Conic sections are among the best-known geometric objects. How can they be constructed with straightedge and compass? Let's begin with the ellipse. The first challenge is to construct a point on it and the tangent at that point.
And perhaps more, if the affinity is right... --------------------
Suppose the two axes of an ellipse are known. To make the diagram easier to draw (without any loss of generality), we shall take the major axis to lie along the x-axis and the minor axis along the y-axis; the center of the ellipse will be O, the origin of the coordinate system. The problem is to construct a point on the ellipse with a given x-coordinate. We shall use the fact that the ellipse is the image of the circle with the major axis as diameter under an affinity (and also the image of the circle with the minor axis as diameter under another affinity). An affinity with axis (D), direction v→\overrightarrow{v} and ratio given by the nonzero real number k is the map that sends each point M in the plane to the point M’ constructed as follows: draw through M the line in direction v→\overrightarrow{v}, which meets (D) at H; M’ is then defined by HM′→=kHM→.\overrightarrow{HM'} = k \overrightarrow{HM}.
When (D) and v→\overrightarrow{v} are perpendicular, the affinity is called orthogonal.