function C = lp_curve(a, nlobe, t)
% LP_CURVE  A smooth closed curve and its first two derivatives.
%
%   r(t) = 1 + a cos(nlobe t),   x = r cos t,   y = r sin t
%
% Returned as one matrix with columns [x y x' y' x'' y''] rather than as six
% outputs, because the drivers call this from inside timing loops and numbl
% compiles a multiple-output call in a loop through its slow path.
%
% The curve is analytic and traversed counterclockwise, which is what makes the
% trapezoid rule spectrally accurate on it and fixes the sign of the outward
% normal (y', -x')/|(x', y')|.
t = t(:);
r = 1 + a * cos(nlobe * t);
rp = -a * nlobe * sin(nlobe * t);
rpp = -a * nlobe^2 * cos(nlobe * t);
c = cos(t);
s = sin(t);
C = [r .* c, ...
     r .* s, ...
     rp .* c - r .* s, ...
     rp .* s + r .* c, ...
     rpp .* c - 2*rp .* s - r .* c, ...
     rpp .* s + 2*rp .* c - r .* s];
end