Deslanges trisectrix

Deslanges trisectrix

The Deslanges trisectrix (sometimes also spelled Delanges) is a curve belonging to the family of Deslanges sectrices, named after the Italian engineer and mathematician Paolo Deslanges (c. 1750–1810), who studied them in 1783. Like other trisectrix curves, it can be used as an auxiliary tool to perform angle trisection with a straightedge and compass, although the use of such auxiliary tools is outside the admissible methods of classical geometry.

Geometric definition

Construction of the Deslanges trisectrix

The geometric construction of the Deslanges trisectrix is based on a circle centered at the origin O, such that the points of the trisectrix form the locus determined by the intersection of two families of lines:

• Radial lines from the origin forming an angle of $2\alpha$ with the x-axis.
• Horizontal lines drawn from the intersection points of the circle and rays forming an angle of $\alpha$ with the x-axis.

According to the diagram, to determine a point P, two radii are drawn: OP₁ (at angle $\alpha$) and OP₂ (at angle $2\alpha$). From point P₁ (the intersection of the first radius and the circle), a line parallel to the x-axis is drawn, which intersects the ray OP₂ at point P.

Equations

According to the geometric definition, the polar equation is of the form:

$\rho ={\frac {a}{\cos(\theta /2)}}$

An equivalent form of the equation in polar coordinates uses the secant function to write the expression compactly: $\rho =a\sec((\theta -\sigma )/2)$ (where $\sigma$ is a parameter to orient the curve with respect to the origin of the angles).

In Cartesian coordinates, the equation of the trisectrix is of the form:

$(x^{2}+y^{2}-2a^{2})^{2}=x^{2}(x^{2}+y^{2})$

Properties

The Deslanges trisectrix and the folium of Dürer are inverse curves to each other with respect to any circle centered at the common center of symmetry of both curves.

Trisection

Construction of the trisection of an arbitrary angle

To determine the trisection of an arbitrary angle $\alpha$, the following construction can be used:

• Draw the circle tangent to the interior of the trisectrix, centered at O with radius a.
• Draw the radius forming an angle of $\alpha$ counterclockwise from the positive x-axis, which determines point P₂ on the circle.
• Draw the tangent to the circle at P₂, which intersects the trisectrix at point P₁.
• The angle P₁OP₂ measures $\alpha /3$.

Deslanges sectrices

Generalizing the trisectrix formula for a value $n$ other than 2 yields the following expression:

$\rho ={\frac {a}{\cos(\theta /n)}}$

from which Deslanges sectrices of order $n+1$ (for integer $n$) can be generated.

Gallery

Deslanges sectrices

The Deslanges trisectrix and its inverse, the folium of Dürer

See also

• Cotes's spiral
• Epispiral

External links

• "DELANGES TRISECTRIX AND SECTRIX". mathcurve. Retrieved March 17, 2021.