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 $$\rho$$

• The azimuthal angle $$\theta$$

• The polar angle $$\varphi$$

The transformation of the point $$P$$ from spherical coordinates $$(\rho ,\theta ,\varphi )$$ to Cartesian coordinates $$(x,y,z)$$ is given by

$$\begin{array}{l}\mathit{x}=\rho \sin \varphi \cos \theta ,\\ \mathit{y}=\rho \sin \varphi \sin \theta ,\\ \mathit{z}=\rho \cos \varphi .\end{array}$$

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 $$(\rho ,\theta ,\varphi )=(1,1.2,0.75)$$.

Transform the spherical coordinates to Cartesian coordinates $$(x_P,y_P,z_P)$$. 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 $$(x_P,y_P,z)$$, with $$z$$ ranging from $$0$$ to $$z_P$$.

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 $$(r\sin \varphi \cos \theta ,r\sin \varphi \sin \theta ,z_P)$$, 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 $$(r\sin \varphi \cos \theta ,r\sin \varphi \sin \theta ,0)$$, with $$r$$ ranging from $$0$$ to $$1$$.

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

Next, plot the plane that shows the span of the azimuthal angle $$\theta$$ 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 $$\varphi$$.

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

Plot a Sphere

Plot a sphere with radius $$r=4$$.

In spherical coordinates, the sphere is parameterized by $$(4,\theta ,\varphi )$$, with $$\varphi$$ ranging from $$0$$ to $$\pi$$ and $$\theta$$ ranging from $$0$$ to $$2\pi$$. 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

Plot a Half Sphere

Plot a half sphere with radius $$r=4$$.

In spherical coordinates, the sphere is parameterized by $$(4,\theta ,\varphi )$$, with $$\varphi$$ ranging from $$0$$ to $$\pi /2$$ and $$\theta$$ ranging from $$0$$ to $$2\pi$$. 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

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 $$\rho =2+\sin (5\varphi +7\theta )$$, with $$\varphi$$ ranging from $$0$$ to $$\pi$$ and $$\theta$$ ranging from $$0$$ to $$2\pi$$. 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)