ORTHOPOLAR OF A CURVE WITH RESPECT TO TWO LINES

 Notion studied by Henri Lazennec in 2015. Homemade name.

 Given two secant lines  and , consider the projection M of a point  on the polar line of  with respect to the lines  and . When the point  describes a curve , the point M describes the orthopolar of the curve  with respect to the lines  and . In the following, we take the orthogonal lines  and , equal to the axes. In this case, if , then . Therefore, we have the simple result in polar coordinates:
 The orthopolar with respect to the axes of the curve  is the curve .

Examples (initial curve in blue, orthopolar in red):

 The orthopolars of lines that do not pass by O are the strophoids. More precisely, the orthopolar with respect to the axes of the line  is the strophoid . It is the right strophoid for . The orthopolar of a circle centred on O is a quadrifolium. The orthopolar with respect to the axes of the circle passing by O is the curve .   For , we get the torpedo... ... and for  we get the regular bifolium. An example with a circle that does not pass through O. The orthopolar of the cross-curve is the circle .

© Robert FERRÉOL 2017