Error using surf, X, Y, Z, and C cannot be complex

i run below code, but get X, Y, Z, and C cannot be complex error, any idea what is wrong?

function simple_math_functions_animation1
clc, close all, clear all

hf1=figure(1);hold on,grid on,axis equal, view([1 -1 1])
set(hf1,'Color','w');set(hf1,'Position',[300, 600, 500, 400]);
xlabel('x');ylabel('y'),zlabel('z');

   k=1;
   u = linspace(0,2*pi,72);
   v = [-3:.2:-1,1:.2:3];
   [U,V] = meshgrid(u,v);
   r=sqrt((4*V.^-k)./(cos(U).^2+k*sin(U).^2));
   X = r.*cos(U);
   Y = r.*sin(U);
   Z = V;

surf(X,Y,Z,'EdgeColor',[0.5 1. 0.2],'FaceColor',[1 0.2 0.8],'FaceAlpha',0.6);

XYZ=[reshape(X,1,prod(size(X)));
     reshape(Y,1,prod(size(Y)));
     reshape(Z,1,prod(size(Z)));
     ones(1,prod(size(Z)))];
phi=[0 : pi/20 : 50*pi];
h=[]; axis([-20 20 -20 20 -20 20]);

for beta=phi   % animation loop  *****************

      T=[cos(beta) -sin(beta) 0    0;     % rotation matrix
         sin(beta)  cos(beta) 0    0;
          0          0        1    0;
          0          0        0    1];

      XYZ1=T*XYZ;  % coordinates changing
      X1=reshape(XYZ1(1,:),size(X));Y1=reshape(XYZ1(2,:),size(Y));Z1=reshape(XYZ1(3,:),size(Z));
      pause(0.1);if ~isempty(h),delete(h);end
      h=surf(X1,Y1,Z1,'EdgeColor',[0.5 1. 0.2],'FaceColor',[0.2 0.2 0.8],'FaceAlpha',0.6);

end       % ******************************************

end

Answer 1

[Thomas Koelen]

you were taking square roots of negative values, thus giving complex values.

 r=abs(sqrt((4*V.^-k)./(cos(U).^2+k*sin(U).^2)));

should work