Given a triangle ABC and triangle A'B'C', there is exactly one affine transformation T for which A'B'C' is the image of ABC. The image of any other point by T can be constructed from the two triangles. By using the Sketchpad Locus construction, one can construct the image by T of various sets in the plane.
In this figure, the transformation T maps the blue triangle ABC to the red triangle A'B'C' and the image of the blue circle is the red ellipse.
If you drag the points ABC and A'B'C' so that the triangles are congruent or similar, the ellipse will become a circle, since the transformation will become an isometry or a similarity transformation.
Suppose angle A is a right angle (you can drag the points so that this is approximately true). If you drag A' to A and B' to B and C' to a point along AC, then you will see how the circle is stretched in the AC direction. If the angle at A is not a right angle, the axis of the ellipse is related more subtly to the triangles.
Finally, keeping A' at A and B' at B, it is interesting to move C' so that CC' is parallel to AB. This is a shear.
James King, 1/8/2003