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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function kappa = curvature(varargin)
%CURVATURE Estimate curvature of a polyline defined by points.
%
% KAPPA = curvature(T, PX, PY, METHOD, DEGREE)
% First compute an approximation of the curve given by PX and PY, with
% the parametrization T. METHOD used for approximation can be only:
% 'polynom', with specified degree
% Further methods will be provided in a future version.
% T, PX, and PY are N*1 array of the same length.
% Then compute the curvature of approximated curve for each point.
%
% For example:
% KAPPA = curvature(t, px, py, 'polynom', 6)
%
% KAPPA = curvature(T, POINTS, METHOD, DEGREE)
% specify curve as a suite of points. POINTS is size [N*2].
%
% KAPPA = curvature(PX, PY, METHOD, DEGREE)
% KAPPA = curvature(POINTS, METHOD, DEGREE)
% compute implicite normalization of the curve, based on euclidian
% distance between 2 consecutive points, and normalized between 0 and 1.
%
%
% See also
% polygons2d, parametrize
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2003-04-07
% Copyright 2003-2023 INRA - TPV URPOI - BIA IMASTE
% default values
degree = 5;
t=0; % parametrization of curve
tc=0; % indices of points wished for curvature
% Extract method and degree
nargin = length(varargin);
varN = varargin{nargin};
varN2 = varargin{nargin-1};
if ischar(varN2)
% method and degree are specified
method = varN2;
degree = varN;
varargin = varargin(1:nargin-2);
elseif ischar(varN)
% only method is specified, use degree 6 as default
method = varN;
varargin = varargin{1:nargin-1};
else
% method and degree are implicit : use 'polynom' and 6
method = 'polynom';
end
% extract input parametrization and curve
nargin = length(varargin);
if nargin==1
% parameters are just the points -> compute caracterization.
var = varargin{1};
px = var(:,1);
py = var(:,2);
elseif nargin==2
var = varargin{2};
if size(var, 2)==2
% parameters are t and POINTS
px = var(:,1);
py = var(:,2);
t = varargin{1};
else
% parameters are px and py
px = varargin{1};
py = var;
end
elseif nargin==3
var = varargin{2};
if size(var, 2)==2
% parameters are t, POINTS, and tc
px = var(:,1);
py = var(:,2);
t = varargin{1};
else
% parameters are t, px and py
t = varargin{1};
px = var;
py = varargin{3};
end
elseif nargin==4
% parameters are t, px, py and tc
t = varargin{1};
px = varargin{2};
py = varargin{3};
tc = varargin{4};
end
% compute implicit parameters
% if t and/or tc are not computed, use implicit definition
if t==0
t = parametrize(px, py);
t = t/t(length(t)); % normalize between 0 and 1
end
% if tc not defined, compute curvature for all points
if tc==0
tc = t;
else
% else convert from indices to parametrization values
tc = t(tc);
end
%% compute curvature for each point of the curve
if strcmp(method, 'polynom')
% compute coefficients of interpolation functions
x0 = polyfit(t, px, degree);
y0 = polyfit(t, py, degree);
% compute coefficients of first and second derivatives. In the case of a
% polynom, it is possible to compute coefficient of derivative by
% multiplying with a matrix.
derive = diag(degree:-1:0);
xp = circshift(x0*derive, [0 1]);
yp = circshift(y0*derive, [0 1]);
xs = circshift(xp*derive, [0 1]);
ys = circshift(yp*derive, [0 1]);
% compute values of first and second derivatives for needed points
xprime = polyval(xp, tc);
yprime = polyval(yp, tc);
xsec = polyval(xs, tc);
ysec = polyval(ys, tc);
% compute value of curvature
kappa = (xprime.*ysec - xsec.*yprime)./ ...
power(xprime.*xprime + yprime.*yprime, 3/2);
else
error('unknown method');
end
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