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RADIAL CURVE OF A GIVEN CURVE


Notion studied by Tucker in 1864.

 
The radial curve of a curve (G), associated to a fixed point O, is the locus of the points P defined by where is the vector joining the current point on (G) to its centre of curvature; in other words, it is the locus of the end of the radius-of-curvature vector, attached to a fixed point.

 
For an initial curve with current point , the radial curve is the set of points .
Cartesian parametrization of the radial curve: .
Complex parametrization: .
If the intrinsic equation 2 of the initial curve is , then the polar equation of its radial curve is .

The radial curve of an algebraic curve is an algebraic curve of the same degree as its evolute.

Examples:
 
initial curve associated radial curve
circle circle
ellipse  sextic
with equation
i.e.  (Loria p. 308)
parabola duplicatrix cubic
cycloid (with rolling circle with radius R) circle with radius 2R
deltoid regular trifolium
astroid quatrefoil
epicycloid with parameter q rose
hypocycloid with parameter q rose 
catenary of equal strength line
catenary kampyle of Eudoxus
tractrix kappa
involute of a circle Archimedean spiral
clothoid lituus
logarithmic spiral logarithmic spiral
Ribaucourt curve of index k
the cases k = -2, -1, and 2 amount to the above cases of the parabola, the catenary and the cycloid.
Clairaut's curve of index k-1
pseudo-spiral of index n
the cases n = 0 , 1, and -1/2 amount to the above cases of the circle, the clothoid and the involute of a circle.
Archimedean spiral of index 

See an application of radial curves for wheel-road couples.
 
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© Robert FERRÉOL  2017