how to use plot to draw a minor arc in matlab without calculating the angle range?

In a two-dimensional surface, for a circle C, if I know its center coordinates, the radius, and two point coordinates on C, then the minor arc between the two points is uniquely determined. How can I draw it using plot or other drawing commands in matlab without calculating the angle range? Or is there any easy way to calculate the angle range?

Answer 1

[Voss]

c = [-1,2]; % center
r = 3;      % radius
p = [-1,5; -3.121,-0.121]; % two points on the circle: (-1,5) and (-3.121,-0.121)

th = atan2(p(:,2)-c(2),p(:,1)-c(1)); % angles from center c to points p

if abs(diff(th)) > pi       % enforce getting the minor arc:
    [~,idx] = min(th);      % if the angles are more than pi apart,
    th(idx) = th(idx)+2*pi; % increment the smaller one by 2*pi
end

% calculate points along the arc
th = linspace(th(1),th(2),100);
x = c(1)+r*cos(th);
y = c(2)+r*sin(th);

% plot
plot(x,y)
axis equal padded

Answer 2

[Bruno Luong]

% Generate A, B, C, C is the center and A B are two points on circle
r = rand();
C = randn(2,1);
phi1 = 2*pi*rand;
A = C + r * [cos(phi1); sin(phi1)];
phi2 = 2*pi*rand;
B = C + r * [cos(phi2); sin(phi2)];

% Compute the minor arc that link A and B
% never call trig function (or even sqrt) to compute angle
CA=A-C;
CB=B-C;
N=[CA(2);-CA(1)];
x = [CA+CB,N]\CB;
tmax=x(2);
n=max(ceil(abs(tmax)/deg2rad(1)),5);
t=linspace(0,tmax,n);
t2 = t.*t;
P=C + ((1-t2).*CA+t.*(2*N))./(1+t2); % points on arc

% Plot the arc
close all
hold on
plot(C(1),C(2),'+b','Markersize', 10);
text(A(1),A(2),'A')
text(B(1),B(2),'B')
text(C(1),C(2),'C')
plot(P(1,:),P(2,:),'r','linewidth',1)
axis equal

[Bruno Luong]

Extension in 3D (still 3 points A, B, C)

% Generate A, B, C
r = rand();
C = randn(3,1);
A = randn(3,1); A=C+A*r/norm(A);
B = randn(3,1); B=C+B*r/norm(B);

% never call trig function to compute angle
CA=A-C;
CB=B-C;
[Q,R]=qr([CA,CB],0);
N=Q(:,2)*R(1);
x = [CA+CB,N]\CB;
tmax=x(2);
n=max(ceil(abs(tmax)/deg2rad(1)),5);
t=linspace(0,tmax,n);
t2 = t.*t;
P=C + ((1-t2).*CA+t.*(2*N))./(1+t2); % points on arc

% Plot the arc
close all
hold on
plot3(C(1),C(2),C(3),'+b','Markersize', 10);
text(A(1),A(2),A(3),'A')
text(B(1),B(2),B(3),'B')
text(C(1),C(2),C(3),'C')
plot3(P(1,:),P(2,:),P(3,:),'r','linewidth',1)
axis equal
view(3)

[Bruno Luong]

Instead of

n=max(ceil(abs(tmax)/deg2rad(1)),5);
t=linspace(0,tmax,n);

(which can be big when A and B almost opposite, since tmax goes to infinity) a more carefully discretization of the arc is

resdeg = 2; % approx resolution of the arc, in degree 
phimax = pi-2/(abs(tmax)+2/pi);
n = max(ceil(phimax/deg2rad(resdeg)),3);
phi = linspace(0,phimax,n);
t = sign(tmax)*2*(1./(pi-phi)-1/pi);

[Bruno Luong]

I pack the method in a function GetPointOnArc.m

% Generate A, B, C
r = rand();
m = 3; % dimension
C = randn(m,1);
A = randn(m,1); A=C+A*r/norm(A);
B = randn(m,1); B=C+B*r/norm(B);
P = GetPointOnArc(A, B, C);
% Plot the arc
close all
hold on
if m == 2
    plot(C(1),C(2),'+b','Markersize', 10);
    text(A(1),A(2),'A')
    text(B(1),B(2),'B')
    text(C(1),C(2),'C')
    plot(P(1,:),P(2,:),'.r-','linewidth',1)
    axis equal
else
    plot3(C(1),C(2),C(3),'+b','Markersize', 10);
    text(A(1),A(2),A(3),'A')
    text(B(1),B(2),B(3),'B')
    text(C(1),C(2),C(3),'C')
    plot3(P(1,:),P(2,:),P(3,:),'.r-','linewidth',1)
    axis equal
    view(3)
end