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[parent] evolute of cycloid (Example)

We shall determine the evolute of the cycloid

$\displaystyle x = a(u-\sin{u}), \quad y = a(1-\cos{u}),$ (1)

where the parametre $ u$ is the rolling angle of the circle with radius $ a$ forming the cycloid.

The parametric equations of the evolute of a curve $ x = x(u),\; y = y(u)$ are

$\displaystyle \xi = x-\varrho\sin{\alpha}, \quad \eta = y+\varrho\cos{\alpha}$ (2)

with $ \alpha$ the slope angle of the tangent and $ \varrho$ the radius of curvature of the given curve in the point $ (x,\,y)$:
$\displaystyle \varrho = \frac{(x'^2+y'^2)^{3/2}}{x'y''-x''y'}$

In the case of the cycloid (1), we have

$\displaystyle x' = a(1-\cos{u}), \quad y' = a\sin{u}, \quad x'' = a\sin{u}, \quad y'' = a\cos{u}.$
Now we get
$\displaystyle x'^2+y'^2 = 2a^2(1-\cos{u}) = 4a^2\sin^2\frac{u}{2},$
$\displaystyle x'y''-x''y' = a^2(\cos{u}-1) = -2a^2\sin^2\frac{u}{2},$
and thus the radius of curvature (red in the diagram) is
$\displaystyle \varrho = -4a\sin\frac{u}{2}.$ (3)

We utilised the identity
$\displaystyle 1-\cos{u} = 2\sin^2\frac{u}{2}$ (4)

(see the half angle formula of sine in the goniometric formulas). It is easy to show that the point, where the circle touches the $ x$-axis, bisects the radius of curvature (which lies on the normal line at the point $ (x,\,y)$ of the cycloid).

Using then the derivative for parametric form, we obtain

$\displaystyle \tan\alpha = \frac{dy}{dx} = \frac{y'}{x'} = \frac{\sin{u}}{1-\cos{u}}.$
which implies
$\displaystyle \sin\alpha = \cos\frac{u}{2}, \quad \cos\alpha = \sin\frac{u}{2},$ (5)

Substituting all needed expressions (1), (3), (5) into (2) and simplifying, we arrive at the result
$\displaystyle \xi = a(u+\sin{u}), \quad \eta = -a(1-\cos{u}).$ (6)

The equation (6) represents the evolute of the given cycloid. But it is also a cycloid, congruent to the original one, which has been translated the distance $ \pi a$ to the left and the distance $ 2a$ downwards; this is seen when one performs in (6) the substitution $ u = v-\pi$; then (6) reads
$\displaystyle \xi = a(v+\sin{v})-\pi a, \quad \eta = a(1-\cos{v})-2a.$
unit=1.5cm
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Cross-references: congruent, expressions, implies, derivative for parametric form, normal line, lies on, goniometric formulas, sine, half angle formula, point, radius of curvature, tangent, slope angle, curve, equations, radius, circle, angle, parametre, cycloid, evolute
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This is version 10 of evolute of cycloid, born on 2008-03-18, modified 2008-03-19.
Object id is 10416, canonical name is EvoluteOfCycloid.
Accessed 337 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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