Affine Image of a Circle

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.

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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


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