221 - Surfaces of Revolution

506 days ago by travis

A surface of revolution around a coordinate axis should be in one of the following forms:

$y^2 + z^2 = [r(x)]^2$, if revolved about the x-axis

$x^2 + z^2 = [r(y)]^2$, if revolved about the y-axis

$x^2 + y^2 = [r(z)]^2$, if revolved about the z-axis

One must be in one of these forms since a slice of a surface of revolution about an axis creates cross sections that must be circles of varying radii which is noted on the left side of each of these

# revolving about the z-axis @interact def _(color=selector(["red","green","blue"],label="Color",default="green")): var("x,y,z") r = 1/z G = implicit_plot3d(x^2+y^2-r(z)^2==0, (x,-2,2), (y,-2,2), (z,0.1,3), color=color) G.show(aspect_ratio=(1,1,1))

 Color red green blue

Click to the left again to hide and once more to show the dynamic interactive window

# revolving about the z-axis @interact def _(color=selector(["red","green","blue"],label="Color",default="green"),f = input_box(1/z)): var("x,y,z,r,theta,u,v") r = 1/z G = implicit_plot3d(x^2+y^2-r(z)^2==0, (x,-2,2), (y,-2,2), (z,0.1,3), color=color) G.show(aspect_ratio=(1,1,1)) # plotted in rectangular coordinates f=v # is this the reciprocal of the actual function given for r(z) with z replaced with v????? urange=(u,0,2*pi) vrange=(v,0.5,3) T = (r*cos(u), r*sin(u), v, [u,v]) plot3d(f, urange, vrange, transformation=T).show(aspect_ratio=(1,1,1)) # plotted in cylindrical coordinates

 Color red green blue f