P1 := x^2+y^2-4:
P2 := y^2-2*x+2:

Original question is find CAD of (some y)[P1 <0 and P2 <0]

how to use maple 12 and maple 2015 to find Q1,Q2,Q3 which are projection of P1 and P2

my book show sample points are [-4,-1-sqrt(7),-3,-2,0,1,3/2,-1+sqrt(7),9/5,2,3]
but FindSamples result is not the same with my book, is it my book wrong or FindSamples function wrong?
I find result of my script is the same as book's quantifier position at 7,8,9 though sample points has little different

how to generalize my following script to multiple variables x, y, z, and more ?

and

I compare with maple 2015 result are different from my book solution, is maple 2015 more advanced version CAD? 

with(ListTools):

P1 := x^2+y^2-4:
P2 := y^2-2*x+2:

Q1 := x^2 + 2*x - 6;
Q2 := x^2 - 4;
Q3 := x - 1;

sourcesamples := sort(evalf([solve(Q1), solve(Q2), solve(Q3)]),`<`);

FindSamples:=proc(sourcesamples)
local N, P;
N:=nops(sourcesamples);
P:=proc(a,b)
local a1, b1, m1, n, m;
if a=b then error "Should be a<>b" fi;
a1,b1:=op(convert(sort([a,b],(x,y)->evalf(x)<evalf(y)),rational));
count := 0:
for n from 1 do
m1:=a1*n;
m:=`if`(type(m1,integer),m1+1,ceil(m1));
count := count + 1:
if is(m/n>a1) and is(m/n<b1) then return m/n fi;
od;
print("count=",count);
end proc:
[ceil(sourcesamples[1])-1, seq(op([sourcesamples[i],P(sourcesamples[i],sourcesamples[i+1])]), i=1..N-1),sourcesamples[N],floor(sourcesamples[N])+1];
end proc:

RemoveComplex := proc(yy)
local result, k:
result := []:
for k in yy do
if Im(k) = 0 then
result := [op(result), k]:
end if:
od:
if result = [] then
result := []:
end if:
return result:
end proc:

Joinsolution := proc(param1, param2group)
local result, k:
result := []:
for k in param2group do
result := [op(result), [param1, k]]:
od:
return result:
end proc:

CADsamples := FindSamples(sourcesamples):
CADresult1 := []:
for mm in CADsamples do
#print(mm):
if MakeUnique(RemoveComplex([solve(subs(x=mm, P1)), solve(subs(x=mm, P2))])) = [] then
CADresult1 := [op(CADresult1), op(Joinsolution(mm,[0]))];
else
CADresult1 := [op(CADresult1), op(Joinsolution(mm,FindSamples(sort(evalf(MakeUnique(RemoveComplex([solve(subs(x=mm, P1)), solve(subs(x=mm, P2))]))),`<`))))];
end if:
od:
CADresult1;

for mm in CADresult1 do
if subs(x=mm[1],subs(y=mm[2], P1)) < 0 and subs(x=mm[1],subs(y=mm[2], P2)) < 0 then
print("solution ", mm, SearchAll(mm[1],CADsamples), evalf(mm)):
end if:
od:

Compare with

with(RegularChains):
with(ChainTools):
with(MatrixTools):
with(ConstructibleSetTools):
with(ParametricSystemTools):
with(SemiAlgebraicSetTools):
with(FastArithmeticTools):
R := PolynomialRing([x,y]):
sys := [x^2+y^2-4,y^2-2*x+2]:
N := []:
P := []: 
H := [x]:
dec := RealTriangularize(sys,N,P,H,R):
proj := Projection(sys, N, P, H, 1, R);
Display(dec, R);

P := SamplePoints(sys, R);
Display(P, R);
cad := CylindricalAlgebraicDecompose(sys, R);
 

is there easy way to simulate lines rotation like robotic arm following point P moving by adjusting angle q1 and q2?

i feel point two lines and simulate static two lines in animate is also not easy.

with(plots):
with(geometry):
#Joint at (5,5)
#length a  2, y=7, x, solve(subs(y=7,2^2 = (y-5)^2+(x-5)^2), x)[1];
#length b 2, y=1, x, solve(subs(y=4,2^2 = (y-5)^2+(x-5)^2), x)[1];
point(A, 5, 5), point(B, solve(subs(y=7,2^2 = (y-5)^2+(x-5)^2), x)[1], 7):
point(C, 5, 5), point(D, solve(subs(y=4,2^2 = (y-5)^2+(x-5)^2), x)[1], 4):
line(L1, [A, B]);
line(L2, [C, D]);
_EnvHorizontalName := x;
_EnvVerticalName := y;
Equation(L1);
Equation(L2);
solve(Equation(L1), y);
solve(Equation(L2), y);
plot([solve(Equation(L1), y),solve(Equation(L2), y)], x = 1 .. 2);
animate(plot, [Equation(L1),Equation(L2), x = 0 .. 2], A = -3 .. 3);

Angle Reference from below

angle of L2 , q1a

angle of L1,  q2a

p1 := vector([a*cos(q1), a*sin(q1)]);
P := evalm(p1 + b*vector([cos(q1+q2), sin(q1+q2)]));
Pe := map(expand, P);
A := {cos(q1) = c1, sin(q1) =s1, cos(q2)=c2, sin(q2)=s2};
P := subs(A, op(Pe));
F1 := [x - P[1], y - P[2], s1^2+c1^2-1, s2^2+c2^2-1 ];
F2 := subs({a=1, b=1}, F1);
with(Groebner):
g1 := Basis(F2, plex(c2, s2, c1, s1, y, x));
g1 := [-4*x^2+x^4+2*x^2*y^2+y^4-4*y*x^2*s1-4*y^3*s1+4*s1^2*x^2+4*s1^2*y^2, 2*y*s1+2*x*c1-x^2-y^2, 4*x-x^3-y^2*x+2*y*x*s1-2*y^2*c1-4*s1^2*x+4*y*s1*c1, s1^2+c1^2-1, x*s1-y*c1+s2, -y^2+2-x^2+2*c2];
map(indets, g1);
 

g2 := Basis(F2, plex(c2, s2, c1, s1));
 

assume point x = 1, y = 0.5
g2spec := subs({x=1, y=1/2}, [g2[1],g2[3]]);
S1 := [solve(g2spec[1])];
q1a := evalf(arcsin(S1[1]));
q1b := evalf(arcsin(S1[2]));

S2 := [solve(subs(s1=S1[1], g2spec[2])), solve(subs(s1=S1[2], g2spec[2])) ];
q2a := evalf(arcsin(S2[1]));
q2b := evalf(arcsin(S2[2]));

q2a := subs({a=2,b=2}, q2a);
if q2a < 0 then
 q2a_pi := evalf(eval(-q2a-Pi));
else
 q2a_pi := evalf(eval(Pi-q2a));
end if:
q2a_pi;

q2b := subs({a=2,b=2}, q2b);
if q2b < 0 then
 q2b_pi := evalf(eval(-q2b-Pi));
else
 q2b_pi := evalf(eval(Pi-q2b));
end if:
q2b_pi;

I follow below question to generate binary sequence
but number of sequence is 6 not 5, why 011001 is wrong?

number of sequence should be catalan number.

how to generate this kind of sequence correctly?

For ecah positive integer n, how many 2n-digit binary sequence
 b1,b2,b2n with n "0" and n "1" are there such that
for each i=1,2,...2n, the number of "0" is larger than or equal 
to the number of "1" in the subsequence b1,b2,...bi?

with(combinat):
with(ListTools):
with(SumTools):
source := permute(Flatten([seq([0,0,0,1,1,1], k=1..1)]), 6);
aavxa := []:
for z in source do
if Summation(z[k], k = 1 .. nops(z)) = 3 then
if z[nops(z)] = 1 and z[1] = 0 then
if Reverse(z) in aavxa then
else
aavxa := [op(aavxa), z];
end if:
end if:
end if:
od:
nops(aavxa);
aavxa;
 

y(t)/diff(y(t),t)

how to dsolve when highest derivative at denominator?

For a from 1 to 2 do

convert(cat("k",a),symbol) := a;

od;

Error : can not do assignment 

how to do ?