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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 .

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© Robert FERRÉOL 2017