Trisectric by Longchamps (red)
The Longchamps trisectrix is a flat curve named after the French mathematician Gohierre de Longchamps (1842–1906) , which can be used to divide angles into three (hence trisectrix ).
definition On a circle with a center and a diameter , the point rotates at a constant speed in a positive angular direction and the point rotates at double speed in the opposite direction. The point starts at the point and the point at the other end of the diameter starts at the point . The circle tangents in the points and intersect in a point . The locus of Punkt is the Trisektrix of Longchamps.
M.
{\ displaystyle M}
A.
B.
{\ displaystyle AB}
B.
′
{\ displaystyle B '}
A.
′
{\ displaystyle A '}
B.
′
{\ displaystyle B '}
B.
{\ displaystyle B}
A.
′
{\ displaystyle A '}
A.
{\ displaystyle A}
A.
′
{\ displaystyle A '}
B.
′
{\ displaystyle B '}
E.
{\ displaystyle E}
E.
{\ displaystyle E}
Equation and parameter form For a circle with a radius whose center is at the origin of the coordinate system, the following equation is obtained in polar coordinates :
a
{\ displaystyle a}
r
=
-
a
cos
(
3
φ
)
{\ displaystyle r = - {\ frac {a} {\ cos (3 \ varphi)}}}
The following equation then results for Cartesian coordinates :
x
(
x
2
-
3
y
2
)
+
a
(
x
2
+
y
2
)
=
0
{\ displaystyle x (x ^ {2} -3y ^ {2}) + a (x ^ {2} + y ^ {2}) = 0}
The following is obtained as a parameter curve in Cartesian coordinates with trigonometric functions:
γ
:
[
0
,
2
π
]
→
R.
2
{\ displaystyle \ gamma: [0.2 \ pi] \ rightarrow \ mathbb {R} ^ {2}}
γ
(
t
)
=
(
x
(
t
)
y
(
t
)
)
=
(
-
cos
(
t
)
cos
(
3
t
)
a
-
cos
(
t
)
sin
(
3
t
)
a
)
{\ displaystyle \ gamma (t) = {\ begin {pmatrix} x (t) \\ y (t) \ end {pmatrix}} = {\ begin {pmatrix} - {\ frac {\ cos (t)} { \ cos (3t)}} a \\ - {\ frac {\ cos (t)} {\ sin (3t)}} a \ end {pmatrix}}}
It is also possible to display it as a parameter curve in Cartesian coordinates with rational functions :
γ
:
(
-
∞
,
∞
)
→
R.
2
{\ displaystyle \ gamma: (- \ infty, \ infty) \ rightarrow \ mathbb {R} ^ {2}}
γ
(
t
)
=
(
x
(
t
)
y
(
t
)
)
=
(
-
1
+
t
2
1
-
3
t
2
a
1
+
t
2
1
-
3
t
2
a
t
)
{\ displaystyle \ gamma (t) = {\ begin {pmatrix} x (t) \\ y (t) \ end {pmatrix}} = {\ begin {pmatrix} - {\ frac {1 + t ^ {2} } {1-3t ^ {2}}} a \\ {\ frac {1 + t ^ {2}} {1-3t ^ {2}}} at \ end {pmatrix}}}
properties
Trisectric by Longchamps (red) with asymptotes (dotted), axes of symmetry (dashed) and trifolium (blue)
Longchamps' trisectrix has three asymptotes and three axes of symmetry.
Asymptotes
x
=
a
3
{\ displaystyle x = {\ frac {a} {3}}}
y
=
-
1
3
x
-
2
a
3
3
{\ displaystyle y = - {\ frac {1} {\ sqrt {3}}} x - {\ frac {2a} {3 {\ sqrt {3}}}}}
y
=
1
3
x
+
2
a
3
3
{\ displaystyle y = {\ frac {1} {\ sqrt {3}}} x + {\ frac {2a} {3 {\ sqrt {3}}}}}
Axes of symmetry
y
=
0
{\ displaystyle y = 0}
y
=
3
2
x
{\ displaystyle y = {\ frac {\ sqrt {3}} {2}} x}
y
=
-
3
2
x
{\ displaystyle y = - {\ frac {\ sqrt {3}} {2}} x}
An inversion of the trisectrix on the circle from its definition yields a trifolium curve .
literature
Gino Loria: Special Algebraic and Transcendent Plane Curves: Theory and History . Teubner, 1902, pp. 87-88
Heinrich Wieleitner: Special level curves . GJ Göschen, Leipzig 1908, p. 47
Vladimir Rovenski: Geometry of Curves and Surfaces with MAPLE . Springer, 2013, ISBN 9781461221289 , p. 70
Eugene V. Shikin: Handbook and Atlas of Curves . CRC Press, 1996, ISBN 9780849389634 , p. 355
Web links
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle \ gamma: [0.2 \ pi] \ rightarrow \ mathbb {R} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bea32becc154f483bda6d73f47426b17ddd93952)
