The conchoidal by Dürer , or mussel line , is a special plane algebraic curve . Albrecht Dürer drew it for the first time in his book Underweysung of Measurement (p. 38) and called it "a muschellini".
Dürer's construction of the shell line
equation
Cartesian coordinates:
(
y
2
+
x
y
+
a
y
-
b
2
)
2
=
(
b
2
-
y
2
)
(
y
-
x
+
a
)
2
{\ displaystyle (y ^ {2} + xy + ay-b ^ {2}) ^ {2} = (b ^ {2} -y ^ {2}) (y-x + a) ^ {2}}
Parameter equation (2 curve branches):
x
(
t
)
=
t
+
b
(
a
-
t
)
a
2
-
2
a
t
+
2
t
2
,
y
(
t
)
=
b
t
a
2
-
2
a
t
+
2
t
2
,
{\ displaystyle x (t) = t + {\ frac {b (at)} {\ sqrt {a ^ {2} -2at + 2t ^ {2}}}} \;, \ y (t) = {\ frac {bt} {\ sqrt {a ^ {2} -2at + 2t ^ {2}}}},}
x
(
t
)
=
t
-
b
(
a
-
t
)
a
2
-
2
a
t
+
2
t
2
,
y
(
t
)
=
-
b
t
a
2
-
2
a
t
+
2
t
2
.
{\ displaystyle x (t) = t - {\ frac {b (at)} {\ sqrt {a ^ {2} -2at + 2t ^ {2}}}} \;, \ y (t) = - { \ frac {bt} {\ sqrt {a ^ {2} -2at + 2t ^ {2}}}}.}
(The second branch of the curve was not discovered by Dürer.)
properties
For the curve degenerates into the pair of lines and a circle .
a
=
0
{\ displaystyle a = 0}
y
=
±
b
/
2
{\ displaystyle y = \ pm b / {\ sqrt {2}}}
x
2
+
y
2
=
b
2
{\ displaystyle x ^ {2} + y ^ {2} = b ^ {2}}
For the two branches of the curve degenerate into a straight line .
b
=
0
{\ displaystyle b = 0}
y
=
0
{\ displaystyle y = 0}
The curve has a peak for .
b
=
2
a
{\ displaystyle b = 2a}
(
x
,
y
)
=
(
-
2
a
,
a
)
{\ displaystyle (x, y) = (- 2a, a)}
See also Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
