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ASTROID : Inversion
Click on the diagram below to open the applet, and then click on the ‘Animate’ button to generate the inverse of the astroid. Click the button again to stop the generation. Clicking the little red x at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly around the circle. Can you describe the inverse construction? Is this a curve we have seen before?
In this applet we have taken the circumcircle as the circle of inversion. As the driver point Q moves around the red circle, the point P generates the astroid, and the point R traces out the inverse curve. This resulting ‘butterfly ’ curve is quite attractive, although it is a curve we have not seen before. The inverse has four cusps, corresponding to the cusps of the astroid. If we compare the inverse obtained here with the inverse of the deltoid curve, we see here another similarity between the astroid and the deltoid.
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We have already looked at the astroid as a type of four-pointed ornamental star. If we wish to take the inverse of this curve, the obvious centre of inversion will be the origin, O, the central point of the astroid.
