#
Geometric Construction 13

Let a,b,c be complex numbers. Construct the curve
ae^{it} + b e^{i2t} + ce^{i3t} , t in [0,
2p].
Construct the tangent to this curve.

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

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

Find a constant c so that
e^{it} + 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 a_{N} 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
ae^{it} + b e^{i2t} , t in [0, 2p].

Construct the osculating circle of a complex curve of the form
ae^{it} + b e^{i2t} + ce^{i3t} , t in [0,
2p].

Construct the osculating circle of a complex curve of the form
e^{it} + ce^{-i2t} , t in [0, 2p],