  A differentiable curve y = y(x) is determined by its tangent lines and is
said to be the envelope of its tangent lines. The coefficients of the curve's
tangent lines form a curve in another space, called the dual space. There is a
transformation between the original x,y-space and the dual space, such that
points on the original curve and the curve of the coefficients of the tangent
lines are transformed into each other.The dual space and the transformation
depend upon the form that is used for the tangent lines. One choice is y = mx +
b, so that the coordinates in the dual space are m and b, where the curve
representing the tangent lines is b = b(m). Each point of the curve b = b(m) in
the dual space corresponds to a tangent line to the curve y = y(x) in
x,y-space. We present other choices of the form for the tangent lines and
explore techniques for finding a curve from the tangent lines, transformations
between the original space and a dual space and among dual spaces, the
differential equation, whose solutions are exactly the equations of the
equations of the tangent lines and the original curve. Geometric constructions
and other tools are used to find dual curves.
