 2, 3 π 2, π 2, 3 π 2, or 0, 0, depending on the value of y. However, if we restrict θ θ to values between 0 0 and 2 π, 2 π, then we can find a unique solution based on the quadrant of the xy-plane in which original point ( x, y, z ) ( x, y, z ) is located. and r 2 = x 2 + y 2 These equations are used to convert from tan θ = y x rectangular coordinates to cylindrical z = z coordinates.Īs when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation tan θ = y x tan θ = y x has an infinite number of solutions. x = r cos θ These equations are used to convert from y = r sin θ cylindrical coordinates to rectangular z = z coordinates. and r 2 = x 2 + y 2 These equations are used to convert from tan θ = y x rectangular coordinates to cylindrical z = z coordinates. X = r cos θ These equations are used to convert from y = r sin θ cylindrical coordinates to rectangular z = z coordinates. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. 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. 2.7.4 Convert from rectangular to spherical coordinates.2.7.3 Convert from spherical to rectangular coordinates.2.7.2 Convert from rectangular to cylindrical coordinates.2.7.1 Convert from cylindrical to rectangular coordinates.Substituting the value of R we found earlier gives x = r*sin(ϕ)*cos(θ).įor y, we use similar logic to get y = R*sin(θ). Construct another triangle in the xy-plane with a hypotenuse of length R, and with an angle of θ between the hypotenuse and x-component.įor x, we find that cos(θ) = x/R. Now that we have the component of r in the xy-plane, we can find the x and y components. The component of r in the xy-plane, which I'll refer to as R, is given by sin(ϕ) = R/r. Then solve for z to find z = r*cos(ϕ).įor x and y, we first have to find the component in the xy-plane, then use θ to solve for the two coordinates. This is the angle between the hypotenuse of the triangle and its z-component.įor z, take cos(ϕ) = z/r. To do this, I find it easier to first find that ϕ is the angle of the triangle opposite the line segment in the xy-plane. To find the values of x, y, and z in spherical coordinates, you can construct a triangle, like the first figure in the article, and use trigonometric identities to solve for the coordinates in terms of r, theta, and phi.

Spherical coordinates graphing calculator