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 . 