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hw5: work

This commit is contained in:
Claudio Maggioni 2021-06-01 16:19:05 +02:00
parent 8922bce42f
commit 95b094f8b0
5 changed files with 1105 additions and 32 deletions
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32
Claudio_Maggioni_5/ex2.m
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@ -1,32 +0,0 @@
syms x1 x2
c1 = 2 * x1 + 3 * x2 - 6;
c2 = -3 * x1 + 2 * x2 - 3;
c3 = 2 * x2 - 5;
c4 = 2 * x1 + x2 - 4;
c1 = solve(c1 == 0, x2, 'Real', true);
c2 = solve(c2 == 0, x2, 'Real', true);
c3 = solve(c3 == 0, x2, 'Real', true);
c4 = solve(c4 == 0, x2, 'Real', true);
i1 = solve(c1 == c2, x1, 'Real', true);
i2 = solve(c1 == c4, x1, 'Real', true);
i3 = solve(c4 == 0, x1, 'Real', true);
px = double([0 0 i1 i2 i3]);
py = double([0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)]);
xl = -0.05;
xh = 2.05;
axis([-0.05 2.05 -0.15 5.15])
hold on
for c = [c1 c2 c3 c4]
plot([xl, xh], [subs(c, x1, xl), subs(c, x1, xh)]);
end
xline(0);
plot([xl, xh], [0, 0]);
patch(px, py, [204/255 255/255 187/255]);
legend('2x1 + 3x2 = 6', '-3x1 + 2x2 = 3', '2x2 = 5', '2x1 + x2 = 4', 'x1 > 0', 'x2 > 0', 'feasible region');
hold off

967
Claudio_Maggioni_5/hatchfill.m Normal file
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@ -0,0 +1,967 @@
function H = hatchfill(A,varargin)
% HATCHFILL2 Hatching and speckling of patch objects
% HATCHFILL2(A) fills the patch(es) with handle(s) A. A can be a vector
% of handles or a single handle. If A is a vector, then all objects of A
% should be part of the same group for predictable results. The hatch
% consists of black lines angled at 45 degrees at 40 hatching lines over
% the axis span with no color filling between the lines.
%
% A can be handles of patch or hggroup containing patch objects for
% Pre-R2014b release. For HG2 releases, 'bar' and 'contour' objects are
% also supported.
%
% Hatching line object is actively formatted. If A, axes, or figure size
% is modified, the hatching line object will be updated accordingly to
% maintain the specified style.
%
% HATCHFILL2(A,STYL) applies STYL pattern with default paramters. STYL
% options are:
% 'single' single lines (the default)
% 'cross' double-crossed hatch
% 'speckle' speckling inside the patch boundary
% 'outspeckle' speckling outside the boundary
% 'fill' no hatching
%
% HATCHFILL2(A,STYL,Option1Name,Option1Value,...) to customize the
% hatching pattern
%
% Name Description
% --------------------------------------------------------------------
% HatchStyle Hatching pattern (same effect as STYL argument)
% HatchAngle Angle of hatch lines in degrees (45)
% HatchDensity Number of hatch lines between axis limits
% HatchOffset Offset hatch lines in pixels (0)
% HatchColor Color of the hatch lines, 'auto' sets it to the
% EdgeColor of A
% HatchLineStyle Hatch line style
% HatchLineWidth Hatch line width
% SpeckleWidth Width of speckling region in pixels (7)
% SpeckleDensity Density of speckle points (1)
% SpeckleMarkerStyle Speckle marker style
% SpeckleFillColor Speckle fill color
% HatchVisible [{'auto'}|'on'|'off'] sets visibility of the hatch
% lines. If 'auto', Visibile option is synced to
% underlying patch object
% HatchSpacing (Deprecated) Spacing of hatch lines (5)
%
% In addition, name/value pairs of any properties of A can be specified
%
% H = HATCHFILL2(...) returns handles to the line objects comprising the
% hatch/speckle.
%
% Examples:
% Gray region with hatching:
% hh = hatchfill2(a,'cross','HatchAngle',45,'HatchSpacing',5,'FaceColor',[0.5 0.5 0.5]);
%
% Speckled region:
% hatchfill2(a,'speckle','HatchAngle',7,'HatchSpacing',1);
% Copyright 2015-2018 Takeshi Ikuma
% History:
% rev. 7 : (01-10-2018)
% * Added support for 3D faces
% * Removed HatchSpacing option
% * Added HatchDensity option
% * Hatching is no longer defined w.r.t. pixels. HatchDensity is defined
% as the number of hatch lines across an axis limit. As a result,
% HatchAngle no longer is the actual hatch angle though it should be
% close.
% * [known bug] Speckle hatching style is not working
% rev. 6 : (07-17-2016)
% * Fixed contours object hatching behavior, introduced in rev.5
% * Added ContourStyle option to enable fast drawing if contour is convex
% rev. 5 : (05-12-2016)
% * Fixed Contour with NaN data point disappearnace issue
% * Improved contours object support
% rev. 4 : (11-18-2015)
% * Worked around the issue with HG2 contours with Fill='off'.
% * Removed nagging warning "UseHG2 will be removed..." in R2015b
% rev. 3 : (10-29-2015)
% * Added support for HG2 AREA
% * Fixed for HatchColor 'auto' error when HG2 EdgeColor is 'flat'
% * Fixed listener creation error
% rev. 2 : (10-24-2015)
% * Added New option: HatchVisible, SpeckleDensity, SpeckleWidth
% (SpeckleDensity and SpeckleWidtha are separated from HatchSpacing and
% HatchAngle, respectively)
% rev. 1 : (10-20-2015)
% * Fixed HG2 contour data extraction bug (was using wrong hidden data)
% * Fixed HG2 contour color extraction bug
% * A few cosmetic changes here and there
% rev. - : (10-19-2015) original release
% * This work is based on Neil Tandon's hatchfill submission
% (http://www.mathworks.com/matlabcentral/fileexchange/30733)
% and borrowed code therein from R. Pawlowicz, K. Pankratov, and
% Iram Weinstein.
narginchk(1,inf);
[A,opts,props] = parse_input(A,varargin);
drawnow % make sure the base objects are already drawn
if verLessThan('matlab','8.4')
H = cell(1,numel(A));
else
H = repmat({matlab.graphics.GraphicsPlaceholder},1,numel(A));
end
for n = 1:numel(A)
H{n} = newhatch(A(n),opts,props);
% if legend of A(n) is shown, add hatching to it as well
% leg = handle(legend(ancestor(A,'axes')));
% hsrc = [leg.EntryContainer.Children.Object];
% hlc = leg.EntryContainer.Children(find(hsrc==A(n),1));
% if ~isempty(hlc)
% hlc = hlc.Children(1); % LegendIcon object
% get(hlc.Children(1))
% end
end
if nargout==0
clear H
else
H = [H{:}];
if numel(H)==numel(A)
H = reshape(H,size(A));
else
H = H(:);
end
end
end
function H = newhatch(A,opts,props)
% 0. retrieve pixel-data conversion parameters
% 1. retrieve face & vertex matrices from A
% 2. convert vertex matrix from data to pixels units
% 3. get xdata & ydata of hatching lines for each face
% 4. concatenate lines sandwitching nan's in between
% 5. convert xdata & ydata back to data units
% 6. plot the hatching line
% traverse if hggroup/hgtransform
if ishghandle(A,'hggroup')
if verLessThan('matlab','8.4')
H = cell(1,numel(A));
else
H = repmat({matlab.graphics.GraphicsPlaceholder},1,numel(A));
end
for n = 1:numel(A.Children)
try
H{n} = newhatch(A.Children(n),opts,props);
catch
end
end
H = [H{:}];
return;
end
% Modify the base object property if given
if ~isempty(props)
pvalold = sethgprops(A,props);
end
try
vislisena = strcmp(opts.HatchVisible,'auto');
if vislisena
vis = A.Visible;
else
vis = opts.HatchVisible;
end
redraw = strcmp(A.Visible,'off') && ~vislisena;
if redraw
A.Visible = 'on'; % momentarily make the patch visible
drawnow;
end
% get the base object's vertices & faces
[V,F,FillFcns] = gethgdata(A); % object does not have its patch data ready
if redraw
A.Visible = 'off'; % momentarily make the patch visible
end
if ~isempty(FillFcns)
FillFcns{1}();
drawnow;
[V,F] = gethgdata(A); % object does not have its patch data ready
FillFcns{2}();
drawnow;
end
% recompute hatch line data
[X,Y,Z] = computeHatchData(handle(ancestor(A,'axes')),V,F,opts);
% 6. plot the hatching line
commonprops = {'Parent',A.Parent,'DisplayName',A.DisplayName,'Visible',vis};
if ~strcmp(opts.HatchColor,'auto')
commonprops = [commonprops {'Color',opts.HatchColor,'MarkerFaceColor',opts.HatchColor}];
end
if isempty(regexp(opts.HatchStyle,'speckle$','once'))
H = line(X,Y,Z,commonprops{:},'LineStyle',opts.HatchLineStyle','LineWidth',opts.HatchLineWidth);
else
H = line(X,Y,Z,commonprops{:},'LineStyle','none','Marker',opts.SpeckleMarkerStyle,...
'MarkerSize',opts.SpeckleSize,'Parent',A.Parent,'DisplayName',A.DisplayName);
end
if strcmp(opts.HatchColor,'auto')
syncColor(H,A);
end
if isempty(H)
error('Unable to obtain hatching data from the specified object A.');
end
% 7. Move H so that it is place right above A in parent's uistack
p = handle(A.Parent);
Hcs = handle(p.Children);
[~,idx] = ismember(A,Hcs); % always idx(1)>idx(2) as H was just created
p.Children = p.Children([2:idx-1 1 idx:end]);
% if HG1, all done | no dynamic adjustment support
if verLessThan('matlab','8.4')
return;
end
% save the config data & set up the object listeners
setappdata(A,'HatchFill2Opts',opts); % hatching options
setappdata(A,'HatchFill2Obj',H); % hatching line object
setappdata(A,'HatchFill2LastData',{V,F}); % last patch data
setappdata(A,'HatchFill2LastVisible',A.Visible); % last sensitive properties
setappdata(A,'HatchFill2PostMarkedClean',{}); % run this function at the end of the MarkClean callback and set NoAction flag
setappdata(A,'HatchFill2NoAction',false); % no action during next MarkClean callback, callback only clears this flag
setappdata(H,'HatchFill2MatchVisible',vislisena);
setappdata(H,'HatchFill2MatchColor',strcmp(opts.HatchColor,'auto'));
setappdata(H,'HatchFill2Patch',A); % base object
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Create listeners for active formatting
addlistener(H,'ObjectBeingDestroyed',@hatchBeingDeleted);
lis = [
addlistener(A,'Reparent',@objReparent)
addlistener(A,'ObjectBeingDestroyed',@objBeingDeleted);
addlistener(A,'MarkedClean',@objMarkedClean)
addlistener(A,'LegendEntryDirty',@(h,evt)[])]; % <- study this later
syncprops = {'Clipping','HitTest','Interruptible','BusyAction','UIContextMenu'};
syncprops(~cellfun(@(p)isprop(A,p),syncprops)) = [];
for n = 1:numel(syncprops)
lis(n+2) = addlistener(A,syncprops{n},'PostSet',@syncProperty);
end
catch ME
% something went wrong, restore the base object properties
if ~isempty(props)
for pname = fieldnames(pvalold)'
name = pname{1};
val = pvalold.(name);
if iscell(val)
pvalold.(name){1}.(name) = pvalold.(name){2};
else
A.(name) = pvalold.(name);
end
end
end
ME.rethrow();
end
end
%%%%%%%%%% EVENT CALLBACK FUNCTIONS %%%%%%%%%%%%
% Base Object's listeners
% objReparent - also move the hatch object
% ObjectingBeingDestroyed - also destroy the hatch object
% MarkedClean - match color if HatchColor = 'auto'
% - check if vertex & face changed; if so redraw hatch
% - check if hatch redraw triggered the event due to object's
% face not shown; if so clear the flag
function objMarkedClean(hp,~)
% CALLBACK for base object's MarkedClean event
% check: visibility change, hatching area change, & color change
if getappdata(hp,'HatchFill2NoAction')
setappdata(A,'HatchFill2NoAction',false);
return;
end
% get the main patch object (loops if hggroup or HG2 objects)
H = getappdata(hp,'HatchFill2Obj');
rehatch = ~strcmp(hp.Visible,getappdata(hp,'HatchFill2LastVisible'));
if rehatch % if visibility changed
setappdata(hp,'HatchFill2LastVisible',hp.Visible);
if strcmp(hp.Visible,'off') % if made hidden, hide hatching as well
if getappdata(H,'HatchFill2MatchVisible')
H.Visible = 'off';
return; % nothing else to do
end
end
end
% get the patch data
[V,F,FillFcns] = gethgdata(hp);
if ~isempty(FillFcns) % patch does not exist, must momentarily generate it
FillFcns{1}();
setappdata(A,'HatchFill2PostMarkedClean',FillFcns{2});
return;
end
if ~rehatch % if visible already 'on', check for the change in object data
VFlast = getappdata(hp,'HatchFill2LastData');
rehatch = ~isequaln(F,VFlast{2}) || ~isequaln(V,VFlast{1});
end
% rehatch if patch data/visibility changed
if rehatch
% recompute hatch line data
[X,Y,Z] = computeHatchData(ancestor(H,'axes'),V,F,getappdata(hp,'HatchFill2Opts'));
% update the hatching line data
set(H,'XData',X,'YData',Y,'ZData',Z);
% save patch data
setappdata(hp,'HatchFill2LastData',{V,F});
end
% sync the color
syncColor(H,hp);
% run post callback if specified (expect it to trigger another MarkedClean
% event immediately)
fcn = getappdata(hp,'HatchFill2PostMarkedClean');
if ~isempty(fcn)
setappdata(hp,'HatchFill2PostMarkedClean',function_handle.empty);
setappdata(hp,'HatchFill2NoAction',true);
fcn();
return;
end
end
function syncProperty(~,evt)
% sync Visible property to the patch object
hp = handle(evt.AffectedObject); % patch object
hh = getappdata(hp,'HatchFill2Obj');
hh.(evt.Source.Name) = hp.(evt.Source.Name);
end
function objReparent(hp,evt)
%objReparent event listener callback
pnew = evt.NewValue;
if isempty(pnew)
return; % no change?
end
% move the hatch line object over as well
H = getappdata(hp,'HatchFill2Obj');
H.Parent = pnew;
% make sure to move the hatch line object right above the patch object
Hcs = handle(pnew.Children);
[~,idx] = ismember(hp,Hcs); % always idx(1)>idx(2) as H was just moved
pnew.Children = pnew.Children([2:idx-1 1 idx:end]);
end
function objBeingDeleted(hp,~)
%when base object is deleted
if isappdata(hp,'HatchFill2Obj')
H = getappdata(hp,'HatchFill2Obj');
try % in case H is already deleted
delete(H);
catch
end
end
end
function hatchBeingDeleted(hh,~)
%when hatch line object (hh) is deleted
if isappdata(hh,'HatchFill2Patch')
% remove listeners listening to the patch object
hp = getappdata(hh,'HatchFill2Patch');
if isappdata(hp,'HatchFill2Listeners')
delete(getappdata(hp,'HatchFill2Listeners'));
rmappdata(hp,'HatchFill2Listeners');
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function varargout = computeHatchData(ax,V,F,opts)
varargout = cell(1,nargout);
if isempty(V) % if patch shown
return;
end
N = size(F,1);
XYZc = cell(2,N);
for n = 1:N % for each face
% 2. get xdata & ydata of the vertices of the face in transformed bases
f = F(n,:); % get indices to the vertices of the face
f(isnan(f)) = [];
[v,T,islog] = transform_data(ax,V(f,:),[]); % transform the face
if isempty(v) % face is not hatchable
continue;
end
% 2. get xdata & ydata of hatching lines for each face
if any(strcmp(opts.HatchStyle,{'speckle','outsidespeckle'}))
xy = hatch_xy(v.',opts.HatchStyle,opts.SpeckleWidth,opts.SpeckleDensity,opts.HatchOffset);
else
xy = hatch_xy(v.',opts.HatchStyle,opts.HatchAngle,opts.HatchDensity,opts.HatchOffset);
end
% 3. revert the bases back to 3D Eucledian space
XYZc{1,n} = revert_data(xy',T,islog).';
end
% 4. concatenate hatch lines across faces sandwitching nan's in between
[XYZc{2,:}] = deal(nan(3,1));
XYZ = cat(2,XYZc{:});
% 5. convert xdata & ydata back to data units
[varargout{1:3}] = deal(XYZ(1,:),XYZ(2,:),XYZ(3,:));
end
function tf = issupported(hbase)
% check if all of the given base objects are supported
supported_objtypes = {'patch','hggroup','bar','contour','area','surface','histogram'};
if isempty(hbase)
tf = false;
else
tf = ishghandle(hbase,supported_objtypes{1});
for n = 2:numel(supported_objtypes)
tf(:) = tf | ishghandle(hbase,supported_objtypes{n});
end
tf = all(tf);
end
end
% synchronize hatching line color to the patch's edge color if HatchColor =
% 'auto'
function syncColor(H,A)
if ~getappdata(H,'HatchFill2MatchColor')
% do not sync
return;
end
if ishghandle(A,'patch') || ishghandle(A,'Bar') || ishghandle(A,'area') ...
|| ishghandle(A,'surface') || ishghandle(A,'Histogram') %HG2
pname = 'EdgeColor';
elseif ishghandle(A,'contour') % HG2
pname = 'LineColor';
end
color = A.(pname);
if strcmp(color,'flat')
try
color = double(A.Edge(1).ColorData(1:3)')/255;
catch
warning('Does not support CData based edge color.');
color = 'k';
end
end
H.Color = color;
H.MarkerFaceColor = color;
end
function [V,F,FillFcns] = gethgdata(A)
% Get vertices & face data from the object along with the critical
% properties to observe change in the hatching area
% initialize the output variable
F = [];
V = [];
FillFcns = {};
if ~isvalid(A) || strcmp(A.Visible,'off')
return;
end
if ishghandle(A,'patch')
V = A.Vertices;
F = A.Faces;
elseif ishghandle(A,'bar')
[V,F] = getQuadrilateralData(A.Face);
elseif ishghandle(A,'area')
[V,F] = getTriangleStripData(A.Face);
set(A,'FaceColor','none');
elseif ishghandle(A,'surface') % HG2
if strcmp(A.FaceColor,'none')
FillFcns = {@()set(A,'FaceColor','w'),@()set(A,'FaceColor','none')};
return;
end
[V,F] = getQuadrilateralData(A.Face);
elseif ishghandle(A,'contour') % HG2
% Retrieve data from hidden FacePrims property (a TriangleStrip object)
if strcmp(A.Fill,'off')
FillFcns = {@()set(A,'Fill','on'),@()set(A,'Fill','off')};
return;
end
[V,F] = getTriangleStripData(A.FacePrims);
elseif ishghandle(A,'histogram') %HG2: Quadrateral underlying data object
[V,F] = getQuadrilateralData(A.NodeChildren(4));
end
end
function [V,F] = getQuadrilateralData(A) % surface, bar, histogram,
if isempty(A)
warning('Cannot hatch the face: Graphics object''s face is not defined.');
V = [];
F = [];
return;
end
V = A.VertexData';
% If any of the axes is in log scale, V is normalized to wrt the axes
% limits,
V(:) = norm2data(V,A);
if ~isempty(A.VertexIndices) % vertices likely reused on multiple quadrilaterals
I = A.VertexIndices;
Nf = numel(I)/4; % has to be divisible by 4
else %every 4 consecutive vertices defines a quadrilateral
Nv = size(V,1);
Nf = Nv/4;
I = 1:Nv;
end
F = reshape(I,4,Nf)';
if ~isempty(A.StripData) % hack workaround
F(:) = F(:,[1 2 4 3]);
end
try
if ~any(all(V==V(1,:))) % not on any Euclidian plane
% convert quadrilateral to triangle strips
F = [F(:,1:3);F(:,[1 3 4])];
end
catch % if implicit array expansion is not supported (<R2016b)
if all(V(:,1)~=V(1,1)) || all(V(:,2)~=V(1,2)) || all(V(:,3)~=V(1,3)) % not on any Euclidian plane
% convert quadrilateral to triangle strips
F = [F(:,1:3) F(:,[1 3 4])];
end
end
end
function [V,F] = getTriangleStripData(A) % area & contour
if isempty(A)
warning('Cannot hatch the face: Graphics object''s face is not defined.');
V = [];
F = [];
return;
end
V = A.VertexData';
I = double(A.StripData);
% If any of the axes is in log scale, V is normalized to wrt the axes
% limits,
V(:) = norm2data(V,A);
N = numel(I)-1; % # of faces
m = diff(I);
M = max(m);
F = nan(N,M);
for n = 1:N
idx = I(n):(I(n+1)-1);
if mod(numel(idx),2) % odd
idx(:) = idx([1:2:end end-1:-2:2]);
else % even
idx(:) = idx([1:2:end-1 end:-2:2]);
end
F(n,1:numel(idx)) = idx;
end
end
% if graphical objects are given normalized to the axes
function V = norm2data(V,A)
ax = ancestor(A,'axes');
inlog = strcmp({ax.XScale ax.YScale ax.ZScale},'log');
if any(inlog)
lims = [ax.XLim(:) ax.YLim(:) ax.ZLim(:)];
dirs = strcmp({ax.XDir ax.YDir ax.ZDir},'normal');
for n = 1:3 % for each axis
if inlog(n)
lims(:,n) = log10(lims(:,n));
end
V(:,n) = V(:,n)*diff(lims(:,n));
if dirs(n)
V(:,n) = V(:,n) + lims(1,n);
else
V(:,n) = lims(2,n) - V(:,n);
end
if inlog(n)
V(:,n) = 10.^V(:,n);
end
end
end
end
function pvalold = sethgprops(A,props)
% grab the common property names of the base objects
pnames = fieldnames(props);
if ishghandle(A,'hggroup')
gpnames = fieldnames(set(A));
[tf,idx] = ismember(gpnames,pnames);
idx(~tf) = [];
for i = idx'
pvalold.(pnames{i}) = A.(pnames{i});
A.(pnames{i}) = props.(pnames{i});
end
props = rmfield(props,pnames(idx));
h = handle(A.Children);
for n = 1:numel(h)
pvalold1 = sethgprops(h(n),props);
ponames = fieldnames(pvalold1);
for k = 1:numel(ponames)
pvalold.(ponames{k}) = {h(n) pvalold1.(ponames{k})};
end
end
else
for n = 1:numel(pnames)
pvalold.(pnames{n}) = A.(pnames{n});
A.(pnames{n}) = props.(pnames{n});
end
end
end
function xydatai = hatch_xy(xydata,styl,angle,step,offset)
%
% M_HATCH Draws hatched or speckled interiors to a patch
%
% M_HATCH(LON,LAT,STYL,ANGLE,STEP,...line parameters);
%
% INPUTS:
% X,Y - vectors of points.
% STYL - style of fill
% ANGLE,STEP - parameters for style
%
% E.g.
%
% 'single',45,5 - single cross-hatch, 45 degrees, 5 points apart
% 'cross',40,6 - double cross-hatch at 40 and 90+40, 6 points apart
% 'speckle',7,1 - speckled (inside) boundary of width 7 points, density 1
% (density >0, .1 dense 1 OK, 5 sparse)
% 'outspeckle',7,1 - speckled (outside) boundary of width 7 points, density 1
% (density >0, .1 dense 1 OK, 5 sparse)
%
%
% H=M_HATCH(...) returns handles to hatches/speckles.
%
% [XI,YI,X,Y]=MHATCH(...) does not draw lines - instead it returns
% vectors XI,YI of the hatch/speckle info, and X,Y of the original
% outline modified so the first point==last point (if necessary).
%
% Note that inside and outside speckling are done quite differently
% and 'outside' speckling on large coastlines can be very slow.
%
% Hatch Algorithm originally by K. Pankratov, with a bit stolen from
% Iram Weinsteins 'fancification'. Speckle modifications by R. Pawlowicz.
%
% R Pawlowicz 15/Dec/2005
I = zeros(1,size(xydata,2));
% face vertices are not always closed
if any(xydata(:,1)~=xydata(:,end))
xydata(:,end+1) = xydata(:,1);
I(end+1) = I(1);
end
if any(strcmp(styl,{'speckle','outspeckle'}))
angle = angle*(1-I);
end
switch styl
case 'single'
xydatai = drawhatch(xydata,angle,1/step,0,offset);
case 'cross'
xydatai = [...
drawhatch(xydata,angle,1/step,0,offset) ...
drawhatch(xydata,angle+90,1/step,0,offset)];
case 'speckle'
xydatai = [...
drawhatch(xydata,45, 1/step,angle,offset) ...
drawhatch(xydata,45+90,1/step,angle,offset)];
case 'outspeckle'
xydatai = [...
drawhatch(xydata,45, 1/step,-angle,offset) ...
drawhatch(xydata,45+90,1/step,-angle,offset)];
inside = logical(inpolygon(xydatai(1,:),xydatai(2,:),x,y)); % logical needed for v6!
xydatai(:,inside) = [];
otherwise
xydatai = zeros(2,0);
end
end
%%%
function xydatai = drawhatch(xydata,angle,step,speckle,offset)
% xydata is given as 2xN matrix, x on the first row, y on the second
% Idea here appears to be to rotate everthing so lines will be
% horizontal, and scaled so we go in integer steps in 'y' with
% 'points' being the units in x.
% Center it for "good behavior".
% rotate first about (0,0)
ca = cosd(angle); sa = sind(angle);
u = [ca sa]*xydata; % Rotation
v = [-sa ca]*xydata;
% translate to the grid point nearest to the centroid
u0 = round(mean(u)/step)*step; v0 = round(mean(v)/step)*step;
x = (u-u0); y = (v-v0)/step+offset; % plus scaling and offsetting
% Compute the coordinates of the hatch line ...............
yi = ceil(y);
yd = [diff(yi) 0]; % when diff~=0 we are crossing an integer
fnd = find(yd); % indices of crossings
dm = max(abs(yd)); % max possible #of integers between points
% This is going to be pretty space-inefficient if the line segments
% going in have very different lengths. We have one column per line
% interval and one row per hatch line within that interval.
%
A = cumsum( repmat(sign(yd(fnd)),dm,1), 1);
% Here we interpolate points along all the line segments at the
% correct intervals.
fnd1 = find(abs(A)<=abs( repmat(yd(fnd),dm,1) ));
A = A+repmat(yi(fnd),dm,1)-(A>0);
xy = (x(fnd+1)-x(fnd))./(y(fnd+1)-y(fnd));
xi = repmat(x(fnd),dm,1)+(A-repmat(y(fnd),dm,1) ).*repmat(xy,dm,1);
yi = A(fnd1);
xi = xi(fnd1);
% Sorting points of the hatch line ........................
%%yi0 = min(yi); yi1 = max(yi);
% Sort them in raster order (i.e. by x, then by y)
% Add '2' to make sure we don't have problems going from a max(xi)
% to a min(xi) on the next line (yi incremented by one)
xi0 = min(xi); xi1 = max(xi);
ci = 2*yi*(xi1-xi0)+xi;
[~,num] = sort(ci);
xi = xi(num); yi = yi(num);
% if this happens an error has occurred somewhere (we have an odd
% # of points), and the "fix" is not correct, but for speckling anyway
% it really doesn't make a difference.
if rem(length(xi),2)==1
xi = [xi; xi(end)];
yi = [yi; yi(end)];
end
% Organize to pairs and separate by NaN's ................
li = length(xi);
xi = reshape(xi,2,li/2);
yi = reshape(yi,2,li/2);
% The speckly part - instead of taking the line we make a point some
% random distance in.
if length(speckle)>1 || speckle(1)~=0
if length(speckle)>1
% Now we get the speckle parameter for each line.
% First, carry over the speckle parameter for the segment
% yd=[0 speckle(1:end-1)];
yd = speckle(1:end);
A=repmat(yd(fnd),dm,1);
speckle=A(fnd1);
% Now give it the same preconditioning as for xi/yi
speckle=speckle(num);
if rem(length(speckle),2)==1
speckle = [speckle; speckle(end)];
end
speckle=reshape(speckle,2,li/2);
else
speckle=[speckle;speckle];
end
% Thin out the points in narrow parts.
% This keeps everything when abs(dxi)>2*speckle, and then makes
% it increasingly sparse for smaller intervals.
dxi=diff(xi);
nottoosmall=sum(speckle,1)~=0 & rand(1,li/2)<abs(dxi)./(max(sum(speckle,1),eps));
xi=xi(:,nottoosmall);
yi=yi(:,nottoosmall);
dxi=dxi(nottoosmall);
if size(speckle,2)>1, speckle=speckle(:,nottoosmall); end
% Now randomly scatter points (if there any left)
li=length(dxi);
if any(li)
xi(1,:)=xi(1,:)+sign(dxi).*(1-rand(1,li).^0.5).*min(speckle(1,:),abs(dxi) );
xi(2,:)=xi(2,:)-sign(dxi).*(1-rand(1,li).^0.5).*min(speckle(2,:),abs(dxi) );
% Remove the 'zero' speckles
if size(speckle,2)>1
xi=xi(speckle~=0);
yi=yi(speckle~=0);
end
end
else
xi = [xi; ones(1,li/2)*nan]; % Separate the line segments
yi = [yi; ones(1,li/2)*nan];
end
% Transform back to the original coordinate system
xydatai = [ca -sa;sa ca]*[xi(:)'+u0;(yi(:)'-offset)*step+v0];
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [h,opts,props] = parse_input(h,argin)
% parse & validate input arguments
patchtypes = {'single','cross','speckle','outspeckle','fill','none'};
% get base object handle
if ~issupported(h)
error('Unsupported graphics handle type.');
end
h = handle(h);
% get common property names
pnames = getcommonprops(h);
% if style argument is given, convert it to HatchStyle option pair
stylearg = {};
if ~isempty(argin) && ischar(argin{1})
try
ptypes = validatestring(argin{1},patchtypes);
stylearg = {'HatchStyle' ptypes};
argin(1) = [];
catch
% STYL not given, continue on
end
end
% create inputParser for options
p = inputParser;
p.addParameter('HatchStyle','single');
p.addParameter('HatchAngle',45,@(v)validateattributes(v,{'numeric'},{'scalar','finite','real'}));
p.addParameter('HatchDensity',40,@(v)validateattributes(v,{'numeric'},{'scalar','positive','finite','real'}));
p.addParameter('HatchSpacing',[],@(v)validateattributes(v,{'numeric'},{'scalar','positive','finite','real'}));
p.addParameter('HatchOffset',0,@(v)validateattributes(v,{'numeric'},{'scalar','nonnegative','<',1,'real'}));
p.addParameter('HatchColor','auto',@validatecolor);
p.addParameter('HatchLineStyle','-');
p.addParameter('HatchLineWidth',0.5,@(v)validateattributes(v,{'numeric'},{'scalar','positive','finite','real'}));
p.addParameter('SpeckleWidth',7,@(v)validateattributes(v,{'numeric'},{'scalar','finite','real'}));
p.addParameter('SpeckleDensity',100,@(v)validateattributes(v,{'numeric'},{'scalar','positive','finite','real'}));
p.addParameter('SpeckleMarkerStyle','.');
p.addParameter('SpeckleSize',2,@(v)validateattributes(v,{'numeric'},{'scalar','positive','finite'}));
p.addParameter('SpeckleFillColor','auto',@validatecolor);
p.addParameter('HatchVisible','auto');
for n = 1:numel(pnames)
p.addParameter(pnames{n},[]);
end
p.parse(stylearg{:},argin{:});
rnames = fieldnames(p.Results);
isopt = ~cellfun(@isempty,regexp(rnames,'^(Hatch|Speckle)','once')) | strcmp(rnames,'ContourStyle');
props = struct([]);
for n = 1:numel(rnames)
if isopt(n)
opts.(rnames{n}) = p.Results.(rnames{n});
elseif ~isempty(p.Results.(rnames{n}))
props(1).(rnames{n}) = p.Results.(rnames{n});
end
end
opts.HatchStyle = validatestring(opts.HatchStyle,patchtypes);
if any(strcmp(opts.HatchStyle,{'speckle','outspeckle'}))
warning('hatchfill2:PartialSupport','Speckle/outspeckle HatchStyle may not work in the current release of hatchfill2')
end
if strcmpi(opts.HatchStyle,'none') % For backwards compatability:
opts.HatchStyle = 'fill';
end
opts.HatchLineStyle = validatestring(opts.HatchLineStyle,{'-','--',':','-.'},mfilename,'HatchLineStyle');
if ~isempty(opts.HatchSpacing)
warning('HatchSpacing option has been deprecated. Use ''HatchDensity'' option instead.');
end
opts = rmfield(opts,'HatchSpacing');
opts.SpeckleMarkerStyle = validatestring(opts.SpeckleMarkerStyle,{'+','o','*','.','x','square','diamond','v','^','>','<','pentagram','hexagram'},'hatchfill2','SpeckleMarkerStyle');
opts.HatchVisible = validatestring(opts.HatchVisible,{'auto','on','off'},mfilename,'HatchVisible');
end
function pnames = getcommonprops(h)
% grab the common property names of the base objects
V = set(h(1));
pnames = fieldnames(V);
if ishghandle(h(1),'hggroup')
pnames = union(pnames,getcommonprops(get(h(1),'Children')));
end
for n = 2:numel(h)
V = set(h(n));
pnames1 = fieldnames(V);
if ishghandle(h(n),'hggroup')
pnames1 = union(pnames1,getcommonprops(get(h(n),'Children')));
end
pnames = intersect(pnames,pnames1);
end
end
function validatecolor(val)
try
validateattributes(val,{'double','single'},{'numel',3,'>=',0,'<=',1});
catch
validatestring(val,{'auto','y','yellow','m','magenta','c','cyan','r','red',...
'g','green','b','blue','w','white','k','black'});
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% axes unit conversion functions
function [V,T,islog] = transform_data(ax,V,ref)
% convert vertices data to hatch-ready form
% - if axis is log-scaled, data is converted to their log10 values
% - if 3D (non-zero z), spatially transform data onto the xy-plane. If
% reference point is given, ref is mapped to the origin. Otherwise, ref
% is chosen to be the axes midpoint projected onto the patch plane. Along
% with the data, the axes corner coordinates are also projected onto the
% patch plane to obtain the projected axes limits.
% - transformed xy-data are then normalized by the projected axes spans.
noZ = size(V,2)==2;
xl = ax.XLim;
yl = ax.YLim;
zl = ax.ZLim;
% log to linear space
islog = strcmp({ax.XScale ax.YScale ax.ZScale},'log');
if islog(1)
V(:,1) = log10(V(:,1));
xl = log10(xl);
if ~isempty(ref)
ref(1) = log10(ref(1));
end
end
if islog(2)
V(:,2) = log10(V(:,2));
yl = log10(yl);
if ~isempty(ref)
ref(2) = log10(ref(2));
end
end
if islog(3) && ~noZ
V(:,3) = log10(V(:,3));
zl = log10(zl);
if ~isempty(ref)
ref(3) = log10(ref(3));
end
end
if noZ
V(:,3) = 0;
end
% if not given, pick the reference point to be the mid-point of the current
% axes
if isempty(ref)
ref = [mean(xl) mean(yl) mean(zl)];
end
% normalize the axes so that they span = 1;
Tscale = makehgtform('scale', [1/diff(xl) 1/diff(yl) 1/diff(zl)]);
V(:) = V*Tscale(1:3,1:3);
ref(:) = ref*Tscale(1:3,1:3);
% obtain unique vertices
Vq = double(unique(V,'rows')); % find unique points (sorted order)
Nq = size(Vq,1);
if Nq<3 || any(isinf(Vq(:))) || any(isnan(Vq(:))) % not hatchable
V = [];
T = [];
return;
end
try % erros if 2D object
zq = unique(Vq(:,3));
catch
V(:,3) = 0;
zq = 0;
end
T = eye(4);
if isscalar(zq) % patch is on a xy-plane
if zq~=0 % not on the xy-plane
T = makehgtform('translate',[0 0 -zq]);
end
else
% if patch is not on a same xy-plane
% use 3 points likely well separated
idx = round((0:2)/2*(Nq-1))+1;
% find unit normal vector of the patch plane
norm = cross(Vq(idx(1),:)-Vq(idx(3),:),Vq(idx(2),:)-Vq(idx(3),:)); % normal vector
norm(:) = norm/sqrt(sum(norm.^2));
% define the spatial rotation
theta = acos(norm(3));
if theta>pi/2, theta = theta-pi; end
u = [norm(2) -norm(1) 0];
Trot = makehgtform('axisrotate',u,theta);
% project the reference point onto the plane
P = norm.'*norm;
ref_proj = ref*(eye(3) - P) + Vq(1,:)*P;
if norm(3)
T = makehgtform('translate', -ref_proj); % user specified reference point or -d/norm(3) for z-crossing
end
% apply the rotation now
T(:) = Trot*T;
% find the axes limits on the transformed space
% [Xlims,Ylims,Zlims] = ndgrid(xl,yl,zl);
% Vlims = [Xlims(:) Ylims(:) Zlims(:)];
% Vlims_proj = [Vlims ones(8,1)]*T';
% lims_proj = [min(Vlims_proj(:,[1 2]),[],1);max(Vlims_proj(:,[1 2]),[],1)];
% xl = lims_proj(:,1)';
% yl = lims_proj(:,2)';
end
% perform the transformation
V(:,4) = 1;
V = V*T';
V(:,[3 4]) = [];
T(:) = T*Tscale;
end
function V = revert_data(V,T,islog)
N = size(V,1);
V = [V zeros(N,1) ones(N,1)]/T';
V(:,end) = [];
% log to linear space
if islog(1)
V(:,1) = 10.^(V(:,1));
end
if islog(2)
V(:,2) = 10.^(V(:,2));
end
if islog(3)
V(:,3) = 10.^(V(:,3));
end
end

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clc
clear
close all
%% Exercise 2.1
syms x1 x2
c1 = 2 * x1 + 3 * x2 - 6;
c2 = -3 * x1 + 2 * x2 - 3;
c3 = 2 * x2 - 5;
c4 = 2 * x1 + x2 - 4;
c1 = solve(c1 == 0, x2, 'Real', true);
c2 = solve(c2 == 0, x2, 'Real', true);
c3 = solve(c3 == 0, x2, 'Real', true);
c4 = solve(c4 == 0, x2, 'Real', true);
i1 = solve(c1 == c2, x1, 'Real', true);
i2 = solve(c1 == c4, x1, 'Real', true);
i3 = solve(c4 == 0, x1, 'Real', true);
px = double([0 0 i1 i2 i3]);
py = double([0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)]);
xl = -0.05;
xh = 2.05;
colors = [1 0 1; 0 0 1; 1 0 0; 0 1 0];
axis([-0.05 2.05 -0.15 5.15])
i = 1;
hold on
for c = [c1 c2 c3 c4]
%plot([xl, xh], [subs(c, x1, xl), subs(c, x1, xh)]);
hatchfill(patch([xl, xl, xh, xh], ...
double([0, subs(c, x1, xl), subs(c, x1, xh), 0]), ...
colors(i, :)), ...
'HatchColor', colors(i, :), 'HatchOffset', (i-1)/5);
i = i + 1;
end
xline(0);
hatchfill(patch([0, 0, xh, xh], ...
double([0, 5.15, 5.15, 0]), ...
'white'), ...
'HatchColor', [1 0.5 0], 'HatchOffset', 4/5);
plot([xl, xh], [0, 0]);
patch(px, py, 'black');
alpha(.05)
legend('','2x1 + 3x2 <= 6', '', '-3x1 + 2x2 <= 3', '', '2x2 <= 5', ...
'', '2x1 + x2 <= 4', '', 'x1 > 0 and x2 > 0', 'feasible region');
hold off
%% Exercise 3.2
G = [6 2 1; 2 5 2; 1 2 4];
c = [-8; -3; -3];
A = [1 0 1; 0 1 1];
b = [3; 0];
[x, lambda] = uzawa(G, c, A, b, [0;0;0], [0;0], 1e-8, 100);
display(x);
display(lambda);

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clc
clear
close all
%% Exercise 2.1
syms x1 x2
c1 = 2 * x1 + 3 * x2 - 6;
c2 = -3 * x1 + 2 * x2 - 3;
c3 = 2 * x2 - 5;
c4 = 2 * x1 + x2 - 4;
c1 = solve(c1 == 0, x2, 'Real', true);
c2 = solve(c2 == 0, x2, 'Real', true);
c3 = solve(c3 == 0, x2, 'Real', true);
c4 = solve(c4 == 0, x2, 'Real', true);
i1 = solve(c1 == c2, x1, 'Real', true);
i2 = solve(c1 == c4, x1, 'Real', true);
i3 = solve(c4 == 0, x1, 'Real', true);
px = double([0 0 i1 i2 i3]);
py = double([0 subs(c2, x1, 0) subs(c1, x1, i1) subs(c4, x1, i2) subs(c4, x1, i3)]);
xl = -0.05;
xh = 2.05;
colors = [224/255 6/255 191/255; 0 0 1; 1 0 0; 0 1 0];
axis([-0.05 2.05 -0.15 5.15])
i = 1;
hold on
for c = [c1 c2 c3 c4]
%plot([xl, xh], [subs(c, x1, xl), subs(c, x1, xh)]);
hatchfill(patch([xl, xl, xh, xh], ...
double([-0.15, subs(c, x1, xl), subs(c, x1, xh), -0.15]), ...
colors(i, :)), ...
'HatchColor', colors(i, :), 'HatchOffset', (i-1)/5, 'HatchAngle', 45);
i = i + 1;
end
%xline(0);
hatchfill(patch([0, 0, xh, xh], ...
double([0, 5.15, 5.15, 0]), ...
'white'), ...
'HatchColor', [249/255 216/255 49/255], 'HatchOffset', 4/5, 'HatchAngle', 45);
%plot([xl, xh], [0, 0]);
hatchfill(patch(px, py, 'black'),'HatchColor', 'black', 'HatchAngle', 90);
alpha(.02)
legend('','2x1 + 3x2 <= 6', '', '-3x1 + 2x2 <= 3', '', '2x2 <= 5', ...
'', '2x1 + x2 <= 4', '', 'x1 > 0 and x2 > 0', '', 'feasible region');
hold off
%% Exercise 3.2
G = [6 2 1; 2 5 2; 1 2 4];
c = [-8; -3; -3];
A = [1 0 1; 0 1 1];
b = [3; 0];
[x, lambda] = uzawa(G, c, A, b, [0;0;0], [0;0], 1e-8, 100);
display(x);
display(lambda);

12
Claudio_Maggioni_5/uzawa.m Normal file
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function [x, lambda] = uzawa(G, c, A, b, x, lambda, tol, max_itr)
w = 1;
old_x = ones(size(G, 1)) * 124000000;
i = 0;
while i < max_itr && norm(x - old_x) > tol
old_x = x;
x = G \ (c - (A' * lambda));
lambda = lambda + w * (A * x - b);
i = i + 1;
end
end