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## The largest incircle and the smallest covering circle (2 procedures)

This post in reply to the Question, plot f(x)-g(x)...????

This message contains two procedures . The first procedure called  LargestIncircle  symbolically looks for the largest circle inscribed in a convex polygon . Polygon must be specified as a list of lists of its vertices . If the polygon does not have parallel sides , the problem has a unique solution , otherwise may be several solutions . It is easy to prove that there is always a maximum circle that touches three sides of the polygon . Procedure code based on this property of the optimal solution.

The second procedure called  MinCoveringCircle  symbolically seeking smallest circle that encloses a given set of points. The set of points can be anyone not necessarily being vertices of a convex polygon. Procedure code based on the following geometric properties of the optimal solution , which can easily be proved :

1) The minimum covering circle is unique.

2) The minimum covering circle of a set Set can be determined by at most three points in  Set  which lie on the boundary of the circle. If it is determined by only two points, then the line segment joining those two points must be a diameter of the minimum circle. If it is determined by three points, then the triangle consisting of those three points is not obtuse.

Code of the 1st procedure:

restart;

LargestIncircle:=proc(L::listlist)

local n, x, y, P, S, T, Eqs, IsInside, OC, Pt, E, R, t, Sol, P0, R1, Circle;

uses combinat, geometry;

n:=nops(L); _EnvHorizontalName := x; _EnvVerticalName := y;

P:=[seq(point(A||i, L[i]), i=1..n)];

if nops(convexhull(P))<n then error "L should be a convex polygon" fi;

S:={seq(line(l||i,[A||i, A||(i+1)]), i=1..n-1), line(l||n,[A||n, A||1])};

Eqs:=map(lhs,{seq(Equation(l||i), i=1..n)});

T:=choose({seq(l||i, i=1..n)}, 3);

IsInside:=proc(Point, OC, Eqs)

convert([seq(is(eval(Eqs[i], {x=Point, y=Point})*eval(Eqs[i], {x=OC, y=OC})>0), i=1..n)], `and`);

end proc;

R:=0; point(Pt,[x,y]);

for t in T do

solve([distance(Pt,t)=distance(Pt,t),      distance(Pt,t)=distance(Pt,t)],[x,y]);

Sol:=map(a->[rhs(a), rhs(a)], %);

P0:=op(select(b->IsInside(b, OC, Eqs), Sol)); if P0<>NULL then R1:=distance(point(E,P0),      t);

if convert([seq(is(R1<=distance(E, i)), i=S minus t)], `and`) and is(R1>R) then R:=R1;      Circle:=[P0, R] fi; fi;

od;

Circle;

end proc:

Code the 2nd procedure:

MinCoveringCircle:=proc(Set::{set, list}(list))

local S, T, t, d, A, B, C, P, R, R1, O1, Coord, L;

uses combinat, geometry;

if type(Set, list) then S:=convert(Set, set) else S:=Set fi;

T:=choose(S, 2);

for t in T do

d:=simplify(distance(point(A, t), point(B, t))); midpoint(C, A, B);

if convert([seq(is(distance(point(P, p), C)<=d/2), p in S minus {t, t})], `and`)   then return            [coordinates(C), d/2] fi;

od;

T:=choose(S, 3);

R:=infinity;

for t in T do

point(A, t); point(B, t); point(C, t);

if is(distance(A, B)^2<=distance(B, C)^2+distance(A, C)^2 and        distance(A,C)^2<=distance(B,        C)^2+distance(A, B)^2 and distance(B, C)^2 <= distance(A, C)^2 + distance(A, B)^2) then

if convert([seq(is(distance(point(P, p), O1)<=R1), p in S minus {t, t, t})], `and`) and is(R1<R) then    R:=R1; L:=[Coord, R] fi; fi;

od;

L;

end proc:

Examples of using the first procedure:

L:=[[0,2],[1,4],[4,4],[5,1],[3,0]]:

C:=LargestIncircle(L);

A:=plottools[disk](op(C), color=green):

B:=plottools[polygon](L, style=line, thickness=2):

plots[display](A,B, scaling=constrained); The procedure can be used to find the largest circle inscribed in a convex shape, bounded by the curves, if these curves are approximated by broken lines:

L:=[seq([k,k^2], k=-1..2, 0.1)]:

C:=LargestIncircle(L);

A:=plottools[disk](op(C), color=green):

B:=plottools[polygon](L, style=line, thickness=2):

plots[display](A,B, scaling=constrained); Examples of using the second procedure:

L:=[[0,2],[1,4],[4,4],[5,1],[3,0]]:

C:=MinCoveringCircle(L);

A:=plottools[circle](op(C), color=red, thickness=2):

B:=plottools[polygon](L, color=cyan):

plots[display](A,B); The smallest circle covering 30 random points:

> roll:=rand(1..10):

L:=[seq([roll(), roll()], i=1..30)]:

C:=MinCoveringCircle(L);

A:=plottools[circle](op(C), color=red, thickness=2):

B:=seq(plottools[disk](L[i],0.07,color=blue), i=1..nops(L)):

C1:=plottools[disk](C, 0.07, color=red):

T:=plots[textplot]([C[1,1]-0.2, C[1,2]-0.2, "C"], font=[TIMES,ROMAN,16]):

plots[display](A, B, C1, T);  ﻿