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ISOPTIC CURVE

The isoptic with angle p/3 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 .

Examples:

- the isoptics of the parabola are hyperbolas (with same focus and directrix as the parabola), with equation: .

- the isoptics of the centred conics are the plane spiric curves
(Loria 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 .

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© Robert FERRÉOL 2017