# Geometric Construction 13

Let a,b,c be complex numbers. Construct the curve
aeit + b ei2t + cei3t , t in [0, 2p].
Construct the tangent to this curve.

Construct the ellipse as a complex curve in the form
aeit + be-it , t in [0, 2p].
Construct the tangent to this curve.

Construct the rose curve of three leaves in the form
eit +  e-i2t  , t in [0, 2p].
Construct the tangent to this curve.

Find a constant c so that
eit +  ce-i2t  , t in [0, 2p],
appears as the deltoid. Construct the tangent to this curve.

For a planar curve, let s denote its arc length, T its unit tangent, N its unit normal, a its acceleration and k its curvature, we have
(Reference: p.811 of Salas/Hille, Calculus, 6th Edition.)
The quantity
is thus the normal component aN of acceleration. We can therefore construct the radius of curvature according to this construction
Construct the osculating circle of a complex curve of the form
aeit + b ei2t  , t in [0, 2p].

Construct the osculating circle of a complex curve of the form
aeit + b ei2t + cei3t , t in [0, 2p].

Construct the osculating circle of a complex curve of the form
eit +  ce-i2t  , t in [0, 2p],