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Transform Spherical Coordinates to Cartesian Coordinates and Plot Analytically

This example shows how to transform a symbolic expression from spherical coordinates to Cartesian coordinates, and plot the converted expression analytically without explicitly generating numerical data.

In the spherical coordinate system, the location of a point P can be characterized by three variables. Different textbooks have different conventions for the variables used to describe spherical coordinates. For these examples, this convention is used:

  • The radial distance ρ

  • The azimuthal angle θ

  • The polar angle ϕ

The transformation of the point P from spherical coordinates (ρ,θ,ϕ) to Cartesian coordinates (x,y,z) is given by

x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕ.

By transforming symbolic expressions from spherical coordinates to Cartesian coordinates, you can then plot the expressions using Symbolic Math Toolbox™ graphics functions, such as fplot3 and fsurf.

Plot a Point and Its Projections

Plot the point P that is located at (ρ,θ,ϕ)=(1,1.2,0.75).

Transform the spherical coordinates to Cartesian coordinates (xP,yP,zP). Because the converted coordinates contain numerical values, use plot3 to plot the point.

rho = 1;
theta = 1.2;
phi = 0.75;
x_P = rho*sin(phi)*cos(theta);
y_P = rho*sin(phi)*sin(theta);
z_P = rho*cos(phi);
plot3(x_P,y_P,z_P,'ko','MarkerSize',10,'MarkerFaceColor','k')
hold on

Label each axis in the plot, change the line of sight, and set the axis scaling to use equal data units.

xlabel('x')
ylabel('y')
zlabel('z')
view([75 40])
axis equal;

Next, plot the line projection of the point P to the origin. This line projection in spherical coordinates is parameterized by (r,1.2,0.75), with r ranging from 0 to 1. Specify this line parameterization as symbolic expressions, and plot it by using fplot3.

syms r
xr = r*sin(phi)*cos(theta);
yr = r*sin(phi)*sin(theta);
zr = r*cos(phi);
fplot3(xr,yr,zr,[0 rho],'k')

Plot the vertical line projection to the xy-plane. This line projection in Cartesian coordinates is parameterized by (xP,yP,z), with z ranging from 0 to zP.

syms z
fplot3(sym(x_P),sym(y_P),z,[0 z_P],'k')

Plot the top horizontal line projection to the z-axis. This line projection in Cartesian coordinates is parameterized by (rsinϕcosθ,rsinϕsinθ,zP), with r ranging from 0 to 1.

fplot3(xr,yr,sym(z_P),[0 rho],'k--')

Plot the bottom horizontal line projection to the z-axis. This line projection in Cartesian coordinates is parameterized by (rsinϕcosθ,rsinϕsinθ,0), with r ranging from 0 to 1.

fplot3(xr,yr,sym(0),[0 rho],'k')

Figure contains an axes object. The axes object contains 5 objects of type line, parameterizedfunctionline.

Next, plot the plane that shows the span of the azimuthal angle θ in the xy-plane with the coordinate z=0.

syms s t
x_azimuthal = s*sin(phi)*cos(t);
y_azimuthal = s*sin(phi)*sin(t);
fsurf(x_azimuthal,y_azimuthal,0,[0 rho 0 theta],'FaceColor','b','EdgeColor','none')

Plot the plane that shows the span of the polar angle ϕ.

syms u v
x_polar = u*sin(v)*cos(theta);
y_polar = u*sin(v)*sin(theta);
z_polar = u*cos(v);
fsurf(x_polar,y_polar,z_polar,[0 rho 0 phi],'FaceColor','g','EdgeColor','none')
hold off

Figure contains an axes object. The axes object contains 7 objects of type line, parameterizedfunctionline, parameterizedfunctionsurface.

Plot a Sphere

Plot a sphere with radius r=4.

In spherical coordinates, the sphere is parameterized by (4,θ,ϕ), with ϕ ranging from 0 to π and θ ranging from 0 to 2π. Transform spherical coordinates to Cartesian coordinates by specifying the surface parameterization as symbolic expressions. Then plot the sphere by using fsurf.

syms phi theta
r = 4;
x = r*sin(phi)*cos(theta);
y = r*sin(phi)*sin(theta);
z = r*cos(phi);
fsurf(x,y,z,[0 pi 0 2*pi])
axis equal

Figure contains an axes object. The axes object contains an object of type parameterizedfunctionsurface.

Plot a Half Sphere

Plot a half sphere with radius r=4.

In spherical coordinates, the sphere is parameterized by (4,θ,ϕ), with ϕ ranging from 0 to π/2 and θ ranging from 0 to 2π. Transform spherical coordinates to Cartesian coordinates by specifying the surface parameterization as symbolic expressions. Then plot the half sphere by using fsurf.

syms phi theta
r = 4;
x = r*sin(phi)*cos(theta);
y = r*sin(phi)*sin(theta);
z = r*cos(phi);
fsurf(x,y,z,[0 pi/2 0 2*pi])
axis equal

Figure contains an axes object. The axes object contains an object of type parameterizedfunctionsurface.

Plot a Parameterized Surface

Plot a parameterized surface whose radial distance in spherical coordinates is related to the azimuthal and polar angles.

The surface has radial coordinates ρ=2+sin(5ϕ+7θ), with ϕ ranging from 0 to π and θ ranging from 0 to 2π. Transform spherical coordinates to Cartesian coordinates by specifying the surface parameterization as symbolic expressions. Then plot the parameterized surface by using fsurf.

syms phi theta
rho = 2 + sin(5*phi + 7*theta);
x = rho*sin(phi)*cos(theta);
y = rho*sin(phi)*sin(theta);
z = rho*cos(phi);
fsurf(x,y,z,[0 pi 0 2*pi])
view(45,50)

Figure contains an axes object. The axes object contains an object of type parameterizedfunctionsurface.