MATLAB: How Do I Perform a Surface of Revolution of a 2D Plot that is not a Function

cylinderMATLABparametricrevolutionvolume

If I have a 2D plot that is defined by two separate functions, such that the resulting plot is not a function, how do I create a surface of revolution of the plot by rotating that curve around the x-axis? For example, consider the plot generated by executing the following:

x = -1:.01:1;
plot(x,acos(x)+4)
grid on
hold on
plot(x,-acos(x)+4)

Best Answer

The function "cylinder" can generate a surface based on a profile by rotating that profile around the x-axis and then aligning the resulting surface with the z- axis:

https://www.mathworks.com/help/matlab/ref/cylinder.html

However, the cylinder function treats the profile that is being passed into it as having monotonically increasing x-values; i.e. that the profile is a function. Thus, when you are trying to rotate a curve that is like a sideways parabola (which is not a function) it will not produce the desired result.

In order to visualize this surface, as well as get the X,Y,and Z coordinates of what the resulting surface would be, you need to parameterize the function in terms of two new variables, "t" and a "theta". This allows you to define an "R(t,x)" that one can use to generate the desired Y and Z values for the resulting surface of revolution.

In this example, you define a "t" to go between 0 and 4. This is because in this curve, the x-values go from -1 to 1 and back to -1. This accounts for a total "distance" traveled of 4. You then define your "x" values in terms of "t". R(t,x) in this case will be "acos(x) + 4" for "t" values less than 2 and will be "-acos(x) + 4" for values of "t" greater than 2.

Once you have your x(t) and R(t,x) defined all that is left is to define the Y and Z coordinates of the surface revolution. This is done by defining a new variable "theta" that goes in between 0 and 2*pi. At every x value, there is a corresponding R(t,x) value that represents the curve at that point in 2D space. When we rotate that around in 3D space, we have Y and Z values that are defined as follows (given to us by basic trigonometry):

Y = R(t,x) * cos(theta)

Z = R(t,x) * sin(theta).

Thus, you have a way to get your X,Y, and Z coordinates. From there, you can use the "surf" function to visualize the surface, or simply use the them for other sorts of processing.

Please see the attached script for more details.

Related Solutions

MATLAB: How to get the new position coordinates of the object after I rotate it using the HGTRANSFORM function

The new position coordinates can be formed by applying the rotation matrices to the initial coordinates. An example of how you can determine the new positions follows:

First, the cone is graphed and transformed using the 'hgtransform' function, and the values of the 'xdata', 'ydata', and 'zdata' properties are queried.

ax = axes('XLim',[-1.5 1.5],'YLim',[-1.5 1.5],'ZLim',[-1.5 1.5]);
view(3); grid on; axis equal
[x y z] = cylinder([.2 0]);
h = surface(x,y,z,'FaceColor','red');
xlabel('x'); ylabel('y'); zlabel('z');
t = hgtransform('Parent',ax);
set(h,'Parent',t)
set(gcf,'Renderer','opengl')
drawnow
x_temp = get(h,'xdata');
y_temp = get(h,'ydata');
z_temp =get(h,'zdata');
Rz = eye(4);
Sxy = Rz;
r = pi;
Rz = makehgtform('xrotate',r);
% Sxy = makehgtform('scale',r/4);
set(t,'Matrix',Rz*Sxy)
drawnow
% end

Then, the transform matrix is used along with the 'maketform' and 'tformfwd' functions of the Image Processing Toolbox to perform the transformation on the coordinates.

T = maketform('affine',Rz)
[xx(1,:), yy(1,:), zz(1,:)] = tformfwd(T, x_temp(1,:),y_temp(1,:),z_temp(1,:));
[xx(2,:), yy(2,:), zz(2,:)] = tformfwd(T, x_temp(2,:),y_temp(2,:),z_temp(2,:));

If you do not have the Image Processing Toolbox, this operation can be performed with a for-loop and standard matrix multiplication.

%calculate new coordinates by multiplying by the rotation matrix.
%       
%  Rz * old_matrix = new_matrix       
%  Where old_matrix =[x_pos;y_pos;z_pos;1]
% 
for i = 1:21
new_first_row(i,:) = (Rz* [x_temp(1,i);y_temp(1,i);z_temp(1,i);1])';
end
for i = 1:21
new_second_row(i,:) = (Rz* [x_temp(2,i);y_temp(2,i);z_temp(2,i);1])';
end
xx = new_first_row(:,1)';
xx(2,:) = new_second_row(:,1)';
yy = new_first_row(:,2)';
yy(2,:) = new_second_row(:,2)';
zz = new_first_row(:,3)';
zz(2,:) = new_second_row(:,3)';

You can now use the coordinates returned by the transformation to create the surface. Comparing this figure to the original figure, you can see that the transformation is as expected.

figure; ax = axes('XLim',[-1.5 1.5],'YLim',[-1.5 1.5],...
'ZLim',[-1.5 1.5]);
view(3); grid on; axis equal
[x y z] = cylinder([.2 0]);
h(1) = surface(xx,yy,zz,'FaceColor','red');
xlabel('x'); ylabel('y'); zlabel('z');

MATLAB: How to set time-varying boundary conditions using the Partial Differential Equation Toolbox

You can use the Partial Differential Equation Toolbox to solve PDEs with time-varying boundary conditions. The following example describes how to define these boundary conditions :

1. Open the PDETOOL GUI by enetering

pdetool

at the MATLAB command prompt.

2. From the PDE menu, select PDE Specifications. In the dialog box, click Hyperbolic. Then click OK to close the dialog box.

3. From the Boundary menu, select Boundary Mode.

4. Double-click one of the boundaries in the figure.

5. In the Boundary Condition dialog box, enter values for the boundary condition parameters. The default values are constants. You can enter any function of time for these parameters, e.g. "sin(t)". You can also enter user-defined functions, e.g. "myBoundary(t)". Note that the function argument must be named "t". Any function you use must

a. Return a scalar for each value of "t" passed to it.
b. Return NaN for any "t" value of NaN.

An example of such a function is shown below:

function f = myFunc(t)
if isnan(t)
      f=NaN;
else
      f = t;
end

6. From the Solve menu, select Solve PDE.

7. You can visualize the time-varying boundary condition as follows:

a. From the Plot menu, select Parameters.
b. Select the Animation checkbox, and click Plot.

You can make the boundary condition a function of space as well as time by specifying a boundary condition such as "myFunc2(t,x,y)" in the Boundary Condition dialog box. The function "myFunc2" must

a. Return NaN for a "t" value of NaN.
b. Return a vector the same size as "x" or "y" for every numerical value of "t".

An example of such a function is shown below:

function f = myFunc2(t, x, y)
if isnan(t)
      f=NaN;
else
      f = (x+y) * t;
end

Best Answer

Related Solutions

MATLAB: How to get the new position coordinates of the object after I rotate it using the HGTRANSFORM function

MATLAB: How to set time-varying boundary conditions using the Partial Differential Equation Toolbox

Related Question