ISOPTIC CURVE

The isoptic with angle  of the cardioid is composed of two limaçons of Pascal.

 Curve studied by La Hire in 1704 and Chasles in 1837. From the Greek Isos "equal" and optikos "relative to sight". Name given by Taylor in 1884.

In differential geometry, the isoptic (curve) with angle  of a curve is the locus of the points through which pass two tangents to the curve forming an angle .

 A method to obtain the equation of the isoptic of the curve f (x, y) = 0: eliminate x1, y1, x2, y2 between the 5 equations: f (x1, y1) = 0, f (x2, y2) ) = 0, p (x1, y1) * (x-x1) + q (x1, y1) * (y-y1) = 0, p (x2, y2) * (x-x2) + q (x2, y2) ) * (y-y2) = 0, and tan (alpha) = (p (x1, y1) -p (x2, y2)) / (1 + p (x1, y1) * q (x2, y2)).

Examples:

 - the isoptics of the parabola are hyperbolas (with same focus and directrix as the parabola), with equation: . On the right, the case of an angle  (the other branch of the hyperbola would give the supplementary angle).

- the isoptics of the centred conics are the plane spiric curves: for the conic , the isoptic curve has equation (Loria 2D tome 2  p. 156).
- the isoptics of the (- epi, - hypo) trochoids are the reunions of (- epi, - hypo) trochoids (see the example of the cardioid above).
- the isoptics of sinusoidal spirals are sinusoidal spirals.

A related notion in metrical geometry, bearing the same name, is that of isoptic with angle  of a part X of the plane: locus of the vertices of the angular domains of angle  that circumscribe X (i.e. that contain X, and the two sides of which intersect with X).

Examples:
- the isoptics of a segment line are the circles a chord of which is this segment line (theorem of the capable arc).

A different notion was also referred to as isoptic: the isoptic of two parts X and Y of the plane is the locus of the points of the plane where X and Y are "seen" under the same angle, more precisely, the locus of the vertices of two angular sectors of same angle, one of which circumscribes X while the other circumscribes Y.

When the two parts X and Y are segment lines, we get the isoptic cubics.

The case of two circles with centres , and radii  is much simpler: we get the circles of Apollonius with equation .