Ribokur Curve

The Ribocour curve is a flat curve, defined as the locus of the points , the constant ratio of the radius of curvature to the length of the normal segment from the intersection with the curve to the intersection with the x-axis .

The curve was investigated by Albert Ribokur in 1880.

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in rectangular coordinates :

x=\int \limits _{0}^{y}{\frac {\mathrm {d} y}{\sqrt {\left({\frac {y}{c}}\right)^{2n}-1}}},

{\ displaystyle x = \ int \ limits _ {0} ^ {y} {\ frac {\ mathrm {d} y} {\ sqrt {\ left ({\ frac {y} {c}} \ right) ^ { 2n} -1}}},}

Where

n

{\ displaystyle n}

- the ratio of the length of the normal to the radius of curvature.

parametric equation:

{\begin{cases}x=(m+1)C\int \limits _{0}^{t}\sin ^{m+1}\tau \;\mathrm {d} \tau \\y=C\sin ^{m+1}t,\end{cases}}

{\ displaystyle {\ begin {cases} x = (m + 1) C \ int \ limits _ {0} ^ {t} \ sin ^ {m + 1} \ tau \; \ mathrm {d} \ tau \\ y = C \ sin ^ {m + 1} t, \ end {cases}}}

Where

m=-(n+1)n,\;\;n={\frac {1}{h}},\;\;h

{\ displaystyle m = - (n + 1) n, \; \; n = {\ frac {1} {h}}, \; \; h}

- whole.

Special Cases

Circumference at $m=0$ ${\ displaystyle m = 0}$ ,
Cycloid with $m=1$ ${\ displaystyle m = 1}$ ,
Chain line at $m=2$ ${\ displaystyle m = 2}$ ,
Parabola at $m=3$ ${\ displaystyle m = 3}$ .
Sinusoidal helix

Literature

Mathematical encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1982.
A. A. Savelov. Flat curves. - M. , 1960.

Ribokur Curve

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Special Cases

Literature

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