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Special Affine Curvature

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Special affine curvature, also called as the equi-affine or affine curvature, is a type of curvature for a plane curve that remains unchanged under a special affine transformation.

For a plane curve parametrized by (x(t),y(t)), the special affine curvature is given by

k(t)=(x^('')y^(''')-x^(''')y^(''))/((x^'y^('')-x^('')y^')^(5/3))-1/2[1/((x^'y^('')-x^('')y^')^(2/3))]^('')
(1)
=(4(x^('')y^(''')-x^(''')y^(''))+(x^'y^('''')-x^('''')y^'))/(3(x^'y^('')-x^('')y^')^(5/3))-5/9((x^'y^(''')-x^(''')y^')^2)/((x^'y^('')-x^('')y^')^(8/3))
(2)

(Blaschke 1923, Guggenheimer 1977), where the prime indicates differentiation with respect to t. This reduces for a curve y=y(x) to

k=-1/2(1/((y^(''))^(2/3)))^('')
(3)
=1/3(y^(''''))/((y^(''))^(5/3))-5/9((y^('''))^2)/((y^(''))^(8/3))
(4)

(Blaschke 1923, Shirokov 1988), where the prime here indicated differentiation with respect to x.

The following table summarizes the special affine curvatures for a number of curves.

curveparametrizationk(t)
catenary(t,acosh(t/a))(4-cosh((2t)/a))/(9a^(4/3)cosh^(8/3)(t/a))
circle(acost,asint)a^(-4/3)
ellipse(acost,bsint)(ab)^(-2/3)
hyperbola(asect,btant)-(ab)^(-2/3)
parabola(2at,at^2)0

See also

Curvature, Special Affine Transformation

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References

Blaschke, W. Affine Differentialgeometrie, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Berlin: Springer-Verlag, 1923.Guggenheimer, H. Differential Geometry. New York: Dover, 1977.Shirokov, A. P. "Affine Curvature." In Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, 1988.

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Special Affine Curvature

Cite this as:

Weisstein, Eric W. "Special Affine Curvature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpecialAffineCurvature.html

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