I need to draw arc through three points in three-dimensional space. Can you please help me out?

x_i = [0,5,20];
y_i = [0,0,0];
z_i = [20,5,0];

P1 = [x_i(1);y_i(1);z_i(1)]

P1 = 3×1

0 0 20

P2 = [x_i(2);y_i(2);z_i(2)]

P2 = 3×1

5 0 5

P3 = [x_i(3);y_i(3);z_i(3)]

P3 = 3×1

20 0 0

v1 = cross(P2-P1,P3-P1);
v1 = v1/norm(v1)

v1 = 3×1

0 -1 0

syms x y z
P0 = [x;y;z];

eqn1 = dot(P0-P1,v1) == 0

eqn1 = 

eqn2 = dot(P0-(P2+P1)/2,P2-P1) == 0

eqn2 = 

eqn3 = dot(P0-(P3+P1)/2,P3-P1) == 0

eqn3 = 

[A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [x, y, z])

A = 

B = 

P_centre = linsolve(A,B)

Warning: Solution does not exist because the system is inconsistent.

P_centre = 

Can you please tell me where I am going wrong?

[Matt J]

x_i = [0,5,20];
y_i = [0,0,0];
z_i = [20,5,0];

line(x_i, y_i,z_i); view(3); grid on

[Siddharth Behal]

Hey Matt, Thanks for your answer but I need to draw a circular arc not straight lines.

[Siddharth Behal]

Can you please elaborate what you are trying to say? The points are P1, P2 and P3.

[Torsten]

syms a b 
syms R positive

P1 = [0 0 20];
P2 = [5 0 5];
P3 = [20 0 0];
P = [a 0 b];
eqn1 = a^2+(20-b)^2-R^2==0

eqn1 = 

eqn2 = (a-5)^2+(b-5)^2-R^2==0

eqn2 = 

eqn3 = (20-a)^2+b^2-R^2==0

eqn3 = 

sol = solve([eqn1, eqn2, eqn3],[a,b,R])

sol = struct with fields:
    a: 35/2
    b: 35/2
    R: (25*2^(1/2))/2

sol.a

ans = 

sol.b

ans = 

sol.R

ans = 

[Siddharth Behal]

Thank you so much Torsten. But can you please tell me where I am going wrong with the code above?

[Torsten]

x_i = [0,5,20];
y_i = [0,0,0];
z_i = [20,5,0];

P1 = [x_i(1);y_i(1);z_i(1)]

P1 = 3×1

0 0 20

P2 = [x_i(2);y_i(2);z_i(2)]

P2 = 3×1

5 0 5

P3 = [x_i(3);y_i(3);z_i(3)]

P3 = 3×1

20 0 0

v1 = cross(P2-P1,P3-P1);
v1 = v1/norm(v1)

v1 = 3×1

0 -1 0

syms x y z real
P0 = [x;y;z];

eqn1 = dot(P0-P1,v1) == 0

eqn1 = 

eqn2 = dot(P0-(P2+P1)/2,P2-P1) == 0

eqn2 = 

eqn3 = dot(P0-(P3+P1)/2,P3-P1) == 0

eqn3 = 

[A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [x, y, z])

A = 

B = 

P_centre = linsolve(A,B)

P_centre = 

[Siddharth Behal]

Thank you so much, Torsten. I really appreciate your help.

[Siddharth Behal]

Also, do you know how can I plot this arc? I just need the arc between three points instead of the circle

Answer 1

[Matt J]

I would suggest this FEX download,

https://www.mathworks.com/matlabcentral/fileexchange/87584-object-oriented-tools-for-fitting-conics-and-quadrics

x_i = [0,5,20];
y_i = [0,0,0];
z_i = [20,5,0];

P1 = [x_i(1),y_i(1),z_i(1)];
P2 = [x_i(2),y_i(2),z_i(2)];
P3 = [x_i(3),y_i(3),z_i(3)];


pfit=planarFit([P1;P2;P3]');

 xy=pfit.R(:,2:3)'*pfit.XYZ;

 cfit=circularFit(xy);

 C=cfit.sample(0:360);

XYZ =num2cell( pfit.R(:,2)*C{1} + pfit.R(:,3)*C{2} + pfit.normal'*pfit.distance  ,2);

scatter3(x_i,y_i,z_i,'filled','r','SizeData',50);
line(XYZ{:}); axis equal
xlabel x, ylabel y, zlabel z

[Siddharth Behal]

Is there a way to draw only the arc between three points instead of the entire circle?

[Matt J]

pfit=planarFit([P1;P2;P3]');

 xy=pfit.R(:,2:3)'*pfit.XYZ;

 cfit=circularFit(xy);

 xy=xy-cfit.center';
 t1=atan2d(xy(2,1),xy(1,1));
 t2=atan2d(xy(2,3),xy(1,3));

 C=cell2mat(cfit.sample(linspace(t1,t2)));

XYZ = num2cell( pfit.R(:,2)*C(1,:) + pfit.R(:,3)*C(2,:) + pfit.normal'*pfit.distance  ,2);

scatter3(x_i,y_i,z_i,'filled','r','SizeData',50);
line(XYZ{:}); axis equal
xlabel x, ylabel y, zlabel z