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NEGATIVE PEDAL or ORTHOCAUSTIC
Let M_{0} be the
current point on (G_{0}).
The current point M on the negative pedal with fixed point O
is defined by
which results in :
in Cartesian coordinates, in complex parametrization, and in polar coordinates, with ; here (see notations). 
The negative pedal of a curve with fixed point O
is the curve whose pedal is the
initial curve. See the bottom of this page
for a kinematic interpretation of this definition.
The negative pedal of a curve (G_{0})
with fixed point O is therefore the envelope
of the lines perpendicular at the point M to the lines (OM),
as M describes (G_{0}),
hence the equivalent name of orthocaustic.
The negative pedal is also the equidistant curve of O and the homothetic transformation of (G_{0}) with ratio 2 and center O. From this point of view, the negative pedal is called the isotel curve of (G_{0}) . 

If the vertex of a try square describes the initial curve
with one of the sides bound to pass through O, then the other side
envelopes the negative pedal.
Remark : the points on the side of the try square that passes through O describe conchoids of the initial curve, and the points on the other side describe isoconchoids. See glissette. 
The negative pedal with fixed point O is also the reciprocal polar of the inverse, for any circle with center O.
Examples :
 the orthocaustics of straight lines
are parabolae (focus at O,
the initial line being the tangent at the vertex)
 the orthocaustics of circles are
ellipses or hyperbolae depending on whether O is inside or outside
the circle (focus at O, the initial circle being the principal circle
of the conic)
 the negative pedal of a parabola
with its focus as fixed point is Tschirnhausen's
cubic
 the negative pedal of an ellipse
with its center as fixed point is Talbot's
curve
 the negative pedal of the involute
of a circle with its center as fixed point is the inverse
caustic of the circle
See more examples at pedal
!
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© Robert FERRÉOL 2017