How plot a grid of rectangles on an overlaid circle?

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JM 2024-5-20

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移动：Matt J 2024-5-28

How to plot a circle of radius R and an overlaid grid of rectangles (with each rectangle of dimensions LxW)?

Choice of dimensions are so that the rectangular grid is larger than the dimeter of the circle and the centre of the circle is on one of the rectangle corners within the grid. The circle fully sits within the rectangular grid.

Secondly, from the plot or otherwise can Matlab help count the full rectangles within the circle?

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Answer 1

Matt J 2024-5-20

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https://ww2.mathworks.cn/matlabcentral/answers/2120706-how-plot-a-grid-of-rectangles-on-an-overlaid-circle#answer_1460431

编辑：Matt J 2024-5-20

在 MATLAB Online 中打开

R = 5; % circle radius

H = 1; % rect height

W = 2; % rect width

circ=nsidedpoly(1000,"Radius",R); %polyshape for the circle

Grid=scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[W,H]);

Grid=arrayfun(@(i)translate(Grid,H*[0,i]),-ceil(R/H):ceil(R/H)-1)';

Grid=arrayfun(@(i)translate(Grid,W*[i,0]),-ceil(R/W):ceil(R/W)-1,'uni',0);

Grid=vertcat(Grid{:}); %polyshape vector for grid rectangles

numContained=sum( area(intersect(Grid,circ))>=area(Grid(1))*0.9999 ) %count contained rectangles

numContained = 28

plot([circ; Grid],'FaceColor','w'); axis equal

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JM 2024-5-21

编辑：JM 2024-5-21

在 MATLAB Online 中打开

@Matt J Thanks, I must have been too tired last night to notice my mistake. Indeed I was plugging in the diameter. All working as expected now and giving me correct numbers out.

I have adapted your code and added a few things which are needed. Next I would like to color the 2 bottom right rectangles within each larger "black" rectangles differently and finally count how many such full differently colored retangle pairs remain inside the inner circle. This completes my requirement.

(If counting the differently colored full rectangle pairs is not possible computationally, I am happy to count them visually)

clear; clc;

wDia = 152400; % waferDia [um]

EdgeExcl = 10000 ; % edge exclusion [um]

wR = wDia/2;

waferRadiusEdgeExluded = wR - EdgeExcl;

R = waferRadiusEdgeExluded;

H = 2250; % die height [um]

W = 5000; % die width [um]

rH = 13500; % reticle height [um]

rW = 15000; % reticle height [um]

circ = nsidedpoly(1000,"Radius",R); %polyshape for the circle

wfr = nsidedpoly(1000,"Radius",wR); %polyshape for wafer

Grid = scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[W,H]);

Grid = arrayfun(@(i)translate(Grid,H*[0,i]),-ceil(R/H):ceil(R/H)-1)';

Grid = arrayfun(@(i)translate(Grid,W*[i,0]),-ceil(R/W):ceil(R/W)-1,'uni',0);

Grid = vertcat(Grid{:}); %polyshape vector for grid rectangles

dieContained=sum( area(intersect(Grid,circ))>=area(Grid(1))*0.9999 ); disp ("Gross die/Wfr"); disp (dieContained);

Gross die/Wfr 1148

% plot die grid on uable wafer

dp = plot([circ; Grid],'FaceColor','m', 'EdgeColor', 'w'); axis equal; hold on; % plot usable wafer

%%% plot reticle grid on full wafer

rGrid = scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[rW,rH]);

rGrid = arrayfun(@(i)translate(rGrid,rH*[0,i]),-ceil(wR/(rH)):ceil(wR/(rH))-1)';

rGrid = arrayfun(@(i)translate(rGrid,rW*[i,0]),-ceil(wR/(rW)):ceil(wR/(rW))-1,'uni',0);

rGrid = vertcat(rGrid{:}); %polyshape vector for grid rectangles

wp = plot([wfr; rGrid],'FaceColor','none'); axis equal; %plot full wafer

rContained=sum( area(intersect(rGrid,circ))>=area(rGrid(1))*0.9999 ); disp ("Gross ReticleShot/Wfr"); disp (rContained);

Gross ReticleShot/Wfr 52

Matt J 2024-5-21

编辑：Matt J 2024-5-22

在 MATLAB Online 中打开

clear; clc;

wDia = 152400; % waferDia [um]

EdgeExcl = 10000 ; % edge exclusion [um]

wR = wDia/2;

waferRadiusEdgeExluded = wR - EdgeExcl;

R = waferRadiusEdgeExluded;

H = 2250; % die height [um]

W = 5000; % die width [um]

rH = 13500; % reticle height [um]

rW = 15000; % reticle height [um]

tH=round(rH/H);

tW=round(rW/W);

%%Form the bounding circles as polyshapes

innerCirc = nsidedpoly(1000,"Radius",R); %polyshape for the circle

outerCirc = nsidedpoly(1000,"Radius",wR); %polyshape for wafer

%%%Form individual cells as polyshapes

smallCell= scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[W,H]);

largeCell=scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[rW,rH]);

subgrid=tileShapes(smallCell,[W,H],[0,tW-1,0,tH-1]); %6x3 tiling of one large cell

cornerCell=union(subgrid([12,18]));

%%%Tile the cells

coarseTiles=tileShapes(largeCell,[rW,rH], getLimits(wR,[rW,rH]));

fineTiles=tileShapes(subgrid,[rW,rH], getLimits(R,[rW,rH]));

cornerTiles=tileShapes(cornerCell,[rW,rH],getLimits(R,[rW,rH]) );

%%%Plot the elements

plot([outerCirc;coarseTiles(:)],'FaceColor','none'); axis equal;hold on

plot([innerCirc; fineTiles(:)],'FaceColor','m', 'EdgeColor', 'w');

plot(cornerTiles,'FaceColor','g'); hold off

%%%Compute containment

dieContained=numInCircle(innerCirc,fineTiles); disp ("Gross die/Wfr"); disp (dieContained);

Gross die/Wfr 1148

rContained=numInCircle(innerCirc,coarseTiles); disp ("Gross ReticleShot/Wfr"); disp (rContained);

Gross ReticleShot/Wfr 52

cornersContained=numInCircle(innerCirc,cornerTiles)

cornersContained = 59

function lims=getLimits(R,xy)

L=ceil(R./xy);

lims=[-L(1),+L(1)-1, -L(2),+L(2)-1];

end

function N=numInCircle(circ,tiles)

V=vertcat(tiles.Vertices);

N=sum( all( reshape(isinterior(circ, V),4,[]) ,1));

end

function Tiles=tileShapes(p,stride,lims)

arguments

p; stride (1,2); lims (1,4) {mustBeInteger}

end

W=stride(1);

H=stride(2);

lims=num2cell(lims);

[Xmin,Xmax,Ymin,Ymax]=deal(lims{:});

Tiles=p;

Tiles = arrayfun(@(i)translate(Tiles,W*[i,0]),Xmin:Xmax,'uni',0);

Tiles=horzcat(Tiles{:});

Tiles = arrayfun(@(i)translate(Tiles,H*[0,i]),(Ymax:-1:Ymin)','uni',0);

Tiles = vertcat(Tiles{:});

end

JM 2024-5-22

@Matt J. This is how I changed the body of your code to count BRCSR as single rectangles. Please see if this is fine or could be better

tH=round(rH/H);

tW=round(rW/W);

%%Form the bounding circles as polyshapes

innerCirc = nsidedpoly(1000,"Radius",R); %polyshape for the circle

outerCirc = nsidedpoly(1000,"Radius",wR); %polyshape for wafer

%%%Form individual cells as polyshapes

smallCell= scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[W,H]);

largeCell=scale(nsidedpoly(4,'Side',1,'Center',[+1,+1]/2),[rW,rH]);

subgrid=tileShapes(smallCell,[W,H],[0,tW-1,0,tH-1]); %6x3 tiling of one large cell

cornerCell_1 = subgrid(12);

cornerCell_2 = subgrid(18);

%%%Tile the cells

coarseTiles=tileShapes(largeCell,[rW,rH], getLimits(wR,[rW,rH]));

fineTiles=tileShapes(subgrid,[rW,rH], getLimits(R,[rW,rH]));

cornerTiles_1=tileShapes(cornerCell_1,[rW,rH],getLimits(R,[rW,rH]));

cornerTiles_2=tileShapes(cornerCell_2,[rW,rH],getLimits(R,[rW,rH]));

%%%Plot the elements

plot([innerCirc; fineTiles(:)],'FaceColor','m', 'EdgeColor', 'w'); axis equal;hold on

plot([outerCirc;coarseTiles(:)],'FaceColor','none', 'EdgeColor', 'b');

plot(cornerTiles_1,'FaceColor','g');

plot(cornerTiles_2,'FaceColor','c'); hold off;

%%%Compute containment

dieContained=numInCircle(innerCirc,fineTiles); disp ("Gross die/Wfr"); disp (dieContained);

rContained=numInCircle(innerCirc,coarseTiles); disp ("Gross ReticleShot/Wfr"); disp (rContained);

cornersContained_1=numInCircle(innerCirc,cornerTiles_1); disp (cornersContained_1);

cornersContained_2=numInCircle(innerCirc,cornerTiles_2); disp (cornersContained_2);

The sum of the last two outputs is 123 which is what I expect.

JM 2024-5-22

编辑：JM 2024-5-22

Upgrade, as much as I would want it, is beyond my control I am afraid

JM 2024-5-22

移动：Matt J 2024-5-28

@Matt J. Thanks

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Answer 2

Voss 2024-5-20

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2120706-how-plot-a-grid-of-rectangles-on-an-overlaid-circle#answer_1460416

在 MATLAB Online 中打开

R = 5; % circle radius

L = 1; % horizontal grid spacing

W = 2; % vertical grid spacing

C = [7 8]; % center of the circle

% calculate some points on the circle

th = linspace(0,2*pi,100);

xc = C(1)+R*cos(th);

yc = C(2)+R*sin(th);

% calculate the grid's x and y coordinates

xg = C(1)+W*(floor(-R/W):ceil(R/W));

yg = C(2)+L*(floor(-R/L):ceil(R/L));

% plot the circle

plot(xc,yc)

axis equal

% create the vertical grid lines

nx = numel(xg);

xdata = [xg;xg;NaN(1,nx)];

ydata = [repmat(yg([1 end]).',1,nx); NaN(1,nx)];

line(xdata,ydata,'Color','k')

% create the horizontal grid lines

ny = numel(yg);

ydata = [yg;yg;NaN(1,ny)];

xdata = [repmat(xg([1 end]).',1,ny); NaN(1,ny)];

line(xdata,ydata,'Color','k')

% count the rectangles completely inside the circle

n_rectangles_inside = nnz(conv2((xg-C(1)).^2+(yg.'-C(2)).^2 <= R^2,ones(2)) == 4)

n_rectangles_inside = 28

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