Examples Involute
1 examples
1.1 involute of circle
1.1.1 curve length
1.2 involute of catenary
1.2.1 derivative
1.3 involute of cycloid
examples
involute of circle
the involute of circle (black) not identical archimedean spiral (red) in trivial manner.
the involute of circle resembles, not, archimedean spiral.
its successive turns parallel curves constant separation distance, property (inaccurately) ascribed archimedean spiral.
in cartesian coordinates x , y, involute of circle has parametric equation
x
=
r
(
cos
θ
+
θ
sin
θ
)
y
=
r
(
sin
θ
−
θ
cos
θ
)
{\displaystyle {\begin{aligned}x&=r\left(\cos \theta +\theta \sin \theta \right)\\y&=r\left(\sin \theta -\theta \cos \theta \right)\end{aligned}}}
where r radius of circle, , θ angle in radians (θ ∈ ℝ). counterclockwise spiral made positive values of θ, , clockwise spiral made negative values of θ.
in polar coordinates r , φ, involute of circle has parametric equation
r
=
a
sec
α
=
a
cos
α
φ
=
tan
α
−
α
{\displaystyle {\begin{aligned}r&=a\sec \alpha ={\frac {a}{\cos \alpha }}\\\varphi &=\tan \alpha -\alpha \end{aligned}}}
where radius of circle , α angle parameter in radians (α ∈ ℝ) equal t − φ (so tan α = t).
with parameter t (with t = tan α; t ∈ ℝ) can written in form
r
=
a
1
+
t
2
φ
=
t
−
arctan
(
t
)
.
{\displaystyle {\begin{aligned}r&=a{\sqrt {1+\ t^{2}}}\\\varphi &=t-\arctan(t).\end{aligned}}}
from cartesian coordinates , first polar coordinates, can seen that
r
(
θ
)
=
a
1
+
θ
2
{\displaystyle r(\theta )=a{\sqrt {1+\theta ^{2}}}}
so
θ
=
t
.
{\displaystyle \theta =t.}
curve length
the arc length of above curve 0 ≤ t ≤ t1 is
l
=
a
2
t
1
2
{\displaystyle l={\frac {a}{2}}{t_{1}}^{2}}
involute of catenary
the involute of catenary, tractrix.
the involute of catenary through vertex tractrix. in cartesian coordinates curve follows
x
=
t
−
tanh
(
t
)
y
=
sech
(
t
)
=
1
cosh
(
t
)
{\displaystyle {\begin{aligned}x&=t-\tanh(t)\\y&=\operatorname {sech} (t)={\frac {1}{\cosh(t)}}\end{aligned}}}
where t parameter , sech hyperbolic secant (1/cosh(t)).
derivative
with
r
(
s
)
=
(
sinh
−
1
(
s
)
,
cosh
(
sinh
−
1
(
s
)
)
)
{\displaystyle r(s)={\big (}\sinh ^{-1}(s),\cosh \left(\sinh ^{-1}(s)\right){\big )}}
we have
r
′
(
s
)
=
(
1
,
s
)
1
+
s
2
{\displaystyle r (s)={\frac {(1,s)}{\sqrt {1+s^{2}}}}}
and
r
(
t
)
−
t
r
′
(
t
)
=
(
sinh
−
1
(
t
)
−
t
1
+
t
2
,
1
1
+
t
2
)
.
{\displaystyle r(t)-tr^{\prime }(t)=\left(\sinh ^{-1}(t)-{\frac {t}{\sqrt {1+t^{2}}}},{\frac {1}{\sqrt {1+t^{2}}}}\right).}
substitute
t
=
1
−
y
2
y
{\displaystyle t={\frac {\sqrt {1-y^{2}}}{y}}}
to get
(
sech
−
1
(
y
)
−
1
−
y
2
,
y
)
.
{\displaystyle \left(\operatorname {sech} ^{-1}(y)-{\sqrt {1-y^{2}}},y\right).}
involute of cycloid
one involute of cycloid congruent cycloid. in cartesian coordinates curve follows
x
=
r
(
t
−
sin
(
t
)
)
y
=
r
(
1
−
cos
(
t
)
)
{\displaystyle {\begin{aligned}x&=r{\bigl (}t-\sin(t){\bigr )}\\y&=r{\bigl (}1-\cos(t){\bigr )}\end{aligned}}}
where t angle , r radius.
Comments
Post a Comment