Unwrapping/unroll a truncated cone

Question

I want to unwrape a pillar (truncated cone shaped) which has cone shape. How can I do it.

Lower radius=1.8506 m (red in pic) Upper radius=1.6849 m (green in pic)

The coordinates of my pillar

center: Bottom circle : xyz: 0,0,0 Top circle : xyz: 0,0,24.6 Please see attached figure.

Cone apex: 0,0,275 (I just calculated) angle (half):0.38 projection radius=1.84 ( WHAT should I give)

The coordinates of cylinder, I can get

r=sqrt(x^2+y^2)
theta=atan(y,x)
z=z

How can I project them on plane.

Answer 1

There's probably many codes around on fileexchange or highly optimized solutions on the web (for texture projection for instance), anyway here is my solution:

I assume pillar data is a matrix of size [tcount x zcount] where tcount is the number of scan positions along angular axis and zcount is the number of scan positions along height axis.

So here is the code:

%
% PURPOSE:
%
%   Project cone data to plane data
%
% INPUT:
%
%   thetaValues: Angular-scan values (in radians !!)
%   zValues: z-scan values (whathever unit)
%   coneData: Scanned data of size thetaCount-by-zCount
%   zminRadius: Radius at zmin position (same unit as zValues)
%   zmaxRadius: Radius at zmax position (same unit as zValues)
%   xvalues: Values to have along x-axis (by default biggestRadius * thetaValues centered around theta = 0)
%
% OUTPUT:
%
%   xvalues: Positions along x-axis (same unit as zValues)
%   zvalues: Positions along z-axis (same values/unit as input)
%   planeData: Plane data of size xcount-by-zcount 
%
function [xvalues, zValues, planeData] = Cone2Plane(thetaValues, zValues, coneData, zminRadius, zmaxRadius, xvalues)
%[
    % Default arguments
    if (nargin < 6), xvalues = []; end

    % Init
    zmin = min(zValues);
    zmax = max(zValues);    
    smallestRadius = min(zminRadius, zmaxRadius);
    biggestRadius = max(zminRadius, zmaxRadius);

    % Ensure thetaValues range in -pi;+pi;
    thetaValues = mod(thetaValues + pi, 2*pi) - pi;
    [thetaValues, si] = sort(thetaValues);
    coneData = coneData(si, :);

    % Intercept theorem (Thales)
    %
    %      A-------+-------\
    %     /|       |        \
    %    / |       |         \
    %   D--E-------+----------\
    %  /   |       |           \
    % B----C-------+------------\
    BC = biggestRadius - smallestRadius; 
    AE = zmax - zValues(:);
    AC = (zmax - zmin);
    DE = (BC * AE) / AC;
    radiuses = smallestRadius + DE;

    % Projection
    if (isempty(xvalues)), xvalues = biggestRadius * thetaValues; end
    xcount = length(xvalues);
    zcount = length(zValues);
    planeData = zeros(xcount, zcount);        
    for zi = 1:zcount,
        localX = radiuses(zi) * thetaValues;
        localValues = coneData(:, zi);
        planeData(:, zi) = interp1(localX, localValues, xvalues, 'linear', 0);
    end
%]
end

• I will not enter into details of Thales theorem to compute radiuses for each z-position.

• I reorder angular positions to be in [-180°;+180°] so as to have symmetric x-positions (i.e. xLocal = radiusLocal * angularPos).

• The tricky part is to re-interpolate xLocal to some xGlobal positions (Which I assume to be xLocal = biggestRadius * angularPos, but you can of course provide your own values as an argument to Cone2Plane routine).

Here some test case:

function [] = TestCone2Plane()
%[
    % Scanning info
    zminRadius = 1.8506;
    zmaxRadius = 1.6849;
    zValues = linspace(0.0, 24.6, 100); 
    thetaValues = linspace(0, 359, 360) * pi / 180;

    % Build dummy cone data
    tcount = length(thetaValues);
    zcount = length(zValues);    
    coneData = rand(tcount, zcount);

    % Convert to plane data
    xvalues = []; % Means automatic x-positions
    [xvalues, zValues, planeData] = Cone2Plane(thetaValues, zValues, coneData, zminRadius, zmaxRadius, xvalues);

    % Display
    [X, Z] = ndgrid(xvalues, zValues);
    pcolor(X, Z, planeData);  
    shading flat;
%]
end

Edit

It is unclear what exact input data you have, but if pillar is a set of (x,y,z, intensity) points, you may reduce data set to coneData as follow:

thetas = atan2(y(:), x(:));
thetaValues = unique(thetas); tcount = length(thetaValues);
zValues = unique(z); zcount = length(zValues);    
coneData = zeros(tcount, zcount);
for ti = 1:tcount,
  logict = (thetas == thetaValues(ti));
  for zi = 1:zcount,
     logicz = (z == zValues(zi));
     coneData(ti, zi) = mean(intensity(logict & logicz));
  end
end

NB In practice, you may want to use some unique with tolerance (+ logic = abs(val-pos) < tol) to further reduce data while accounting for noise in scan positions.

Answer 2

1. so you want get (x,y,z)=f(u,v) ?

   • u=<0.0,1.0> ... angle coordinate
   • v=<0.0,1.0> ... height coordinate let v=0.0 be bottom
   • you got 24 periods of |sin| waves along the perimeter
   • with amplitudes: a0 at bottom and a1 at the top
   • pillar has radius r0 at bottom and r1 at the top
   • pillar has height h
   • 

2. first basic cylinder

   x=r*cos(2.0*M_PI*u)
   y=r*sin(2.0*M_PI*u)
   z=h*v
   

3. now cone

   r=r0+(r1-r0)*v
   x=r*cos(2.0*M_PI*u)
   y=r*sin(2.0*M_PI*u)
   z=h*v
   

   • just linear interpolation of radius is added

4. now the |sin| waves

   r=r0+(r1-r0)*v
   a=a0+(a1-a0)*v
   r-=a*fabs(sin(2.0*M_PI*12.0*u))
   x=r*cos(2.0*M_PI*u)
   y=r*sin(2.0*M_PI*u)
   z=h*v
   

   • just linear interpolation of amplitude of |sin| wave is added
   • and then the |sin| wave is applied to radius

Now the u,v coordinates are the coordinates inside your plane (texture for example)