Algebraic Curves∗

curve is
( )
fy
(f y , −f x )H
−f x
κ = ±
‖∇f‖ 3
= ± f2 yf xx − 2f x f y f xy + f 2 xf yy
(f 2 x + f 2 y) 3 2
= ± f2 y f xx − 2f x f y f xy + fx 2f yy
(fx 2 + ,
f2 y )3 2
where the sign ‘−’ is chosen in case the motion along the curve is in the direction of the vector
(f y , −f x ) and where the sign ‘+’ is chosen in case the motion along the curve is in the direction of
the vector (−f y ,f x ).
Proof By Theorem 1, the algebraic curve can be given a regular local parameterization near a
non-singular point by α(t) = (x(t),y(t)). Differentiating the equation
( )
f x(t),y(t) = 0,
using the chain rule, we obtain
α ′ · (f x ,f y ) = 0.
Therefore α ′ is orthogonal to (f x ,f y ), and we have α ′ = (x ′ ,y ′ ) = λ(f y , −f x ) where λ ≠ 0 is a
function of t. On differentiating once more we obtain
( )
α ′′ · (f x ,f y ) + (x ′ ,y ′ x
′
)H
y ′ = 0.
Therefore we have
x ′ y ′′ − y ′ x ′′ = α ′′ · (−y ′ ,x ′ )
Since ‖α ′ ‖ = |λ| · ‖(f y , −f x )‖, we have that
= λα ′′ · (f x ,f y )
( )
= −λ(x ′ ,y ′ x
′
)H
y ′ ( )
= −λ 3 fy
(f y , −f x )H .
−f x
κ = α′ × α ′′
‖α ′ ‖ 3 = x′ y ′′ − y ′ x ′′
‖α ′ ‖ 3
= ± f2 yf xx − 2f x f y f xy + f 2 xf yy
‖∇f‖ 3 .
Corollary 3 The algebraic curve has a point of inflection at a non-singular point (a,b) if and only
if,
f 2 y f xx − 2f x f y f xy + f 2 x f yy
is zero at (a,b) and changes sign as (x,y) moves through (a,b) along the curve.
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